Synthetic control charts with two-stage sampling for monitoring bivariate processes

Abstracts

In this article, we consider the synthetic control chart with two-stage sampling (SyTS chart) to control bivariate processes. During the first stage, one item of the sample is inspected and two correlated quality characteristics (x;y) are measured. If the Hotelling statistic T1² for these individual observations of (x;y) is lower than a specified value UCL1 the sampling is interrupted. Otherwise, the sampling goes on to the second stage, where the remaining items are inspected and the Hotelling statistic T2² for the sample means of (x;y) is computed. When the statistic T2² is larger than a specified value UCL2, the sample is classified as nonconforming. According to the synthetic control chart procedure, the signal is based on the number of conforming samples between two neighbor nonconforming samples. The proposed chart detects process disturbances faster than the bivariate charts with variable sample size and it is from the practical viewpoint more convenient to administer.

bivariate processes; synthetic control chart; two-stage sampling

Este artigo apresenta um gráfico de controle com regra especial de decisão e amostragens em dois estágios para o monitoramento de processos bivariados. No primeiro estágio, um item da amostra é inspecionado e duas características de qualidade correlacionadas (x;y) são medidas. Se a estatística de Hotelling T1² para as observações individuais de (x;y) for menor que um valor especificado UCL1 a amostragem é interrompida. Caso contrário, a amostragem segue para o segundo estágio, onde os demais itens da amostra são inspecionados e a estatística de Hotelling T2² para as médias de (x;y) é calculada. Quando a estatística T2² é maior que um valor especificado UCL2, a amostra é classificada como não conforme. De acordo com a regra especial de decisão, o alarme é baseado no número de amostras entre duas não conformes. O gráfico proposto é mais ágil e mais simples do ponto de vista operacional que o gráfico de controle bivariado com tamanho de amostras variável.

processos bivariados; regra especial de decisão; amostragens em dois estágios

Synthetic control charts with two-stage sampling for monitoring bivariate processes

Antonio F. B. Costa^*; Marcela A. G. Machado

Production Department – São Paulo State University (UNESP) – Guaratinguetá – SP – Brazil – [e-mail] ; [e-mail]

ABSTRACT

In this article, we consider the synthetic control chart with two-stage sampling (SyTS chart) to control bivariate processes. During the first stage, one item of the sample is inspected and two correlated quality characteristics (x;y) are measured. If the Hotelling statistic T₁² for these individual observations of (x;y) is lower than a specified value UCL₁ the sampling is interrupted. Otherwise, the sampling goes on to the second stage, where the remaining items are inspected and the Hotelling statistic T₂² for the sample means of (x;y) is computed. When the statistic T₂² is larger than a specified value UCL₂, the sample is classified as nonconforming. According to the synthetic control chart procedure, the signal is based on the number of conforming samples between two neighbor nonconforming samples. The proposed chart detects process disturbances faster than the bivariate charts with variable sample size and it is from the practical viewpoint more convenient to administer.

Keywords: bivariate processes; synthetic control chart; two-stage sampling.

RESUMO

Este artigo apresenta um gráfico de controle com regra especial de decisão e amostragens em dois estágios para o monitoramento de processos bivariados. No primeiro estágio, um item da amostra é inspecionado e duas características de qualidade correlacionadas (x;y) são medidas. Se a estatística de Hotelling T₁² para as observações individuais de (x;y) for menor que um valor especificado UCL₁ a amostragem é interrompida. Caso contrário, a amostragem segue para o segundo estágio, onde os demais itens da amostra são inspecionados e a estatística de Hotelling T₂² para as médias de (x;y) é calculada. Quando a estatística T₂² é maior que um valor especificado UCL₂, a amostra é classificada como não conforme. De acordo com a regra especial de decisão, o alarme é baseado no número de amostras entre duas não conformes. O gráfico proposto é mais ágil e mais simples do ponto de vista operacional que o gráfico de controle bivariado com tamanho de amostras variável.

Palavras-chave: processos bivariados; regra especial de decisão; amostragens em dois estágios.

