What shift from the target do you want to be able to detect?

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

A. PRACTICE A1. CUSUM chart:

. What shift from the target do you want to be able to detect?

This shift is usually ½ or 1 standard deviation of the parameter being plotted. This shift is often referred to as the “allowable slack” in the process.

A CUSUM control chart then tracks the deviations from the target. If a sample result is within the allowable slack, the process is considered to be operating on target. If a sample result is beyond the allo wable slack, the process is considered to be operating off target.

Definition: 𝐶𝑖 = Σ𝑖 (𝑥𝑖 − 𝜇0)

j=1

𝐶𝑖 = Σ𝑖 (𝑥𝑖 − 𝜇0) = (𝑥𝑖 − 𝜇0) + Σ𝑖−1 (𝑥𝑖 − 𝜇0) = (𝑥𝑖 − 𝜇0) + 𝐶𝑖−1

j=1 𝑗= 1

Upper and Lower statistics

𝐶𝑖+ = max [ 𝟎, (𝑥𝑖 − 𝜇0) + 𝐶𝑖−1 ] [ Upper Cusum statistic]

𝐶𝑖− = max[ 0, −(𝑥𝑖 − 𝜇0) + 𝐶𝑖−1 ] [Lower Cusum statistic]

𝐶0+ = 𝐶0− = 0

𝐶𝑖 + , 𝐶𝑖 − measure the deviations from the target.(positive and negative)

𝑆ℎ𝑖𝑓𝑡 𝑜𝑓 𝜇 𝑓𝑟𝑜𝑚 𝜇0 𝑡𝑜 𝜇1: 𝜇1 = 𝜇0 + 𝛿𝜎 = > 𝛿 = (𝜇1 − 𝜇0)/𝜎

𝐴𝑙𝑙𝑜𝑤𝑎𝑏𝑙𝑒 𝑆𝑙𝑎𝑐𝑘: 𝐾 = ( 𝛿) 𝜎 (𝑎𝑙𝑙𝑜𝑤𝑎𝑏𝑙𝑒 𝑠𝑙𝑎𝑐𝑘) = (𝜇1 − 𝜇0) ; 𝐾 = 𝑘𝜎     2 2

𝐷𝑒𝑐𝑖𝑠𝑖𝑜𝑛 𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙: 𝐻 = 4𝜎 𝑜𝑟 5𝜎 ; 𝐻 = ℎ𝜎

Upper and Lower statistics

𝐶𝑖+ = max [ 𝟎, (𝑥𝑖 − (𝜇0 + 𝐾) ) + 𝐶𝑖−1 ] [ Upper Cusum statistic]

𝐶𝑖− = max[ 0, −(𝑥𝑖 − (𝜇0 + 𝐾) ) + 𝐶𝑖−1 ] [Lower Cusum statistic]

Note: There are 2 Cusum charts, plotted together

> library(SixSigma)

> data("ss.data.thickness2")

> head(ss.data.thickness2)

day shift thickness ushift flaws 1 1 1 0.713 1.1 9

2 1 1 0.776 1.1 NA

3 1 1 0.743 1.1 NA

4 1 1 0.713 1.1 NA

5 1 1 0.747 1.1 NA

6 1 1 0.753 1.1 NA

 

 

# CUSUM chart[detecting shifts in the sample means]

> thickness.cusum <- cusum(thickness, nsigmas=3)

> summary(thickness.cusum) Call:

cusum(data = thickness, nsigmas = 3) cusum chart for thickness

Summary of group statistics:

Min. 1st Qu. Median Mean 3rd Qu. Max. 0.6770 0.7395 0.7535 0.7585  0.7780 0.8220

Group sample size: 1

Number of groups: 84

Center of group statistics: 0.7585238

Standard deviation: 0.02314577

Decision interval (std.err.): 5 Shift detection (std. err.): 1

Attachments:

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