Lecture #26. Gage R&R Study. Oct. 29, 2003

Transcription

1 Lecture #26 Gage R&R Study Oct. 29, 2003

2 Background - Measurement: Integral part of manufacturing Estimate the contribution attributable to the measurement system itself SPC requires accurate and precise data - Variation: Study and predict system variation Decompose the variation into components

3 Gage R&R R&R - Repeatability & Reproducibility Gage - A gage is any device that is used to obtain measurements. What is gage R&R? - Statistical approach of determing if a gage/gaging system is suitable for the process under measurement.

4 Repeatability: Definition Variation in measurements when an operator measures identical characteristics on the same part using the same measuring instrument. Reproducibility: Variation in measurements when different operators measure identical characteristics on the same part using the same measuring instrument.

5 Sources of Variation Part to Part Variation Overall Variation Measurement System Variation Variation due to Gage Repeatability Variation due to Operators Reproducibility Operators Operator by Part

6 A Statistical Model for Describing the Gage R&R Study y ijk = µ + α i + β j + (αβ) ij + ε ijk σ y 2 = σ 2 part + σ2 op + σ2 op*part + σ2 y ijk - kth measurement by operator j on part i µ - Unknown grand average α i ~ N(0,σ part ), Part effect. β j ~ N(0,σ op. ), Operator effect ( between operator effect ) (αβ) ij ~ N(0,σ op.*part ), Operator-Part interaction ε ijk ~ N(0,σ), Pure error ( within-operator effect )

7 Repeatability & Reproducibility Repeatability: [σ repeat ] 2 = σ 2 Reproducibility [σ reprod ] 2 = σ 2 op + σ2 op*part Repeatability & Reproducibility [σ r&r ] 2 = [σ repeat ] 2 + [σ reprod ] 2 = σ 2 + σ 2 op + σ2 op*part Total Variance σ y 2 = σ 2 part + σ2 r&r

8 Methods to Determine Gage R&R Average and Range method Analysis of Variance (ANOVA)

9 Example Operator A Operator B Operator C Part 1 Part 2 Part 3 Part

10 Analysis of Variance (ANOVA) Part 1 Part 2 Part 3 Part 4 Mean Operator A Operator B Operator C Mean

11 Analysis of Variance (ANOVA) S. of S. for S. of S. for Part Operator 32.60*9= *12= x 12 x 12 x 12 Mean Mean x 9 x 9 x 9 x 9

12 Analysis of Variance (ANOVA) Part 1 Part 2 Part 3 Part 4 Operator A Operator B Operator C S. of S. Pure Error: = 60.82

13 Analysis of Variance (ANOVA) Part 1 Part 2 Part 3 Part 4 Operator A Operator B Operator C S. of S. of interacton: 3*[ ( ) 2 + (0.275) (0.425) 2 ]=17.965

14 Analysis of Variance (ANOVA) No. of parts = 4 (i) No. of operators = 3 (j) No. of replications = 3 (k) Results of ANOVA DOF Sum of Squares Mean Squares F calc F table(.95) Part Operator Op * Part Error

15 Two Way Random Effect ANOVA Table DOF Sum of Squares Mean Squares EMS Part i-1 SS part MS part kjσ 2 part + kσ2 op*part + σ2 Operator j-1 SS op. MS op. kiσ 2 op. + kσ2 op*part + σ2 Op * Part (i-1)*(j-1) SS op.*part MS op.*part kσ 2 op*part + σ2 Error i*j*(k-1) SS error MS error σ 2 Part 3? 9σ 2 part + 3σ2 op*part + σ2 Operator 2? 12σ 2 op. + 3σ2 op*part + σ2 Op * Part 6? 3σ 2 op*part + σ2 Error 24? σ 2

16 Analysis of Variance (ANOVA) Repeatability [σ repeat ] 2 =σ 2 = Reproducibility σ 2 op*part = 1/3( ) = σ 2 op. = 1/12( ) = [σ reprod ] 2 = σ 2 op + σ2 op*part = Repeatability & Reproducibility [σ r&r ] 2 = [σ repeat ] 2 + [σ reprod ] 2 = 3.419

17 Gage Capability Ratio GCR If > 1? = 6σ r&r UpperSpec LowerSpec %GCR_repeat If > 5 %? if > 10%? %GCR_reprod. = = If > 20 %? if > 30%? 6σ repeat UpperSpec LowerSpec X 100% 6σ reprod UpperSpec LowerSpec X 100%

18 Average and Range Method [σ r&r ] 2 = [σ reprod ] 2 + [σ repeat ] 2 Repeatability - Equipment variation EV = Rdbar ( k) d 2 Rdbar = (Rbar A + Rbar B + Rbar C )/ No. of operators Reproducibility - Operator variation OV = X d d 2 ( j) 2 ( EV) k Xd = Max. Xbar - Min Xbar