1. Introduction

The control charts have been widely used for process surveillance because of their operational simplicity. However, this operational simplicity, that is, taking samples of fixed size at regular time intervals and searching for an assignable cause when a point falls outside the control limits, makes the control chart slow in detecting small to moderate shifts in the process parameter being controlled. Many innovations have been proposed to improve the charts’ performance, such as the adaptive schemes, the supplementary run rules, or the two-stage sampling procedure. When the standard Shewhart control charts are in use, the rate of sampling is kept fixed, once samples of equal size are taken from the process at fixed-length sampling intervals. A more elaborated scheme, called adaptive scheme, has been proposed, in which the size of the samples and/or the length of the sampling intervals and/or the false alarm risk are allowed to vary in an adaptive manner, that is, in real time, based on current sample information. The adaptive scheme enhances the ability of the control chart to signal process disturbances (Reynolds et al. (1988); Prabhu et al. (1993); Costa (1994, 1997, 1998, 1998a, 1999, 1999a); Costa & De Magalhães (2007); De Magalhães et al. (2001, 2002, 2006); De Magalhães & Moura Neto (2005); Epprecht & Costa (2001); Epprecht et al. (2003, 2005); Michel & Fogliatto (2002)). In the multivariate SPC, the adaptive schemes have been applied for Hotelling’s T² control charts (Aparasi (1996); Aparasi & Haro (2001, 2003); Chou et al. (2006); Yeh & Lin (2002)).

Champ & Woodall (1987) determined the properties of control charts with supplementary runs rules. More recently, Wu & Spedding (2000) presented the synthetic control chart, where the signal is given after the occurrence of a second point on the action region. According to Davis & Woodall (2002), the synthetic control chart is a runs rule chart with a head start feature. The growing interest in the synthetic charts (see Wu & Spedding (2000, 2000a), Wu & Yeo (2001), Wu et al. (2001), Calzada & Scariano (2001), Davis & Woodall (2002), Machado & Costa (2005), Costa & Rahim (2006)) may be explained by the fact that many practitioners prefer waiting until the occurrence of a second point beyond the control limits before looking for an assignable cause.

In this paper, we consider a synthetic control chart with two-stage sampling (SyTS chart) to control bivariate processes. As in the case of Shewhart control charts, samples of fixed size are taken from the process at regular time intervals; however, the sampling is performed in two stages. During the first stage, one item of the sample is inspected and two correlated quality characteristics (x;y) are measured. If the Hotelling statistic for these individual observations of (x;y) is lower than a specified value UCL₁ the sampling is interrupted. Otherwise, the sampling goes on to the second stage, where the remaining items are inspected and the Hotelling statistic for the sample means of (x;y) is computed and compared with UCL₂, the upper control limit. When the synthetic control chart is in use, the signal is given by the occurrence of a second point on the action region and under the condition they can not be far from each other.

Like the adaptive schemes, the two-stage sampling enhances the ability of the control chart in detecting process disturbances. According to Costa & Rahim (2004, 2006a), when the process is stable, the monitoring becomes monotonous because the sample values rarely fall outside the control limits. The natural consequence is that the user pays less and less attention to the steps required to obtain these values, such as the sample mean, the sample variance, or the sample range. When the two-stage sampling scheme is in use, the second stage is frequently discarded, consequently most of the time the user will be dealing with the direct information from the parameters under surveillance, specifically, with their (x;y) values. The go-no-go gauge devices are appropriate to decide if the sampling should be interrupted or not.

2. The properties of the synthetic control chart with two-stage sampling

A synthetic control chart with two-stage sampling (SyTS chart) is proposed for simultaneously monitoring of two correlated quality characteristics (x;y), described by a bivariate normal distribution with mean vector m = (µ_x; µ_y) and a known covariance matrix , where s_xy = s_yx = rs_xs_y is the covariance between x and y. The process is considered to start with the mean vector on target, m₀ = (µ_0x; µ_0y), but at some random time in the future an assignable cause shifts the mean vector from m₀ to m₁, where (m₁ - m₀) =d_xs_x; d_ys_y) , with d_x ¹ 0 and/or d_y ¹ 0. During the in-control period m = m₀ and during the out-of-control period m = m₁. The objective of process monitoring is the detection of any assignable cause that shifts m'.

