Comparing the Performance of Cpk and X Control Charts With the Generalized Multiple Dependent State Sampling Supplementary Signaling Rule

The <inline-formula> <tex-math notation="LaTeX">$C_{pk}$ </tex-math></inline-formula> index was originally created to measure the ability of the processes to produce products meeting specifications but, more recently, Rao and team proposed the use of the <inline-formula> <tex-math notation="LaTeX">$C_{pk}$ </tex-math></inline-formula> index to control processes. In reality, they investigated the performance of the <inline-formula> <tex-math notation="LaTeX">$C_{pk}$ </tex-math></inline-formula> chart with the GMDS (Generalized Multiple Dependent State Sampling) supplementary signaling rule. With the GMDS supplementary run rule, the <inline-formula> <tex-math notation="LaTeX">$C_{pk}$ </tex-math></inline-formula> chart signals when a point falls in the action region or when a point falls in the warning region after a sequence of <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> points, with less than <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>, in the central region. The points on the <inline-formula> <tex-math notation="LaTeX">$C_{pk}$ </tex-math></inline-formula> chart are the estimated values of the <inline-formula> <tex-math notation="LaTeX">$C_{pk}$ </tex-math></inline-formula> index obtained with the mean and the variance of the samples. Rao and team obtained the <italic>ARLs</italic> of the GMDS <inline-formula> <tex-math notation="LaTeX">$C_{pk}$ </tex-math></inline-formula> chart by simulation; reminding the <italic>ARL</italic> is the average number of samples the chart requires to signal a change in the process. This present work uses the Markov chain approach to obtain the <italic>ARLs</italic>, once this approach leads to exact results. The main conclusion is that the GMDS <inline-formula> <tex-math notation="LaTeX">$\bar {X}$ </tex-math></inline-formula> chart is not only simpler to use than the GMDS <inline-formula> <tex-math notation="LaTeX">$C_{pk}$ </tex-math></inline-formula> chart, but it is also more sensitive to process changes.


I. INTRODUCTION
Control charts are monitoring tools specially designed to detect changes in the process parameters. In general, the control charts have lower and upper control limits (LCL and UCL) and require the withdrawal of samples from the process. The values of the monitoring statistic, obtained with the sample observations, generate the charting points and, according to the standard signaling rule, if one of them falls beyond the control limits, then the control chart triggers an alarm [1].
In recent studies, the C pk index has been used as the monitoring statistic of the control chart, that is, at each sampling point the C pk index is calculated with the sample mean and standard deviation as the estimates of mean and standard deviation of the process. These sample values of the C pk index are the charting points of the C pk chart. For instance, The associate editor coordinating the review of this manuscript and approving it for publication was Giambattista Gruosso .
Rao et al. [2] proposed the use of the C pk chart with the GMDS (Generalized Multiple Dependent State Sampling) supplementary signaling rule, that means the C pk chart signals when a point falls in the action region or, yet, when a point falling in the warning region is preceded by a sequence of m points with less than k in the central region. If k = m, the GMDS reduces to the MDS (Multiple Dependent State Sampling) [3]- [6]. The multiple dependent state sampling has also been used in combination with repetitive sampling, MDSRS [7]- [14].
So, supplementary run rules enhance the sensitivity of the control charts in signaling process shifts. It is possible to observe three recent articles focusing on the use of supplementary signaling rules [15]- [17].
Control charts based on capability index are drawing researchers' attention. Reference [18] proposed the use of the C pk chart with the repetitive sampling scheme, that is, three different decisions are taken according to the sample point position; if the point falls outside (inside) the outer (inner) control limits, the process is declared out of control (in control). Alternatively, a point falling outside the inner control limits, but not outside the outer control limits, leads to the withdrawal of a new sample, then, the information of this new sample is used to decide the state of the process or to postpone the decision to the next sample. Reference [19] proposed the two-stage sampling, with the use of the npchart during the first stage and the C pk chart during the second stage. Reference [20] considered the charting points as the estimated values of the C pk index obtained with the median and the variance values of the samples. Reference [21] considered the charting points as the estimated values of the C pk index obtained with the Gini's mean difference. Other capability indices have also been used to control processes [22]- [26].
Thus, the aim of this paper is to compare the performance of the C pk andX charts, when the standard signaling rule of a single point in the action region is used in combination with the GMDS supplementary run rule. Therefore, section 2 presents the C pk andX charts with the GMDS supplementary signaling rule. Section 3 describes the Markov Chains approach with which the exact ARLs and steady-states ARLs of the GMDS charts are computed. In Section 4, the ARLs of the two GMDS charts, C pk andX , are compared. In Section 5, the comparisons are based on their SSARLs. Section 6 displays a real example. Finally, section 7 brings some conclusions.

