量測重複性與再現性分析之研究 - scu.edu.tw

1 2006 10 MAY 27, pp577-594 -577- QS 9000 ISO/IEC 17025 Gauge Repeatability and Reproducibility GR&R GR&R GR&R npk n p k The Study of Gauge Repeatability

2 And Reproducibility Jeng-Jung Fang Peng-Sen Wang Yu-Li Lee Department of Management and Information Technology Southern Taiwan University of Technology Department of Management and Information Technology Southern Taiwan University of TechnologyGraduate Program of Industrial Management Southern Taiwan University of Technology Abstract Due to the QS 9000 and ISO/IEC 17025 quality standards put emphasis on the requirement of measurement system, the recent researches of gauge repeatability and reproducibility GR&R in gauge variation study have caused much attention.

3 The current measurement system usually utilizes the method of the analysis of variance (ANOVA) to analyze the measurement system while there is interaction between the products and operators. There are other methods can be used to analyze the measurement systems such as Classical GR&R and Long Form methods. Although they are easy in calculation, they are not suitable to analyze the measurement system while there is interaction between the products and operators. Based on the disadvantage of Classical GR&R and Long Form methods, the research would like to modify the methods to make them are suitable no matter there is interaction between products and operators or not.

4 In the mean time, the research will compare the repeatability and reproducibility variance estimators of eight different measurement methods by simulating a measurement system based on the criterion of the biasness, varance, and mean squares error MSE of the qauge variance estimators. The research will evaluate the accuracy and precision of the variance estimators of the five methods by changing the parameters (n, p or k) of the measurement system subject to the total number of the combination being fixed. Keywords Gauge Analysis, Repeatability, Reproducibility, Interaction 2006 10 -578- GR&R [2][9][10][11][13][18][20][30] Mandel [22] ANOVA Analysis of Variance [22][23][25][26][27] [28][29] GR&R Classical GR&R Studies [19][23][25][30] GR&R QS9000 MSA [12] Long Form [1][4][6][8][13][18][19]

5 GR&R GR&R n p k npk n p k [3][15][16] 1. 2. 3. 4. 5. 6. 7. -579- 1. Measuring System 2.

6 True Value 3. Accuracy 4. Precision 5. Repeatability 6. Reproducibility 7. Gauge Repeatability and Reproducibility GR&R 8. Interaction 1. ANOVA Montgomery and Runger [24] ANOVA P n O p Random Effect [13][23][24] Random Effects Model k 1 ===++++=k kpjniRPOOPX ijkijjiijk, 2, 1, , 2, 1, , 2, 1, 1 ijkX j i k 2006 10 -580- Pi Oj POij Rijk 2P 2O 2PO 2R 2 ()[]() ()

7 []nkMSknMSnMSnknMSMSnMSMSRPOO ilityreproducibityrepeatabilgaugeRPOOPOO ilityreproducibRRityrepeatabil11 1 22222222 + +=+= +=+=== 2 0 2<po 0 2=po [23][27] Reduced Model 3 ===+++= kp jn iROPX ijkjiijk,2,1,,2,1,,2,1 3 1 4 ()()[]nkMSnkMSnkMSMSMSRO ilityreproducibityrepeatabilgaugeROOilit yreproducibRRityrepeatabil1 2222222 +=+= ==== 4 2. GR&R Montgomery and Runger [23] Classical Gauge Repeatability and Reproducibility Study 1 5 d2 A k 1 Classical GR&R 1 2.

8 P x111 x112 1 .. x11k 11X 11R ..x12k 12X12R ..x1pk pX1 pR1x211 x212 x221x222 x2p1x2p2 2 .. x21k 21X 21R ..x22k 22X22R ..x2pk pX2 pR2 -581- .. xn11 xn12 xn21xn22 xnp1xnp2 n .. xn1k 1nX 1nR ..xn2k 2nX2nR ..xnpk npX npR 1X 1 R 2X 2 pX pR 2 dRityrepeatabil= 5 6 2 dRXilityreproducib= 6 =jjjjXXXR minmax 7 jX j 6 d2 p 3. QS 9000 MSA [12] Long Form 8 9 2d B g=1 m= 2 dRityrepeatabil= 8 ()nkdRdRXilityreproducib222*2 = 9 2 GR&R GR&R 2 ityrepeatabil ilityreproducib RMS ()[]nknMSMSnMSRPOO +1( ) ()nkMSMSRO ( ) 2006 10 -582- GR&R 2dR 2dRX 2dR ()

9 NkdRdRX222*2 GR&R GR&R Classical GR&R Classical GR&R 10 nRRniXXijij = = 1 10 == = = pjnikXXkkijkij,,1,,1,1 11 niXXRijjijjXij,,1,minmax = = 12 [5] GR&R 13 14 2' dRijXilityreproducib = 13 ()nkdRdRijXilityreproducib222*2" = 14 Montgomery and Runger [23] ()21222') (EnRPOO ilityreproducib ++= GR&R 15 16 ()ndRdRijXilityreproducib2222' = 15 -583- ()ndRdRijXilityreproducib222*2" = 16 Burdick Larsen [15] [3] p GR&R 17 ()npkdRdRijXilityreproducib222*2"' = 17 GR&R N Mean Ratio of Estimated Gauge Variance (Variance of Estimated Gauge Variance)

10 Mean Squares Error of Estimated Gauge Variance, MSE 18 1 NNigaugegauge = = 122 18 19 ()1 121222 ==NNNNiNigaugegauge 19 MSE 20 MSE () Var Bias MSE 2006 10 -584- ()()()()()[]()[]()[]()22222 Bias Var MSE+= + = + = = EEEEEEEE 20 21 MSE ()()