New Capability Indices for Non Normal Processes

Grau, Daniel

doi:10.1080/03610920903156839

Cited by 5 publications

(7 citation statements)

References 11 publications

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“…Noticing that the right and left skewed distributions cannot be differentiated by PCIs

C_{Np}^{″}

C_{Npk}^{″}

C_{Npm}^{″}

, and

C_{Npmk}^{″}

, Grau proposed indices

C_{p}^{#}

as follows:

{italicC}_{italicp}^{#} = min \{\frac{{italicd}_{italicU}}{P_{99.865} - italicM}, \frac{{italicd}_{italicL}}{italicM - P_{0.135}}\} .

…”

Section: Some Existing Process Capability Indices and Their Problems mentioning

confidence: 99%

“…Using the same algebraic relations in , Grau presented indices

C_{pk}^{#}

C_{pm}^{#}

, and

C_{pmk}^{#}

as follows:

{italicC}_{italicpk}^{#} = (1 - α) {italicC}_{italicp}^{#}, {italicC}_{italicpm}^{#} = {((), 1 + δ^{2})}^{- 1 / 2} {italicC}_{italicp}^{#}, {italicC}_{italicpmk}^{#} = (1 - α) {((), 1 + δ^{2})}^{- 1 / 2} {italicC}_{italicp}^{#} .

…”

Section: Some Existing Process Capability Indices and Their Problems mentioning

confidence: 99%

“…Considering the asymmetry of tolerances and the asymmetry of the process distribution from the viewpoint of probability density, similar to the definition of index

C_{p}^{#}

in the study of Grau, a new potential capability index

C_{Hp}^{#}

can be defined as

{italicC}_{italicHp}^{#} = min (\frac{{italicd}_{italicU}}{P_{italich 2} - italicM}, \frac{{italicd}_{italicL}}{italicM - P_{italich 1}})

where M is the process median. From the relationship in and , the new proposed indices

C_{Hpk}^{#}

C_{Hpm}^{#}

, and

C_{Hpmk}^{#}

have the following relationships:

{italicC}_{italicHpk}^{#} = (1 - α) {italicC}_{italicHp}^{#}, {italicC}_{italicHpm}^{#} = {((), 1 + δ_{H}^{2})}^{- 1 / 2} {italicC}_{italicHp}^{#}, {italicC}_{italicHpmk}^{#} = (1 - α) {((), 1 + δ_{H}^{2})}^{- 1 / 2} {italicC}_{italicHp}^{#}

where α = max{( M − T )/ d U , ( T − M )/ d L } and δ H = 6 A /( P h 2 − P h 1 ).…”

Section: Process Capability Indices Based On the Highest Density Intementioning

confidence: 99%

“…Noticing that the right and left skewed distributions cannot be differentiated by PCIs C ″ Np ,C ″ Npk , C ″ Npm , and C ″ Npmk , Grau 14 proposed indices C # p as follows:…”

Section: Some Existing Process Capability Indices For Non-normal Procmentioning

confidence: 99%

“…proposed a family of PCIs

C_{Np}^{″}

C_{Npk}^{″}

C_{Npm}^{″}

, and

C_{Npmk}^{″}

, which outperforms the PCIs C Np , C Npk , C Npm , and C Npmk in detecting process shifts. With the consideration of both the asymmetry of tolerances and the asymmetry of the process distribution, Grau presented a family of PCIs

C_{p}^{#}

C_{pk}^{#}

C_{pm}^{#}

, and

C_{pmk}^{#}

. On the other hand, data transformation is another alternative to deal with non‐normal processes.…”

Section: Introductionmentioning

confidence: 99%

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Process Capability Indices Based on the Highest Density Interval

Yang

Gang

Cheng

et al. 2014

Quality & Reliability Eng

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For process capability indices (PCIs) of non‐normal processes, the natural tolerance is defined as the difference between the 99.865 percentile and the 0.135 percentile of the process characteristic. However, some regions with relatively low probability density may still be included in this natural tolerance, while some regions with relatively high probability density may be excluded for asymmetric distributions. To take into account the asymmetry of process distributions and the asymmetry of tolerances from the viewpoint of probability density, the highest density interval is utilized to define the natural tolerance, and a family of new PCIs based on the highest density interval is proposed to ensure that regions with high probability density are included in the natural tolerance. Some properties of the proposed PCIs and two algorithms to compute the highest density interval are given. A real example is given to show the application of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.

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“…Noticing that the right and left skewed distributions cannot be differentiated by PCIs

C_{Np}^{″}

C_{Npk}^{″}

C_{Npm}^{″}

, and

C_{Npmk}^{″}

, Grau proposed indices

C_{p}^{#}

as follows:

{italicC}_{italicp}^{#} = min \{\frac{{italicd}_{italicU}}{P_{99.865} - italicM}, \frac{{italicd}_{italicL}}{italicM - P_{0.135}}\} .