II. THE C pk ANDX CHARTS WITH GMDS SUPPLEMENTARY RUN RULE
The control chart proposed by Rao et al. [2] combines the use of the process capability index, as the monitoring statistic, with the generalized multiple dependent state sampling, as the supplementary signaling rule. The monitoring statistic of the C pk chart is the C pk index in (1): Obtained with the mean and the standard deviation of the sample, respectively (2) and (3): where n is the size of the samples. In (2), the USL and the LSL are, respectively, the upper and the lower specification limits. The C pk and theX charts with the GMDS supplementary run rule have inner (LCL 2 , UCL 2 ) and outer (LCL 1 , UCL 1 ) control limits shown in (4)-(7): In (4)- (7), the MS is the monitoring statistic of the chart; if the chart is the C pk chart, then E(MS) = E Ĉ pk and Var(MS) = Var Ĉ pk . Alternatively, if the chart is thē X chart, then E(MS) = E X = µ 0 and Var (MS) = Var X = σ 2 n, where µ 0 and σ 2 are the in-control mean and variance of the process.
The opening parameters k 1 and k 2 are adjusted to meet the required ARL 0 -average number of samples between false alarms; it has been adopted an ARL 0 of 370.4. Rao et al. [2] obtained k 1 and k 2 by simulation, and they also used simulation to obtain the ARL 1 -average number of samples the chart requires to signal process mean increases (or decreases). The increase or decrease is expressed in units of σ and represented by δ; if µ 1 is the out-of-control mean, then (8): Without losing generality, adopting µ 0 = 0 and σ = 1. It was also considered the symmetric case where LSL = -USL. The process is declared to be in control when (9): And out of control when (10) or (11): If the current monitoring statistic point falls in the warning region (12) or (13): Then the state of the process is decided by the previous m points: if less than k are in the central region (in between LCL 2 and UCL 2 ), then the process is declared to be out of control; otherwise, the process is declared to be in control.

III. THE MARKOV CHAIN TO OBTAIN THE EXPRESSION OF THE STEADY-STATE ARL
The work developed by Rao et al. [2] calculated the ARLs using simulation. Since it was not displayed some required inputs to run the simulation, such as E(MS) and Var (MS), it was not possible to recalculate the ARLs. As an alternative, it was used the Markov chains approach, that leads to exact ARLs and steady-state ARLs of the GMDS charts; reminding the steady-state ARL (SSARL) is the average number of samples the chart requires to signal a change in the process, under the assumption that until the monitoring statistic reaches its stationary distribution the process mean (µ) remains unaltered, with µ = µ 0 .
The number of states with which the Markov chain is built is a function of m and k. To illustrate, let m = k = 3, then the states are defined by the position of the last four sample points. Fig. 1 presents the Markov chain with the states defined by the sequence (a, b, c, d), in which a, b, and c are the sample points positions of the third, second and first points before the last one, and d is the position of the last sample point; for instance (C, W, C, A) means the last  point is in the action region (represented by A), and the third, second and first points before the last one are, respectively, in the central (represented by C), in the warning (represented by W) and in the central regions. Fig. 2 shows that, if m = k = 3, then the Markov chain can be represented by five transient states (in white) and one absorbing state (in red).
Using the arrows of Fig. 1 as guide, the transition probability matrix is given by (14): where p ij denotes the transition probability in which i is the prior state and j is the current state. So: Reference [27] displays that the first step to obtain the steady-state ARL is the construction of the matrix (I-Q), in which I is the 5 by 5 identity matrix and Q is the transition matrix given in (14), with the last row and column removed as displayed in (15): The steady-state ARL is given by S (I-Q) −1 , in which S is the vector with the stationary probabilities of being in each transient state. The inverse of (I-Q) is described in (16), as shown at the bottom of the next page.
Reference [27] presents that the vector S is the solution of S = P * S, in which P * is the transition probability matrix P with the absorbing state 6 transformed into a return to the state 1; in other words, after a false alarm the monitoring always returns to state 1 (the return to state 1 is equivalent to assume that the last four sample points are in the central region). Matrix P * is shown in (17): The solution of S = P * S is given by (18): In which a is calculated through (19): Consequently, the steady-state ARL of the GMDSX chart is computed with the S (I-Q) −1 expression. The ARL of the GMDSX chart is also computed with the S (I-Q) −1 expression, but now with the first element of vector S being equal to one, and all the other elements being equal to zero. Table 1 presents the ARLs of the GMDSX chart given by S (I-Q) −1 with S'= (1,0. . . 0) and also the ARLs of the GMDS C pk chart published by Rao et al. [2].

IV. COMPARING THE ARLs OF THE GMDSX AND C pk CHARTS
The results show that depending on the (m, k) combination one chart is better than the other (the winning chart is in bold). For instance, if (m, k) = (3,2) the GMDS C pk chart is more sensitive than theX chart, but with (m, k) = (4,2) the charts alter their positions: now the GMDSX chart is more sensitive than the GMDS C pk chart. If (m, k) = (3,3) or (4,4), the GMDS C pk chart signals faster small shifts (δ ≤ 0.67), but for larger shifts (δ ≥0.89), the GMDSX chart signals faster. In terms of overall performance, the GMDS C pk chart with (m, k) = (3,2) is the winner of all control charts under comparison. The authors were not able to verify the simulated ARLs obtained by Rao et al. [2] because it was not provided all the required inputs to run the simulations, that is, E(MS), Var (MS) and USL values. As an alternative, it was decided to investigate the properties of the GMDS C pk chart by using the S (I-Q) −1 expression. In the next section, it will be compared the SSARLs of the GMDS C pk chart with the SSARLs of the GMDSX chart; both SSARLs were obtained by using S (I-Q) −1 expression.