Similar to the Shewhart control charts, samples of size n₀ + 1 are taken from the process at regular time intervals. The sampling is performed in two stages. At the first stage, one item of the sample is inspected and two correlated quality characteristics (x;y) are measured; if the statistical distance between X' = (x,y) and = (µ_0x; µ_0y), given by = (X - µ₀)'S^-1(X - µ₀), is lower than a specified value UCL₁ the sampling is interrupted. Otherwise, the sampling goes on to the second stage, where the remaining n₀ items are inspected and the statistical distance between and = (µ_0x; µ_0y), given by = n( - µ₀)'S^-1( - µ₀) is computed; being and the sample mean of the two correlated quality characteristics (x;y) taking into account the entire sample of size n₀+ 1. When the statistic is larger than a specified value UCL₂, the sample is classified as nonconforming. According to the synthetic procedure, the signal is based on the Conforming Run Length (CRL). The CRL is the number of samples taken from the process since the previous nonconforming sample until the occurrence of the next nonconforming sample or, in case of the absence of previous nonconforming sample, the CRL is the number of samples taken from the beginning of the process monitoring until the occurrence of a nonconforming sample. The signal is given when the CRL is smaller than or equal to L, where L is a specified positive integer.

In order to obtain the properties of the SyTS control chart we consider the following theorem (Wexler, 1962): If a point P has rectangular coordinates (x,y) then the polar coordinates of P is (r,q) where

hence, for any disk D(0,a) centered at the origin, (x,y) Î D(0,a) if and only if r² < a². Based on that we define T² = n( - µ)'S^-1( - µ) = (n,m) + (n,m), where

with sin 2j = s_xy / (s_xs_y), , , and . According to Theorem A1, see the ^appendix, g(n,m) = ( (n,m), (n,m)) has normal distribution with zero mean vector and unit covariance matrix e = . Consequently,

where f_N(0,e)(x,y) = exp[-0,5(x² + y²)]. When the process is in control, p₀ = Pr[ < UCL₁] is the probability that the two-stage sampling ends at the first stage:

During the in-control period, the rate of inspected items per sampling, , is given by

If the parameters n₀ and UCL₁ are designed to make equal to n, the size of the samples when the standard bivariate T² control chart is in use, then both, the SyTS and the standard T² control charts will have the same rate of inspected items per sampling.

When the sample of size n is split in two sub-samples of size n₁ and n₂, with n₁ + n₂ = n, it follows that

In the case of the two-stage sampling, n₁ = 1, n₂ = n₀, and the sample is classified as nonconforming if g(n₁,m) Ï D(0, ) and g(n,m) Ï D(0, ). According to (8), the false alarm risk is given by

where C(-x/; -y/) and R = .

From (3) and (4), follows that

consequently, the power of the control chart will be given by

where a = (;) and C(-x/ + ; -y/ + ).

3. The performance of the synthetic control chart with two-stage sampling

According to the synthetic procedure, the signal is based on the Conforming Run Length (CRL). The CRL is the number of samples taken from the process since the previous nonconforming sample until the occurrence of the next nonconforming sample or, in case of the absence of previous nonconforming sample, the CRL is the number of samples taken from the beginning of the process until the occurrence of a nonconforming sample. The signal is given when the CRL is smaller than or equal to L, where L is a specified positive integer. According to Wu & Spedding (2000),

During the in-control period Q = a and the ARL is called ARL₀, and during the out-of-control period, Q = p. Figure 1 shows a SyTS bivariate control chart with the values plotted. The point on the action region triggers the inspection of the whole sample, in which case the statistic is computed and compared with UCL₂. If > UCL₂ then the point is surrounded by a circle. In other words, the nonconforming samples have their points surrounded by circles, see in Figure 1, 9th and 13th sample points. This simple procedure has a practical appeal, once the signal is based on the Conforming Run Length. According to Figure 1, the previous nonconforming sample is the 9th sample, the next nonconforming sample is the 13th, and the CRL is equal to four (13-9). As the CRL value is lower than L(=5), the SyTS bivariate control chart signals an out-of-control condition.

When a process is in control, it is desirable that the expected number of samples taken since the beginning of the monitoring until a signal (ARL₀) be large, to guarantee few false alarms. When a process is out of control, it is desirable that the expected number of samples taken since the occurrence of the assignable cause until a signal (ARL) be small, in order to guarantee fast detection of process changes (see Costa et al. (2005)).

