Consider a process with observed values following a normal distribution with ex-pected value µ and variance σ2. When FP control chart (control chart with fixed parameters) is used to monitor the aforementioned process, a random sample (n0) is taken every hour (denoted byh0).
In the case of a VSSI control chart, two different levels can be determined for the contol chart parameters (n, h). The first level represents a parameter set with loose control (n1,h1) including smaller sample size and longer sampling interval, and the second level means a strict control policy (n2,h2) with a larger sample size and shorter sampling interval. Nevertheless, n and h must satisfy the following relations:n1 < n0 < n2andh2 < h0 < h1, wheren0is the sample size andh0is the sampling interval of the FP control chart. The switch rule between the parameter levels is determined by a warning limit coefficientwindicating the specification of central and warning regions (Chen et al.,2007):
I1(i) =
wherei=1, 2... is the number of the sample,I1denotes the central region, andI2 the warning region. During the control process, the following decisions can be made (Lim et al.,2015):
1. Ifxi ∈ I1, the manufacturing process is in "in-control" state. Sample sizen1and sampling intervalh1are used to computexi+1.
2. If xi ∈ I2, the monitored process is "in-control" but xi falls in the warning region; thus,n2andh2are used for the(i+1)thsample.
3. If xi ∈/ I1 and xi ∈/ I2, the process is out of control, and corrective actions must be taken. After the corrective action, xi+1 falls into the central region (assuming that the correction was successful), but there is no previous sample to determinen(i+1)andh(i+1). Therefore, as Prabhu et al. (1994) and Costa (1994) proposed, the next sample size and interval are selected randomly with probability p0. p0 denotes the probability that the sample mean falls within
point falls within the warning region.
Appendix F
Examples for measurement process monitoring techniques
MSA Handbook (Measurement System Analysis) determines five categories of mea-surement system error: bias, repeatability, reproducibility, stability and linearity (AIAG,2010). Different statistical methods can be used to assess measurement sys-tem performance considering the five aforementioned categories. This section clari-fies the meaning of the categories and discusses the suggested methods for analysis.
Bias
Bias is the difference between the reference value and the detected average of multiple measurements when considering the same characteristic of the same part (Pyzdek, 2003, AIAG, 2010). Bias can be determined through experimental mea-surements using a certified etalon. Detailed guidance and practical examples are provided by Shaji,2006, Sibalija and Majstorovic (2007), AIAG (2010), Sahay (2010), Yu (2012).
Stability
Stability is the change in bias over an extended time period. It is the variation of the measurement result when same characteristic is measured on the same part (by the same person) over a time period. Stability can be analyzed by X-R charts, for practical example, see Shaji (2006), Sibalija and Majstorovic (2007), Sahay (2010), Pai et al. (2015).
Linearity
Similarly to stability, linearity is associated with the examination of bias however, linearity refers to the bias throughout the expected operating range. AIAG (2010) suggests to use at least five parts for the experiment that cover the operating range of the examined gage. Each part should be measured at least ten times and average bias values must be calculated against the reference values. Linear fitting can be conducted if average bias values are plotted with respect to the reference values:
biasi = axi+b (F.1)
where biasi is the bias average and xi is the reference value, a is the slope and b is the intercept of the fitted line. Gage linearity is acceptable if "bias=0" line is located entirely within the confidence bounds of the fitted curve. Detailed numerical examples are provided by Shaji (2006), Sibalija and Majstorovic (2007), AIAG (2010), Sahay (2010), Yu (2012), Pai et al. (2015), Mat-Shayuti and Adzhar (2017).
Repeatability and reproducibility
words, it is the variation in measurements when a single characteristic is measured several times on the same part by the same appraiser and using the same device.
Despite the repeatability, reproducibility aims to characterize variation "between appraisers". In this case a single characteristic is measured several times on the same part using the same device but the measurement is conducted by different appraisers (AIAG,2010). Gage R&R (or GRR) is the proposed method to analyze the variation regarding repeatability and reproducibility. The study can be conducted based on different approaches:
• Range method
• Average and Range method
• ANOVA method Range method
This is an approximation of measurement variability. Usually two appraisers participates in the study who measure the same part (5 parts) once with the same instrument. Range method does not decompose variability into repeatability and reproducibility, it focuses on the ratio of average range of obtained measurements and process standard deviation:
whereRiis the range of the obtained measurements by appraiser "A" and "B" regard-ing partiandd2is the correction constant. Based onF.2, the result can be expressed related to the process variation:
%GRR=100∗
GRR
Process Standard Deviation
(F.3) For details and practical examples, see AIAG (2010), Sahay (2010).
Average and Range method
Despite Range method, this approach provides information about repeatability and reproducibility. Three appraisers are recommended to participate in the study.
They need to measure at least ten parts, each part is measured three times by each appraiser (without seeing each others’ results) (AIAG,2010). In this case the GRR value can be expressed by equipment variation (repeatability) and appraiser varia-tion (reproducibility):
GRR =pEV2+AV2 (F.4)
whereEVis the equipment variation andAVis the appraiser variation respectively.
GRR can be also represented relative to the total variation (TV):
%GRR =100∗
GRR TV
(F.5) Average and Range method was applied by several scholars to investigate mea-surement system repeatability and reproducibility: Mohamed and Davahran (2006),
Sibalija and Majstorovic (2007), AIAG (2010), Sahay (2010), Dalalah and Diabat (2015), Mat-Shayuti and Adzhar (2017).
ANOVA method
Analysis Of Variance method provides more information than Average and Range method, since it is also able to characterize the interaction between parts and ap-praisers. The data collection procedure is the same as it is described by Average and Range method however, ANOVA table is used in order to decompose the vari-ance into specific components: parts, appraisers, interaction between appraisers and parts, and finally, repeatability due to the measurement device (AIAG, 2010). As outcome, the components’ contribution to total variance can be expressed:
%Contribution=100∗ σ
2
(components)
σ(2total)
!
(F.6) Numerical examples are provided by Senol (2004), Mohamed and Davahran (2006), Sibalija and Majstorovic (2007), Kazerouni (2009), AIAG (2010), Mat-Shayuti and Adzhar (2017).