V. COMPARING SSARLs OF THE GMDSX AND C pk CHARTS
In order to obtain the cumulative distribution of theĈ pk statistic it was used the recent result presented by Costa [28] that evidenced (20): (20) In (20), Z ∼ N (0,1) and f (s) is the density function of the sample standard deviation given by (21): The density function of theĈ pk statistic is totally asymmetric; Fig. 3 shows the case for which n = 5 and USL = 1.5. Based on that, it was chosen to use symmetric warning and action limits regarding the in-control probabilities (a, b), in which (a) is the probabilities to obtain â C pk point in the action region and (b) is the probabilities to obtain aĈ pk point in the warning region. If LCL 1 = ka 1 (in which ka indicates the opening parameter for action region), LCL 2 = kw 1 (in which kw indicates the opening parameter for warning region), UCL 2 = kw 2 and UCL 1 = ka 2 , it is possible to calculate (22)- (24):  Pr[Ĉ pk > ka 2 ] = a/2 (23) Consequently, the probabilities p 1 , p 2 and p 3 with which the transition probability matrix P is built and used to obtain the SSARLs of the GMDS C pk chart are given by (25)- (27): Table 2 presents the SSARLs of the GMDSX chart (n = 5) and the SSARLs of the GMDSC pk chart (n = 5, UCL = 1.5); the GMDS C pk chart reaches its best overall performance with UCL = 1.5.
The results show the superiority of the GMDSX chart; that is, regardless of the (m, k) combination, the SSARLs values of the GMDSX chart are always the lower ones. It is worthy to note that all SSARLs values were checked by simulation. Table 2  , the second signaling rule always reduces the ability of the C pk chart to detect mean increases or decreases, and slightly improves the ability of thē X chart in signaling moderate or small mean increases or decreases (δ ≤ 1); c) Comparing the signaling rules (m, k) = (4, 3) and (m, k) = (4, 2), the second one reduces the ability of the C pk and theX charts to detect mean increases or decreases.

VI. REAL EXAMPLE
Suppose that a company, which supplies shafts, is facing considerable losses due to the production of defective items. The diameter of the shafts has been the cause of rejection. As the specifications are very tight, a minor shift in the process mean leads to manufacturing the shafts with diameters beyond the specifications; so, the shift should be detected as fast as possible. Past data shows the standard deviation (σ ) of the diameters is pretty close to 0.0010 inches. This example was originally explored by Costa and Rahim [29]. Table 3 presents the diameter X of 25 samples with 5 shafts each; the data of the first 15 samples were simulated with the process mean free of shifts, whereas the data of the last 10 samples were simulated with an increased process mean of one standard deviation. Fig. 4 and fig. 5 present, respectively, the C pk and theX charts with the GMDS supplementary run rule (m, k) = (3, 2); in both charts, the eighteenth sample signals the mean shift. The C pk chart signals with the eighteenth sample because the sixteenth and seventeenth samples are in-between the LCL 1 and LCL 2 lower limits. On the other hand, theX chart signals with the eighteenth sample because the sixteenth and seventeenth samples are in-between the UCL 2 and UCL 1 upper limits.
As a result, both the C pk and theX charts are considered practical tools for process surveillance subject to moderate  TABLE 3. The X s ,X andĈ pk values for the real example. disturbances. Depending on the parameters, one chart can be perceived as more efficient than the other one. In this real example, both charts displayed the same performance for the supplementary run rule (m, k) = (3, 2).

VII. CONCLUSION
Thanks to the Markov chains properties, it was obtained the exact expression of the SSARLs. To illustrate how exact expressions of the SSARLs can be obtained by the Markov chain approach, it was considered the control chart with the GMDS supplementary run rule (m, k) = (3, 3). It is important to notice that this expression is independent of the statistic in use to control the process.
Inspired by the work proposed by Rao et al. [2] in which the authors investigate the performance of the C pk control chart with the supplementary run rule GMDS, this article broadens the investigation to also include theX control chart with GMDS run rule. Consequently, the ARL and SSARL values calculated demonstrate insightful analysis on the comparison of the performance of GMDS C pk andX control charts.
In terms of comparison, it was proved that the GMDS C pk chart should be considered with care, because it is pretty slow in signaling mean shifts. The GMDSX chart with the supplementary run rule (m, k) = (4, 3) has the best overall performance.
It is worthy to note that, when the monitoring statistic of the control chart is theĈ pk , that is, the C pk index obtained with theX and S sample values, the specification limits can no longer be seen as the limits beyond which a product is considered defective. In fact, they are now tuning parameters of the control chart because they deeply affect the speed in which the C pk chart signals. The GMDS C pk chart reaches its best overall performance with UCL = 1.5. VOLUME 10, 2022