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Chapter 12 Factorial and Fractional Factorial Experiments for Process Design and Improvement Introduction to Statistical Quality Control, 4th Edition General model of a process Controllable input factors x1 x2 . . . xp Input Output, y Process z1 z2 ... zq Uncontrollable input factors Introduction to Statistical Quality Control, 4th Edition 12-1. What is Experimental Design? Objectives of Experimental Design – Determine which variables (x’s) are most influential on the response, y – Determine where to set the influential x’s so that y is near the nominal requirement – Determine where to set the influential x’s so that variability is small – Determine where to set the influential x’s so that the effects of the uncontrollable variables z are minimized Introduction to Statistical Quality Control, 4th Edition 12-1. What is Experimental Design? Results of Experimental Design (used early in process development): 1. Improved yield 2. Reduced variability and closer conformance to nominal 3. Reduced development time 4. Reduced overall costs Introduction to Statistical Quality Control, 4th Edition 12-2. Examples Example 12-1 Characterizing a Process • SPC has been applied to a soldering process. Through u-charts and Pareto analysis, statistical control has been established and the number of defective solder joints has been reduced to 1%. The average board contains over 2000 solder joints, 1% may still be too large. • Desired to reduce the defects level more. Introduction to Statistical Quality Control, 4th Edition 12-1. What is Experimental Design? Example 12-1 Characterizing a Process • Note: since the process is in statistical control, not obvious what machine adjustments will be necessary. There are several variables that may affect the occurrence of defects: – Solder temp, preheat temp, conveyor speed, flux type, flux specific gravity, conveyor angle. • A designed experiment involving these factors could help determine which factors could help significantly reduce defects. (Screening experiment) . Introduction to Statistical Quality Control, 4th Edition 12-2. Guidelines for Designing Experiments Procedure for designing an experiment 1. Recognition of and statement of the problem. 2. Choice of factors and levels. 3. Selection of the response variable. 4. Performing the experiment 5. Data analysis 6. Conclusions and recommendations • #1-#3 make up pre-experimental planning • #2 and #3 often done simultaneously, or in reverse order. Introduction to Statistical Quality Control, 4th Edition 12-3. Factorial Experiments • When there are several factors of interest in an experiment, a factorial design should be used. • A complete trial or replicate of the experiment for all possible combinations of the levels of the factors are investigated. Introduction to Statistical Quality Control, 4th Edition 12-3. Factorial Experiments • Main effect is the change in response produced by a change in the level of a primary factor. • An interaction is present among factors if a change in the levels of one factor influences the effect of another factor. • Consider an experiment with two factors A & B – Interested in • Main effect of A • Main effect of B • Interaction effect of AB Introduction to Statistical Quality Control, 4th Edition 12-3. Factorial Experiments 12-3.2 Statistical Analysis • Completely randomized design with two factors (A and B) and n replicates. • The model is y ij k i j ( ) ij ij k where = overall mean i = effect of ith level of factor A j = effect of jth level of factor B ()ij = effect of the interaction between A and B = random error component Introduction to Statistical Quality Control, 4th Edition 12-3. Factorial Experiments 12-3.2 Statistical Analysis Factor B 1 2 b 1 y111, y112, …, y11n y121, y122, …, y12n y1b1, y1b2, …, y1bn 2 y211, y212, …, y21n y221, y222, …, y22n … y2b1, y2b2, …, y2bn Factor A a ya11, ya12, …, ya1n ya21, ya22, …, ya2n … yab1, yab2, …, yabn Introduction to Statistical Quality Control, 4th Edition 12-3. Factorial Experiments 12-3.2 Statistical Analysis • Total corrected sum of squares SSB decomposition a b n a SSA b ( y ijk y... ) bn ( y i.. y... ) an ( y. j. y... ) 2 2 2 i 1 j1 k 1 i 1 j1 a b n ( y ij. y i.. y. j. y... ) 2 SST i 1 j1 a