P10

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Second Graduate Research Symposium, November 16, 2007

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 Contents Introduction to SPC & Control Charts Traditional Attribute CC Problems in High Quality Processes Geometric Control Charts (GCC) in High Quality Processes Runs Rules Geometric Control Charts Optimized Runs Rules Geometric Control Charts Comparisons & Conclusions

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 SQC Statistical Quality Control (SQC) The application of statistical techniques to control and improve quality

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 A chart with upper and lower control limits on which values of some statistical measure for a series of samples or subgroups are plotted. Control Charts Shewhart (1924):

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 Other Control Charts: EWMA, CUSUM, MVCC, MACC, Probability Limits CC Control Charts

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 What is a high-quality process? P (‘a unit doesn’t not conform to specifications ’) = Very small value (e.g., 1000 PPM=0.001) Electronics Manufacturing Medical Supplies Sterilization Medicine And all of the enhanced processes by Six Sigma methodology

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 Inefficiency of p, np, c, u Consider :n=50 , p=0.0001 (100 PPM)

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 Consider :n=50 , p=0.0001 (100 PPM) To avoid this pitfall, we can increase the sample size (n). Inefficiency of p, np, c, u

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 Proper sample size Montgomery (2005) & Duncan (1987) Calculated sample size by mentioned methods is not practical

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 Geometric Control Charts

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 Geometric Control Charts With Probability Limits (Xie & Goh 1997 ) To calculate these control limits we use inverse of geometric CDF.

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 Average Run Length (ARL) ARL is the average number of points that must be plotted before a point indicates an out-of-control condition. When observations are independent, ARL can be calculated by:

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 Runs Rules Control Charts Derman and Ross (1997) & Klein (2000) Two consecutive points outside the control limits Two of three consecutive points outside the control limits

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 Runs Rules Geometric Control Charts State 1: No points beyond either control limits State 2: A point above the upper control limit State 3: A point below the lower control limit State 4: Two successive points are beyond just one of the control limits. (Absorbing State) (Markov Chain Definition for 2 successive points)

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 Runs Rules Geometric Control Charts ARL Calculation for 2 successive points:

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 State (OO) has two successive points between both control limits; State (OU) has a first point between both control limits and the second above the UCL; State (OL) has a first point between both control limits and the second below the LCL; State (UL) has its first point above the UCL and its second below the LCL; State (UO) has its first point above the UCL and its second between the control limits; State (LO) has its first point below the LCL and its second between the control limits; State (LU) has its first point below the LCL and its second above the UCL; State (OOC) the absorbing state, has two of three points either below the LCL or above the UCL. Runs Rules Geometric Control Charts (Markov Chain Definition for 2 out of 3 points)

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 Conditional Decision Procedure (Kuralamani et al., 2002) The number of conforming items must be counted until a nonconforming one is found. This number is drawn on control charts. The process is announced under statistical control if: the count of conforming items are plotted within the lower and upper control limits; or the current count of conforming items are not within the control limits given the s previous observations were plotted within the control limits.

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 Comparing RRGCC with Conditional Decision Procedure (Kuralamani 2002) using ARL

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 Comparing RRGCC with Conditional Decision Procedure (Kuralamani 2002) using ARL

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 Optimizing RRGCC using a Constraint Nonlinear Programming

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 Optimized Conditional Decision Procedure (Kuralamani et al., 2002)

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 Comparing optimized RRGCC with optimized Conditional Decision Procedure (Kuralamani 2002)

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 Comparing optimized RRGCC with optimized Conditional Decision Procedure (Kuralamani 2002)

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Kamran Paynabar, Kami@wayne.edu Second Graduate Research Symposium, November 16, 2007 Conclusions The Geometric control chart is a good substitute for traditional attribute control charts (p, np, c, u). To improve the performance of G-charts it’s better to use the information of past observation. Runs Rules Geometric Control Chart use these information It performs better than the conditional decision procedure suggested by Kuralamani (2002). Using a Constraint Non-linear programming model, RRGCC can be optimized.