When the sampling interval is fixed, the ARL provides a measure of the time required to detect an assignable cause that is affecting the process since the beginning of the monitoring (zero-state ARL) or since the occurrence of the assignable cause (steady-state ARL). However, it is more realistic to assume that the assignable cause only occurs after the process has been running for some time. As a result, the assignable cause will occur between two nonconforming samples. The steady-state ARL measures how long the synthetic control chart will take to signal in this situation. We used the Markov chain model of the synthetic control chart described in Davis & Woodall (2002) to obtain the steady-state ARL for the SyTS bivariate chart.

The steady-state ARL values are presented in Table 1 for the synthetic bivariate chart with two-stage sampling, = 4, n₀ = 9, UCL₁=2.1973 and L=1, 5, 10, 20, 50, 100. As L increases from 1 to 50, all the steady-state ARL values decrease. The process disturbance is measured by l = , with . For example, if l = 0.5, the steady-state ARL decreases from 84.79 to 59.74. The effect of varying L from 50 to 100 almost not affects the speed with which the chart detects small or large disturbances. For example, if l = 1.5, the steady-state ARL increases from 4.80 to 4.88 as L increases from 50 to 100. And if a larger shift is considered, for example, l = 3.0, the steady-state ARL decreases from 1.62 to 1.59 as L increases from 50 to 100. Table 1 also brings the ARL values for the standard bivariate chart (T² chart). Inspection of Table 1 reveals that the SyTS Chart always outperforms the T² chart, except the case when L=1. For example, if l = 1.5 and L=20, the steady-state ARL value for the SyTS Chart is 4.86 and the equivalent ARL value for the T² chart 5.79.

Table 2 brings the steady-state ARL values for the synthetic chart with two-stage sampling, , L=10, and n₀= 6, 9, 12. The choice of n₀ affects the speed with which the SyTS chart signals. Larger values of n₀ are better for detecting smaller process changes and worse for detecting larger process changes. For example, one can see from Table 4 that, when l = 1.5, as n₀ increases from 6 to 12, the steady-state ARL value decreases from 5.91 to 5.30. On the other hand, when l = 3.0, as n₀ increases from 6 to 12, the steady-state ARL value increases from 1.55 to 2.13. Table 2 also brings the ARL values for the standard bivariate chart T² chart and the ARL values for the bivariate control chart with two-stage sampling (BTS chart). Inspection of Table 2 reveals the SyTS chart, when L=10, always outperforms the T² chart and outperform the BTS chart for small disturbances. For example, if l = 1.0 and n₀=9, the steady-state ARL value for the SyTS chart is 13.93. The equivalent ARL values for the T² chart and for the BTS chart are respectively 41.97 and 17.59. When l increases from 1.5 to 3.0, most of the time the BTS chart has a better performance than the proposed chart (the SyTS chart). For example, if l = 2.5 and n₀=9, the ARL value for the BTS chart is 1.94 and the steady-state ARL value for the SyTS chart is 2.18

4. Comparing charts

As the SyTS control chart was designed to detect small changes in the process parameters, it seems reasonable to compare its performance with the performance of the Hotelling's T² control chart with variable sample sizes (VSS chart).

The design of univariate Shewhart charts with adaptive sample sizes has been studied by Daudin (1992), Prabhu et al. (1993) and Costa (1994). Aparisi (1996) has extended these studies for the Hotelling's T² control chart. The monitoring procedure is as follows: the statistic is charted with upper control limit UCL_VSS. Two samples sizes are used, n₁ and n₂, n₂ > n₁. A warning limit w is defined, where 0 < w < UCL_VSS as a threshold limit that identifies when to shift from one sample size to another (Fig. 2). The sample size of subgroup i depends on the value of . If < w the ith subgroup sample size is n₁, and corresponds to the black points in the chart. If w < < UCL_VSS the ith subgroup sample size is n₂, and correspond to the white points in the chart (see sample points 6, 8, 10 and 12). If > UCL_VSS then the control chart signals.

Table 3 provides the ARL for the SyTS chart and for the Hotelling's T² control chart with adaptive sample sizes. Similarly to Costa (1994), Aparisi (1996) obtained the ARL values for the Hotelling's T² control chart with adaptive sample sizes using Markov chains.