b n ( y ijk y ij. ) 2 SSAB i 1 j1 k 1 SSE Introduction to Statistical Quality Control, 4th Edition 12-3. Factorial Experiments 12-3.2 Statistical Analysis • Total corrected sum of squares decomposition, notation: SST = SSA + SSB + SSAB + SSE • The corresponding degree of freedom decomposition is abn – 1 = (a – 1) + (b – 1) + (a – 1)(b – 1) + ab(n – 1) Introduction to Statistical Quality Control, 4th Edition 12-3. Factorial Experiments 12-3.2 Statistical Analysis Source of Sum of Degrees of Variation Squares Freedom Mean Square F0 A SSA a-1 SS MSA MSA A F0 a 1 MSE B SSB b-1 SS B MSB MSB F0 b 1 MSE Interaction SSAB (a – 1)(b – 1) SS AB MSAB MS AB F0 (a 1)( b 1) MSE Error SSE ab(n-1) SS E MSE ab(n 1) Total SST abn - 1 Introduction to Statistical Quality Control, 4th Edition 12-3. Factorial Experiments 12-3.2 Statistical Analysis Sum of Squares Computing Formulas 2 a b n y... SS T y ijk 2 i 1 j1 k 1 abn 2 a y2 2 y... b y 2 y... Main Effects SS A i.. SS B . j. i 1 bn abn j1 an abn 2 2 a b y y... SS AB SS A SS B ij. Interaction i 1 j1 n abn Error SSE = SST – SSA - SSB - SSAB Introduction to Statistical Quality Control, 4th Edition Adhesion Force Data Methods Primer Dipping Spraying yi. Type 1 4.0, 4.5,4.3 5.4, 4.9,5.6 28.7 12.8 15.9 2 5.6,4.9,5.4 5.8,6.1,6.3 34.1 15.9 18.2 3 3.8,3.7,4.0 5.5,5.0,5.0 15.5 27.0 11.5 y.j 40.2 49.6 89.8 Introduction to Statistical Quality Control, 4th Edition 12-3. Factorial Experiments Example 12-5 Aircraft Primer Example 2 a b n y... SST yijk 2 i 1 j1 k 1 abn (89.8) 2 (4.0) 2 (4.5) 2 (5.0) 2 10.72 18 a y i2.. y... 2 SSprimers i 1 bn abn (28.7) 2 (34.1) 2 (27.0) 2 (89.8) 2 4.58 6 18 2 2 b y y... SSmethods . j. j1 an abn (40.2) 2 (49.6) 2 (89.8) 2 4.91 9 18 Introduction to Statistical Quality Control, 4th Edition 12-3. Factorial Experiments Example 12-5 Aircraft Primer Example 2 2 a b y ij. y... SSint eraction SSprim ers SSm ethods i 1 j1 n abn (12.8) 2 (15.9) 2 (11.5) 2 (15.9) 2 (18.2) 2 (15.5) 2 3 (89.8) 2 4.58 4.91 0.24 18 SSE = 10.72 – 4.58 – 4.91 – 0.24 = 0.99 Introduction to Statistical Quality Control, 4th Edition 12-3. Factorial Experiments Example 12-5 Aircraft Primer Example – Table 12-4. Analysis of Variance Sum of Degrees of Source of Variation Squares Freedom Mean Square F0 P-value Primer types 4.58 2 2.29 28.63 2.71 x 10-5 Application methods 4.91 1 4.91 61.38 4.65 x 10-6 Interaction 0.24 2 0.12 1.5 0.269 Error 0.99 12 0.08 Total 10.72 17 Introduction to Statistical Quality Control, 4th Edition 12-3. Factorial Experiments Example 12-5 Aircraft Primer Example – Figure 12-12. Graph of average adhesion force versus primer types Introduction to Statistical Quality Control, 4th Edition 12-3. Factorial Experiments 12-3.3 Residual Analysis • Residuals are important in accessing model adequacy • The residuals from a two-factor factorial are eijk yijk yijk ˆ yijk yijk Introduction to Statistical Quality Control, 4th Edition Residuals Primer Type Dipping Spraying 1 -.26, .23, .03 .10, -.40, .30 2 .30, -.40, .10 -.26, .04, .23 3 -.03, -.13, .16 .34, -.17, -.17 Introduction to Statistical Quality Control, 4th Edition 12-3. Factorial Experiments 12-3.3 Residual Analysis (Table 12.6 also) Residuals Versus Primer (response is Force) 0.4 0.3 0.2 0.1 Residual 0.0 -0.1 -0.2 -0.3 -0.4 1 2 3 Primer Introduction to Statistical Quality Control, 4th Edition 12-3. Factorial Experiments 12-3.3 Residual Analysis Normal Probability Plot .999 .99 .95 Probability .80 .50 .20 .05 .01 .001 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 Residuals Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design 12-4.1 The 22 Design • 2k is the notation used to indicate that a certain experimental design has k factors of interest, each at two levels. • 22 design: Two factors A and B, each at two levels A B Low -1 -1 High +1 +1 There are a total of four possible combinations. Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design 12-4.1 The 22 Design • The simplest design involves two factors A and B and n replicates. • Interested in the main effect of A, the main effect of B, and the interaction between A and B. • Effects are calculated by: Average Response at high level - Average Response at the low level. • A large effect would indicate a significant factor (or interaction). (How large is large?) • Contrasts can be calculated and used to estimate the effects and the sums of squares. Introduction to Statistical Quality Control, 4th Edition High (+) b ab B Low (1) a (-) Low High (-) A (+) Introduction to Statistical Quality Control, 4th Edition Understanding the effect of A a+ab-b-(1) High (+) b ab - + B Low (1) a (-) - Low High + (-) A (+) Introduction to Statistical Quality Control, 4th Edition Understanding the effect of B b+ab-a-(1) High (+) b ab + + B Low (1) a (-) - Low High - (-) A (+) Introduction to Statistical Quality Control, 4th Edition Understanding the effect of AB ab+(1)-a-b High (+) b ab - + B Low (1) a (-) + Low High - (-) A (+) Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design 12-4.1 The 22 Design • Let the letters (1), a, b, and ab represent the totals of all n observations taken at these design points. • Effect estimate of A: A yA yA a ab b (1) 2n 2n 1 a ab b (1) 2n Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design 12-4.1 The 22 Design • Effect estimate of B: B y B y B b ab a (1) 2n 2n 1 b ab a (1) 2n • Effect estimate of AB: ab (1) a b AB 2n 2n 1 ab (1) a b 2n Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design 12-4.1 The 22 Design • For the previous effects formulas, the quantities in brackets are called contrasts. • For example, ContrastA = a + ab – b – (1) • The contrasts are used to calculate the sum of squares for the factors and interaction. (contrast ) 2 SS n (contrast coefficien ) 2 ts Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design 12-4.1 The 22 Design • The sum of squares for A, B, and AB are: SSA a ab b (1) 2 4n SSB b ab a (1)2 4n SSAB ab (1) a b2 4n Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design Example 12-6 • The effect estimates for A, B, and AB for the router example are: A a ab b (1) 2n 1 96.1 161.1 59.7 64.4 16.64 2( 4) B b ab a (1) 2n 1 59.7 161.1 96.1 64.4 7.54 2( 4) Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design Example 12-6 • The effect estimates for A, B, and AB for the router example are: AB ab (1) a b 2n 1 161 .1 64 .4 96 .1 59 .7 8.71 2( 4) Introduction to Statistical Quality Control, 4th Edition Sums of squares SSA=[a+ab-b-(1)]2/abn SSA=(96.1+161.1-59.9-64.4)2/16=1107.226 SSB=[b+ab-a-(1)]2/abn SSB=(59.7+161.1-96.1-64.4)2/16=227.256 Introduction to Statistical Quality Control, 4th Edition Sums of squares SSAB=[ab+(1)-a-b]2/abn SSAB=(161.1+64.4-96.1-59.7]2/16=303.631 SST=(18.22 +18.92+…+39.92) – (64.4+96.1+59.7+161.1)2/16 = 1709.836 Introduction to Statistical Quality Control, 4th Edition Sums of squares SSE by subtraction: SSE= 1709.836-1107.226-227.256-303.631 =71.723 Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design Example 12-6 • The ANOVA table for the router example is then Source of Sum of Degrees of P-value Variation Squares Freedom Mean Square F0 Bit Size (A) 1107.226 1 1107.226 185.25 1.17 x 10-8 Speed (B) 227.256 1 227.256 38.03 4.82 x 10-5 AB 303.631 1 303.631 50.80 1.20 x 10-5 Error 71.723 12 5.977 Total 1709.836 15 F.025,1,12 = 6.55 2-tailed at .05 1-tailed at .05 F.05,1,12 = 4.75 Introduction to Statistical Quality Control, 4th Edition Part of MiniTab Output Source DF Seq SS Adj SS Adj MS F P Main Effects 2 1334.48 1334.48 667.241 111.64 0.000 2-Way Interactions 1 303.63 303.63 303.631 50.80 0.000 Residual Error 12 71.72 71.72 5.977 Pure Error 12 71.72 71.72 5.977 Total 15 1709.83 Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design Regression Model • A regression model could be fit to data from a factorial design y 0 1x1 2 x 2 12 x1x 2 • where 0 is the grand average of all observations and each coefficient, j is effect estimate. • For Example 12-6, the fitted regression model is 16 .64 7.54 8.71 ˆ 23 .83 y x1 x 2 x1x 2 2 2 2 Introduction to Statistical Quality Control, 4th Edition Using the regression model • Predict the value with small bit (x1=-1) and low speed (x2=-1) • yest = 23.83 + [16.64/2](-1) + [7.54/2](-1) +[8.71/2](-1)(-1) = 16.1 Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design Residual Analysis • Residual plots are used to access the adequacy of the model once again. • Residuals are calculated using the fitted regression model. • The residual plots versus the factor levels, interactions, predicted values, and a normal probability plot are