…”

Section: Some Existing Process Capability Indices and Their Problems mentioning

confidence: 99%

“…Using the same algebraic relations in , Grau presented indices

C_{pk}^{#}

C_{pm}^{#}

, and

C_{pmk}^{#}

as follows:

{italicC}_{italicpk}^{#} = (1 - α) {italicC}_{italicp}^{#}, {italicC}_{italicpm}^{#} = {((), 1 + δ^{2})}^{- 1 / 2} {italicC}_{italicp}^{#}, {italicC}_{italicpmk}^{#} = (1 - α) {((), 1 + δ^{2})}^{- 1 / 2} {italicC}_{italicp}^{#} .

…”

Section: Some Existing Process Capability Indices and Their Problems mentioning

confidence: 99%

“…Considering the asymmetry of tolerances and the asymmetry of the process distribution from the viewpoint of probability density, similar to the definition of index

C_{p}^{#}

in the study of Grau, a new potential capability index

C_{Hp}^{#}

can be defined as

{italicC}_{italicHp}^{#} = min (\frac{{italicd}_{italicU}}{P_{italich 2} - italicM}, \frac{{italicd}_{italicL}}{italicM - P_{italich 1}})

where M is the process median. From the relationship in and , the new proposed indices

C_{Hpk}^{#}

C_{Hpm}^{#}

, and

C_{Hpmk}^{#}

have the following relationships:

{italicC}_{italicHpk}^{#} = (1 - α) {italicC}_{italicHp}^{#}, {italicC}_{italicHpm}^{#} = {((), 1 + δ_{H}^{2})}^{- 1 / 2} {italicC}_{italicHp}^{#}, {italicC}_{italicHpmk}^{#} = (1 - α) {((), 1 + δ_{H}^{2})}^{- 1 / 2} {italicC}_{italicHp}^{#}

where α = max{( M − T )/ d U , ( T − M )/ d L } and δ H = 6 A /( P h 2 − P h 1 ).…”

Section: Process Capability Indices Based On the Highest Density Intementioning

confidence: 99%

“…Noticing that the right and left skewed distributions cannot be differentiated by PCIs C ″ Np ,C ″ Npk , C ″ Npm , and C ″ Npmk , Grau 14 proposed indices C # p as follows:…”

Section: Some Existing Process Capability Indices For Non-normal Procmentioning

confidence: 99%

“…proposed a family of PCIs

C_{Np}^{″}

C_{Npk}^{″}

C_{Npm}^{″}

, and

C_{Npmk}^{″}

C_{p}^{#}

C_{pk}^{#}

C_{pm}^{#}

, and

C_{pmk}^{#}

. On the other hand, data transformation is another alternative to deal with non‐normal processes.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Process Capability Indices Based on the Highest Density Interval

Yang

Gang

Cheng

et al. 2014

Quality & Reliability Eng

View full text Add to dashboard Cite

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Model‐based process capability indices: The dry‐etching semiconductor case study

Borgoni

Zappa

2020

Quality & Reliability Eng

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Process capability indices are widely used to check quality standards both at the production level and for business activity. They consider the location and the deviation from specification limits and targets. The literature contains many contributions on this topic both in the univariate and the multivariate context. Motivated by a real semiconductor case study, we discuss the role of rational subgroups and the challenge they present in the computation of capability indices, especially when data refer to lots of products. In addition, our context involves a mix of problems: unilateral specification limit, nonsymmetric distribution of the data, evidence of data from a mixture of distributions, and the need to filter one component of the mixture. After solving the previous issues and because of the peculiar characteristics of semiconductor processes based on the so called "wafers," we contribute to the literature a proposal on how to compute capability indices in the case of heteroscedastic spatial processes. With a generalized additive model, we show that it is possible to estimate a capability surface that allows the identification of regions expected to not be fully compliant with the desired quality standards. K E Y W O R D S dry-etching semiconductor processes, mixture distributions, process capability indices, sampling network 1 | INTRODUCTION Process capability indices (PCIs) are well-known tools to estimate the mean-deviance performance of key product characteristics with respect to both targets and specification limits. The main aim of PCIs was originally to evaluate the instant quality of a process given specification limits and prior to manufacturing. Next, because of their simple interpretation, they have also become a fundamental tool for commercial activity. Nowadays, PCIs are unavoidable instruments for managers and engineers; hence, when assumptions upon which these indices are grounded can be violated, much effort has been made to extend and adapt them to such situations (see Montgomery 1). A flow of contributions is available in the literature for both univariate and multivariate cases. The capability indices that researchers have proposed are so many that, since the masterful work of Kotz and Johnson, 2 the word "avalanche" has been coined. A detailed and somewhat critical review on PCIs is available in Kotz and Lovelace, 3 whereas Vännman 4 proposes a unified approach. A thorough review of contributions is reported in de-Felipe and Benedito 5 and Bong-Jin and Kwan-Woo 6 among others). Most of the papers focus on the assumption of normality of the random components, but how capability indices could be efficiently estimated when non normal distributions are involved remains

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