The Case Against the Use of Synthetic Control Charts

Knoth, Sven

doi:10.1080/00224065.2016.11918158

Cited by 35 publications

(51 citation statements)

References 28 publications

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“…The time T ∗ is equal to a sum of a random number of run lengths, which could consist of in‐control and out‐of‐control observations. More specific, as in Knoth, we have

\begin{matrix} T_{*} & = T_{1} + T_{2} + ⋯ + T_{I_{m} - 1} + T_{I_{m}}, \\ I_{m} & = \min ({}, i \geq 1 : true \sum_{j = 1}^{i} T_{j} \geq m) . \end{matrix}

Finally, the cyclical S S A R L is given by

{scriptD}_{*} = \lim_{m \to \infty} E_{m} (T^{*} - m + 1) .

To calculate the value of

D_{*}

for T fix , we adopt the EWMA sequence Z i in the following way:

S_{i} = ({, \begin{matrix} center \\ array {\tilde{S}}_{i} & array, L C L \leq {\tilde{S}}_{i} \leq U C L \\ array z_{0} & array, ({\tilde{S}}_{i} > U C L) \cup ({\tilde{S}}_{i} < L C L) . \end{matrix}), {\overset{S}{true}}_{i} : = false(1 - λ false) S_{i - 1} + λ {\overset{Z}{true}}_{i}, i = 1, 2, ⋯

…”

Section: Computation Of Arl Of Proposed Control Chartsmentioning

confidence: 81%

“…To study the long‐term properties of ARL‐unbiased EWMA‐RZ control charts, it is appropriate to investigate the steady‐state A R L ( S S A R L ). The S S A R L measures the average number of samples required to signal when the effect of head start has disappeared, see Knoth for more details. In this paper, we consider the conditional and cyclical steady state by using both numerical attempts.…”

Section: Computation Of Arl Of Proposed Control Chartsmentioning

confidence: 99%

“…Moreover, D 1 coincides with the latter, E 1 ( T ). The so‐called conditional S S A R L is then defined as limit

D = \lim_{m \to \infty} {scriptD}_{m} .

To calculate

scriptD

, like in Knoth, we make use of the left eigenfunction ψ ( y ) of the two‐sided EWMA‐RZ control charts as follows:

η ψ (y) = \int_{L C L}^{U C L} ψ (s) \frac{1}{λ} f_{{\hat{Z}}_{i}} (\frac{y - (1 - λ) s}{λ}| n, γ_{X}, γ_{Y}, τ ω_{0}, ρ_{0}) d s

for the dominant (hence largest in magnitude and unique) eigenvalue η <1. Then we can calculate the conditional S S A R L through

D = \frac{\int_{L C L}^{U C L} ψ (s) L (s) d s}{\int_{L C L}^{U C L} ψ (s) d s} .

For the cyclical steady state, it is assumed that the change point m may happen after a sequence of false alarms, where the control chart is restarted, in z 0 , after each false alarm.…”

Section: Computation Of Arl Of Proposed Control Chartsmentioning

confidence: 99%

“…In this case, we can use a slightly different left eigenfunction ψ ∗ ( y ) of two‐sided EWMA‐RZ control charts as follows:

\begin{matrix} ψ_{*} false(y false) & = Ψ_{*} \frac{1}{λ} f_{{\overset{Z}{true}}_{i}} ((), \frac{y - false(1 - λ false) z_{0}}{λ} true| n, γ_{X}, γ_{Y}, τ ω_{0}, ρ_{0}) \\ + \int_{L C L}^{U C L} ψ_{*} false(s false) \frac{1}{λ} f_{{\overset{Z}{true}}_{i}} ((), (|, \frac{y - false(1 - λ false) s}{λ}) n, γ_{X}, γ_{Y}, τ ω_{0}, ρ_{0}) d s, \end{matrix}

where Ψ ∗ denotes the probability that, at a certain point in time, the EWMA statistic resides at the restarting value z 0 . According to Knoth, the pattern defined in is equivalent to a renewal point. Hence, re‐starting fabricates a renewal process and the corresponding EWMA sequence is a regenerative process.…”

Section: Computation Of Arl Of Proposed Control Chartsmentioning

confidence: 99%

See 3 more Smart Citations

Steady‐state ARL analysis of ARL‐unbiased EWMA‐RZ control chart monitoring the ratio of two normal variables

Tran

Knoth

2018

Quality & Reliability Eng

Self Cite

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Very recently, control charts for monitoring the ratio of 2 normal variables have been investigated in statistical process control. In the two‐sided case, however, these control charts tend to be average run length (ARL) biased, in the sense that some out‐of‐control ARL values are larger than the in‐control ARL. This paper proposes an ARL‐unbiased EWMA control chart for monitoring of this kind of ratio with each subgroup consisting of n⩾1 sample units. Also, to study the long‐term properties of ARL‐unbiased EWMA‐RZ control chart, we investigate the steady‐state ARL. Several tables and figures are given to show the statistical properties of the proposed control charts. The comparison results show that the proposed ARL‐unbiased chart outperforms other two‐sided control charts in terms of the zero‐state and steady‐state ARL. An example illustrates the use of this chart on a real quality control problem from the food industry.

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“…The time T ∗ is equal to a sum of a random number of run lengths, which could consist of in‐control and out‐of‐control observations. More specific, as in Knoth, we have

\begin{matrix} T_{*} & = T_{1} + T_{2} + ⋯ + T_{I_{m} - 1} + T_{I_{m}}, \\ I_{m} & = \min ({}, i \geq 1 : true \sum_{j = 1}^{i} T_{j} \geq m) . \end{matrix}

Finally, the cyclical S S A R L is given by

{scriptD}_{*} = \lim_{m \to \infty} E_{m} (T^{*} - m + 1) .