According to Table 3 the SyTS chart has a better overall performance than its competitor, the VSS chart.

5. Illustrative example

In this section the proposed model is applied to the example considered by Montgomery (2004) to explain the use of the bivariate T² control chart. The two quality characteristics are the tensile strength (psi) and the diameter of a textile fiber (x10^-2 inch). He considered a set of twenty samples of size ten to estimate the in-control values of the mean vector and the covariance matrix:

Based on these estimations, ten samples of size n₀ + 1, with n₀ = 6, were generated using the DRNMVN sub routine available on the IMSL FORTRAN library (1995). After fixing = 3, the rate of inspected items per sampling, a = 0.005, the false alarm risk, and L = 2, the expressions (6) and (11) were used to obtain the control chart limits, UCL₁ = 2.197 and UCL₂ = 4.911. The general expression of the Hotelling T² statistic for two quality characteristics is:

According to the proposed sampling scheme, the sampling is performed in two stages. At the first stage, one item of the sample is inspected and the two correlated quality characteristics (x;y) are measured. The statistical distance between X' = (x;y) and = (µ_0x; µ_0y) is given by

If the statistic is lower than UCL₁ the sampling is interrupted. Otherwise, the sampling goes on to the second stage, where the remaining n₀ items are inspected and the statistical distance between and , given by

is computed; being and the sample mean of the two correlated quality characteristics (x;y) taking into account the entire sample of size n₀ + 1. When the statistic is larger than UCL₂ = 4.911, the sample is classified as nonconforming. According to the synthetic procedure, the signal is based on the Conforming Run Length (CRL). The CRL is the number of samples taken from the process since the previous nonconforming sample until the occurrence of the next nonconforming sample or, in case of the absence of previous nonconforming sample, the CRL is the number of samples taken from the beginning of the process monitoring until the occurrence of a nonconforming sample. The signal is given when the CRL is smaller than or equal to L.

Table 4 presents the data of (x;y), the sample means, the statistics and and the sampling status.

Figure 3 presents the SyTS control chart for the illustrative example. According to this figure, the first nonconforming sample is the 5^th sample, the second nonconforming sample is the 9^th, and the CRL is equal four (9-5). As the CRL value is larger than L (= 2), the SyTS bivariate control chart does not signal an out-of-control condition.

6. Conclusions

In this article, a synthetic control chart with two-stage sampling (SyTS chart) to control bivariate processes was proposed for detecting assignable causes. An important advantage of the proposed chart is that the practitioner can control the process by looking at only one chart. The synthetic control chart with two stage sampling (SyTS), the standard control chart, the control chart with two-stage sampling and the VSS control charts for bivariate processes were compared, in terms of the speed they detect changes in the process parameters. The numerical results we obtained show a better overall performance of the SyTS chart if compared with the other control charts.

In order to obtain the properties of the synthetic control chart with two-stage sampling, the IMSL FORTRAN library (1995) was used, where the subroutine DCSNDF is available to evaluate the non-central chi-square distribution function. The SyTS chart was conceived to attend the user because frequently they will be dealing with the situation where the inspection is reduced to the first item of the sample. Moreover, they feel more secure in stopping the process only after the occurrence of a second point beyond the control limits.

Acknowledgements

This work was supported by CAPES – High Level Staff Improvement Coordination – and CNPq – National Council for Scientific and Technological Development. The authors are indebted to two anonymous referees for the suggestions, which significantly contributed to the improvement of this paper.

Recebido em 03/2006; aceito em 09/2006

Received March 2006; accepted September 2006

Proof: The random functions g₁(n,m) and g₂(n,m) are normal, because they are linear combinations of normal variables x_i and y_i. We know that

After some calculation we obtain

By symmetry with respect to indices x and y, we have

As sin 2j = s_xy /(s_xs_y) and follows

Similarly

Theorem A1 is proved.

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appendix

.

Theorem A1: The random functions g₁(n,m) and g₂(n,m) have normal distribution with zero mean vector and unit covariance matrix e = .

*

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Publication Dates

• Publication in this collection
  21 June 2007
• Date of issue
  Apr 2007

History

• Accepted
  Sept 2006
• Received
  Mar 2006