all useful in determining the adequacy of the model and satisfaction of assumptions. Introduction to Statistical Quality Control, 4th Edition Computing the residuals • For (1) with yest = 16.1 – estimate = actual – yest – e1 = 18.2 – 16.1 = 2.1 – e2 = 18.9 – 16.1 = 2.8 – e3 = 12.9 – 16.1 = -3.2 – e4 = 14.4 – 16.1 = -1.7 • See Fig. 12-19 and Fig. 12-20 Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design Analysis Procedure for Factorial Designs 1. Estimate the factor effects 2. Form preliminary model 3. Test for significance of factor effects 4. Analyze residuals 5. Refine model, if necessary 6. Interpret results Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design 12-4.2 The 2k Design for k 3 Factors • When k 2, you could have a single replicate, but some assumptions need to be made. (Can’t estimate all interactions). • For k = 3, the main effects and interactions of interest are A, B, C, AB, AC, BC, ABC. • The main effects are again represented by a, b, c, ab, ac, bc, abc, and (1) Introduction to Statistical Quality Control, 4th Edition See Fig. 12-24 for understanding the effects Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design 12-4.2 The 2k Design for k 3 Factors Effect Estimate for A: A y A y A 1 a ab ac abc b c bc (1) 4n Effect Estimate for B: B y B y B 1 b ab bc abc a c ac (1) 4n Effect Estimate for C: C y C y C 1 c ac bc abc a b ab (1) 4n Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design 12-4.2 The 2k Design for k 3 Factors Effect Estimate for AB: AB y AB y AB 1 ab (1) abc c b a bc ac 4n Effect Estimate for AC: AC y AC y AC 1 ac (1) abc b a c ab bc 4n Effect Estimate for BC: BC y BC y BC 1 bc (1) abc a b c ab ac 4n Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design 12-4.2 The 2k Design for k 3 Factors Effect Estimate for ABC: ABC y ABC y ABC 1 abc bc ac c ab b a (1) 4n In general, the effects can be estimated using Contrast Effect n 2k 1 The sum of squares for any effect is SS Contrast 2 n 2k Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design Example 12-7 • An experiment was performed to investigate the surface finish of a metal part. The experiment is a 23 factorial design in the factors feed rate (A), depth of cut (B), and tool angle (C), with n = 2 replicates. Introduction to Statistical Quality Control, 4th Edition Effect estimates • A = (1/8)(22+27+23+30-20-21-18-16) =3.375 • B = 1.625 • C = 0.875 • AB = 1.375 • AC = 0.125 • BC = -0.625 • ABC = 1.125 Introduction to Statistical Quality Control, 4th Edition Sums of squares • A = (27)2/[2(8)] = 45.5625 • B= (13)2/[2(8)] = 10.5625 • : • Total = 92.9375 • Error by subtraction Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design Example 12-7 Design Factors Surface Run A B C Finish Totals 1 (1) -1 -1 -1 9, 7 16 2 a 1 -1 -1 10, 12 22 3 b -1 1 -1 9, 11 20 4 ab 1 1 -1 12, 15 27 5 c -1 -1 1 11, 10 21 6 ac 1 -1 1 10, 13 23 7 bc -1 1 1 10, 8 18 8 abc 1 1 1 16, 14 30 Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design Example 12-7 – Analysis of Variance Table Source of Sum of Degrees of P-value Variation Squares Freedom Mean Square F0 A 45.5625 1 45.5625 18.69 2.54 x 10-3 B 10.5625 1 10.5625 4.33 0.07 C 3.0625 1 3.0625 1.26 0.29 AB 7.5625 1 7.5625 3.10 0.12 AC 0.0625 1 0.0625 0.03 0.88 BC 1.5625 1 1.5625 0.64 0.45 ABC 5.5625 1 5.5625 2.08 0.19 Error 19.5000 8 2.4375 Total 92.9375 15 Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design Regression Model • For Example 12-7, the fitted regression model involving only those factors found significant (A, B) and the next significant interaction (AB) is 3.375 1.625 1.375 y 11.0625 ˆ x1 x 2 x1 x 2 2 2 2 11.0625 1.6875x1 0.8125x 2 0.6875x1x 2 • Developing a model from a designed experiment can be a valuable tool in determining optimal settings for the factors. Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design Other Methods for Judging the Significance of Effects • The standard error of any effect estimate in a 2k design is 2 ˆ s.e.(Effect) n 2k 2 • Two standard deviation limits on any estimated effect is Effect estimate 2[s.e.(Effect)] Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design Other Methods for Judging the Significance of Effects Effect estimate 2[s.e.