To calculate the value of

D_{*}

for T fix , we adopt the EWMA sequence Z i in the following way:

S_{i} = ({, \begin{matrix} center \\ array {\tilde{S}}_{i} & array, L C L \leq {\tilde{S}}_{i} \leq U C L \\ array z_{0} & array, ({\tilde{S}}_{i} > U C L) \cup ({\tilde{S}}_{i} < L C L) . \end{matrix}), {\overset{S}{true}}_{i} : = false(1 - λ false) S_{i - 1} + λ {\overset{Z}{true}}_{i}, i = 1, 2, ⋯

…”

Section: Computation Of Arl Of Proposed Control Chartsmentioning

confidence: 81%

Section: Computation Of Arl Of Proposed Control Chartsmentioning

confidence: 99%

“…Moreover, D 1 coincides with the latter, E 1 ( T ). The so‐called conditional S S A R L is then defined as limit

D = \lim_{m \to \infty} {scriptD}_{m} .

To calculate

scriptD

, like in Knoth, we make use of the left eigenfunction ψ ( y ) of the two‐sided EWMA‐RZ control charts as follows:

η ψ (y) = \int_{L C L}^{U C L} ψ (s) \frac{1}{λ} f_{{\hat{Z}}_{i}} (\frac{y - (1 - λ) s}{λ}| n, γ_{X}, γ_{Y}, τ ω_{0}, ρ_{0}) d s

for the dominant (hence largest in magnitude and unique) eigenvalue η <1. Then we can calculate the conditional S S A R L through

D = \frac{\int_{L C L}^{U C L} ψ (s) L (s) d s}{\int_{L C L}^{U C L} ψ (s) d s} .

For the cyclical steady state, it is assumed that the change point m may happen after a sequence of false alarms, where the control chart is restarted, in z 0 , after each false alarm.…”

Section: Computation Of Arl Of Proposed Control Chartsmentioning

confidence: 99%

“…In this case, we can use a slightly different left eigenfunction ψ ∗ ( y ) of two‐sided EWMA‐RZ control charts as follows:

\begin{matrix} ψ_{*} false(y false) & = Ψ_{*} \frac{1}{λ} f_{{\overset{Z}{true}}_{i}} ((), \frac{y - false(1 - λ false) z_{0}}{λ} true| n, γ_{X}, γ_{Y}, τ ω_{0}, ρ_{0}) \\ + \int_{L C L}^{U C L} ψ_{*} false(s false) \frac{1}{λ} f_{{\overset{Z}{true}}_{i}} ((), (|, \frac{y - false(1 - λ false) s}{λ}) n, γ_{X}, γ_{Y}, τ ω_{0}, ρ_{0}) d s, \end{matrix}

Section: Computation Of Arl Of Proposed Control Chartsmentioning

confidence: 99%

See 2 more Smart Citations

Steady‐state ARL analysis of ARL‐unbiased EWMA‐RZ control chart monitoring the ratio of two normal variables

Tran

Knoth

2018

Quality & Reliability Eng

Self Cite

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show abstract

“…We also studied the effects of measurement errors on the performance of the Synthetic

\overset{X}{true}

control chart by assuming a linear covariate error model. As has been advised in the literature (see, for example, Davis and Woodalland Knoth), both the zero‐state and the steady‐state performance of the Synthetic

\overset{X}{true}

control chart have been studied. On the basis of the presented results, it is obvious that measurement errors greatly affect the performance of Synthetic

\overset{X}{true}

chart compared with no measurement errors case.…”

Section: Discussionmentioning

confidence: 99%

A Synthetic median control chart for monitoring the process mean with measurement errors

Tran

Rakitzis

2019

Quality & Reliability Eng

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Recent studies show that the Shewhart median chart is widely used for detecting shifts in a process, but it is often rather inefficient in detecting small or moderate process shifts. In order to overcome this problem, a Synthetic chart can be used. This chart outperforms the Shewhart‐type chart because it uses the information about the time interval between two consecutive nonconforming samples. In this paper, we propose and study the Phase II Synthetic median control chart. A Markov chain methodology is used to evaluate the statistical performance of the proposed chart. Moreover, its performance is investigated in the presence of measurement errors, which are modelled by a linear covariate error model. We provide the results of an extensive numerical analysis with several tables and figures in order to show the statistical performance of the investigated chart, for both cases of measurement errors and no measurement errors. Finally, an example illustrates the use of the Synthetic median chart.

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Control Charts, Synthetic

Chakraborti

Rakitzis

2021

Wiley StatsRef: Statistics Reference Online

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It is well known that the Shewhart control charts are not very sensitive for detecting small and moderate shifts in the process parameters. A synthetic chart, which is a combination of a Shewhart control chart and a conforming run‐length (CRL) chart, has been proposed and studied as a viable alternative for that purpose. The advantage of a synthetic chart is its ease of design, operation, and interpretation. In this contribution, we discuss the most common types of synthetic charts, along with some illustrations of their practical implementation. Both parametric and nonparametric synthetic charts are considered.

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The Case Against the Use of Synthetic Control Charts

Cited by 35 publications

References 28 publications

Steady‐state ARL analysis of ARL‐unbiased EWMA‐RZ control chart monitoring the ratio of two normal variables

Steady‐state ARL analysis of ARL‐unbiased EWMA‐RZ control chart monitoring the ratio of two normal variables

A Synthetic median control chart for monitoring the process mean with measurement errors

Control Charts, Synthetic

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