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SMU EMIS 7364

SMU EMIS 7364. NTU TO-570-N. Statistical Quality Control Dr. Jerrell T. Stracener, SAE Fellow. More Control Charts Material Updated: 3/24/04. Operating Characteristic (OC) Function for the x - Chart The OC curve describes the ability of the x-chart to detect shifts in process quality.

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SMU EMIS 7364

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  1. SMU EMIS 7364 NTU TO-570-N Statistical Quality Control Dr. Jerrell T. Stracener, SAE Fellow More Control Charts Material Updated: 3/24/04

  2. Operating Characteristic (OC) Function for the x - Chart • The OC curve describes the ability of the x-chart to detect shifts in process quality. • For an x-chart with s known & constant mean m shifts from in-control value, m0 to another value m1, where • m1 = m0 + Ks

  3. Operating Characteristic (OC) Function for the x – Chart (continued)

  4. Operating Characteristic (OC) Function for the x – Chart (continued) where and L is usually 3, the three-sigma limits

  5. Example If n=5 & L=3, determine & plot the OC function vs K, where m1= m0 + Ks.

  6. Example - Solution

  7. OC Function of the Fraction Nonconforming Control Chart

  8. OC Function of the Fraction Nonconforming Control Chart Where [nUCL] denotes the largest integer  nUCL and <nLCL> denotes the smallest integer  nLCL Note: The OC curve provides a measure of the sensitivity of the control chart – i.e., its ability to detect a shift in the process fraction nonconforming from the nominal value p to some other value p.

  9. Example For a fraction nonconforming control chart with parameters n = 50, LCL = 0.0303, and UCL = 0.3697, Determine & plot the OC curve.

  10. Example - Solution

  11. Example - Solution OC(p) p

  12. OC Function for c-charts and u-charts • For the c-chart

  13. OC Function for c-charts and u-charts • For the u-chart

  14. Example Determine & plot the OC function for a u-chart with parameter. LCL = 6.48, and UCL = 32.22.

  15. Example - Solution

  16. Example - Solution OC(u) u

  17. Average Run Length for x-Charts • Performance of Control Charts can be characterized by their run length distribution. • Run Length (RL) of a control chart is defined to be the number of samples until the process characteristic exceeds the control limits for the first time. • Run Length, RL, is a random variable and therefore has a probability distribution

  18. Average Run Length for x-Charts Let p = P(x falls outside control limits) Then P(RL = 1) = P(x1 falls outside CL)=p P(RL = 2) = P(x1 falls inside CL & x2 falls outside of CL) = (1-p)p P(RL = 3) = P(x1, x2 fall inside CL & x3 falls outside of CL) = (1-p)(1-p)p P(RL = i) = P(x1, x2, …xi-1 fall inside CL & xi falls outside of CL) = (1-p)i-1p

  19. Average Run Length for x-Charts Therefore, the probability mass function of RL is The mean or expected value of RL is

  20. Average Run Length for x-Charts • The Average Run Length, ARL, indicates the number of samples needed, on the average before x will exceed the control chart limits.

  21. Probability of Out-Of-Control Signal and ARL • Process in control with mean m0 • p = 1 – P(LCL  x  UCL) • = 0.0027 • ARL • i.e., one the average we would expect 1 out-of-control signal out of 370 samples.

  22. Probability of Out-Of-Control Signal and ARL • Process in control with mean m=m1=m0+ds with constant s • What happens if the process goes out of control? • How long does it take until the control charts detects the shift? • Probability of detecting shift

  23. Probability of Out-Of-Control Signal and ARL

  24. Example For example, if n = 5, and d = 1, and

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