(Effect)] • This interval is an approximate 95% confidence interval on the estimated effect. • Interpretation: If zero is contained within the 95% confidence interval, then that effect is essentially zero and the corresponding factor is not significant at the = 0.05 level. Introduction to Statistical Quality Control, 4th Edition Example • In example 12-7 – MSE = est2 = 2.4375 – s.e.(Effect) = SQRT{2.4375/[(2)(23-2)]} = .78 Introduction to Statistical Quality Control, 4th Edition Example • 2 std dev limits on the effects: – A: 3.375 + 1.56 (does not include 0) – B: 1.625 + 1.56 (does not include 0) – C: 0.875 + 1.56 (does include 0) – AB: 1.375 + 1.56 (does include 0) – AC: 0.125 + 1.56 (does include 0) – BC: -0.625 + 1.56 (does include 0) – ABC: 1.125 + 1.56 (does include 0) Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design 12-4.3 A Single Replicate of the 2k Design • As the number of factors in a factorial experiment increases, the number of effects that can be estimated also increases. • In most situations, the sparsity of effects principle applies. • For a large number of factors, say k > 5, it is common practice to run only a single replicate of the 2k design and pool or combine the higher- order interactions in the estimate of error. Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design Example 12-8 • Nitride etch process on a single-wafer plasma etcher. There are four factors of interest. The response is etch rate for silicon nitride. A single replicate is used. Gap Pressure C2F6 Flow Power Level A (cm) B (m Torr) C (SCCM) D (W) Low (-) 0.80 450 125 275 High (+) 1.20 550 200 325 Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design Example 12-8 A B C D Etch Rate Run (Gap) (Pressure) (C2F6 Flow) (Power) (Ao/min) 1 -1 -1 -1 -1 550 2 1 -1 -1 -1 669 3 -1 1 -1 -1 604 4 1 1 -1 -1 650 5 -1 -1 1 -1 633 6 1 -1 1 -1 642 7 -1 1 1 -1 601 8 1 1 1 -1 635 9 -1 -1 -1 1 1037 10 1 -1 -1 1 749 11 -1 1 -1 1 1052 12 1 1 -1 1 868 13 -1 -1 1 1 1075 14 1 -1 1 1 860 15 -1 1 1 1 1063 16 1 1 1 1 729 Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design Example 12-8 • The estimated effects are found to be A = -101.625 AD = -153.625 B = -1.625 BD = -0.625 AB = -7.875 ABD = 4.125 C= 7.375 CD = -2.125 AC = -24.875 ACD = 5.625 BC = -43.875 BCD = -25.375 ABC = -15.625 ABCD = -40.125 D = 306.125 Introduction to Statistical Quality Control, 4th Edition Table 12-16 has the contrast constants Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design Example 12-8 • Normal probability plot of effects Normal Probability Plot of the Effects (response is Etch, Alpha = .05) A: A D B: B C: C D: D 1 Normal Score Significant 0 -1 A Significant AD -100 0 100 200 300 Effect Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design Example 12-8 • Normal probability plot reveals that A, D, and AD appear to be significant. • To be sure that other main factors or two factor interactions are not significant, pool the three- and four- factor interactions to form the error mean square. • (NOTE: if the normal probability plot had indicated that any of these interactions were important, they would not be included in the error term.) Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design Example 12-8- Analysis of Variance Source of Sum of Degrees of Variation Squares Freedom Mean Square F0 A 41,310.563 1 41,310.563 20.28 B 10.563 1 10.563 <1 C 217.563 1 217.563 <1 D 374,850.063 374,850.063 183.99 AB 248.063 1 248.063 <1 AC 2,475.063 1 2,475.063 1.21 AD 94,402.563 94,402.563 48.79 BC 7,700.063 1 7,700.063 3.78 BD 1.563 1.563 <1 CD 18.063 1 18.063 <1 Error 10,186.815 5 2,037.363 Total 531,420.936 15 Introduction to Statistical Quality Control, 4th Edition 12-4. 2k Factorial Design Example 12-8- Analysis of Variance • Factors A, D, and the interaction AD are significant. • The fitted regression model for this experiment is 101 .625 306 .125 153 .625 y 776 .0625 ˆ x1 x 2 x1x 2 2 2 2 where x1 represents A, x2 represents D. Introduction to Statistical Quality Control, 4th Edition Assignment • We only covered pages 569 – 615 • Understand those designs and analyses in the included pages • Be able to work exercises like 12-1 and 12-2 Introduction to Statistical Quality Control, 4th Edition End Introduction to Statistical Quality Control, 4th Edition

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