Advanced Six-Sigma Statistical Tools

Transcription

1 Six Sigma: The Statistical Tool Box Advanced Six-Sigma Statistical Tools ASQ-RS Meeting, March 2003 Dr. Joseph G. Voelkel, RIT for material BB Six-Sigma Statistical Tools for material Examples Gage R&R Studies Short/Long Control Charts X-bar/R, p, c Experimental Design 2 k, 2 k p, simple RSM Taguchi/Robust control, noise, S/N Analysis of Variance one-, two-way Regression one or two predictors Basic Level, mostly Rev: 3/25/03 2

2 Why Are We Here? Show example problems where basic Black Belt tools do not perform well. Second, show methods that might be used to solve such problems. Third, show how the methods can indeed solve such problems. Rev: 3/25/03 3 People Solve Problems Good methods provide insight, often visual quantify the extent of the problem may predict results of improvements Rev: 3/25/03 4

3 Case Study 1: Reducing Dimensional Variation in Radiator Cores Radiator Core Want dimensional stability across the core ( positions ) Not being achieved Control factors CS, PS Noise factors Position, Frame Rev: 3/25/03 5 Radiator-Core Experiment CS PS Seven frames used Fr Two cores tested, for each CS/PS/Frame combination Core Taguchi Approach? Each core measured at 3 positions Each position measured twice Pos Meas Rev: 3/25/03 6

4 Radiator-Core: Taguchi Approach Taguchi approach After averaging data over two measurements Noise Factors Frame Control Factors Pos s CS PS Rep Rev: 3/25/03 7 Radiator-Core: Taguchi Approach Main Effect Plots bar CS PS s sd CS PS Rev: 3/25/03 8

5 Radiator-Core: Taguchi Approach ANOVA/Regression and % contributions Term Coef T P %Contr Constant CS PS CS*PS Error 41 s Term Coef T P %Contr Constant CS PS CS*PS Error 25 Rev: 3/25/03 9 Radiator-Core: Critique of Taguchi Approach 2 CS 2 PS 2 Reps 7 Frames 3 Pos = 168 data points So, 168 d.f. In Taguchi approach: df are grouped less information Noise Factors Frame Control Factors Pos df s df CS PS Rep Rev: 3/25/03 10

6 Radiator-Core: ANOVA Approach Split apart d.f. More Information Step 1: ESD (Extended Structure Diagram) tool used CS TF (2) PS TF (2) Frame TF (7) Core UA (2) Pos CF (3) CS PS Section UA (1) Meas CA (2) Rev: 3/25/03 11 Radiator-Core: ANOVA Approach Step 2: translate this to an ANOVA model CS TF (2) PS TF (2) Frame TF (7) = CS PS Fr Core (CS PS Fr) Pos Pos*Core Error [Meas(Core Pos)] Core UA (2) Section UA (1) Meas CA (2) Pos CF (3) Rev: 3/25/03 12

7 Radiator-Core: ANOVA Approach Step 3a: Run the ANOVA model Source DF SS MS F P CS PS Fr Pos CS*PS CS*Fr CS*Pos PS*Fr PS*Pos Fr*Pos CS*PS*Fr CS*PS*Pos CS*Fr*Pos PS*Fr*Pos CS*PS*Fr*Pos Core(CS PS Fr) Pos*Core(CS PS Fr) Error Total Rev: 3/25/03 13 Radiator-Core: ANOVA Approach Step 3b: Find Components of Variance Full Model Source MS 10000*VC % CS % PS % Fr % Pos % CS*PS % CS*Fr % CS*Pos % PS*Fr % PS*Pos % Fr*Pos % CS*PS*Fr % CS*PS*Pos % CS*Fr*Pos % PS*Fr*Pos % CS*PS*Fr*Pos % Core(CS PS Fr) % Pos*Core(CS PS Fr) % Error % % Rev: 3/25/03 14

8 Radiator-Core: ANOVA Approach Step 3b: Find Components of Variance Full Model Source MS *VC % Fr % Pos % Fr*Pos % Pos*Core % Error % Rest 0.0 2% % Rev: 3/25/03 15 Radiator-Core: ANOVA Approach How come we didn t see this using the Taguchi approach? Taguchi approach depends totally on control (CS, PS) & control x noise effects Full Model Source MS *VC % Fr % Pos % Fr*Pos % Pos*Core % Error % Rest 0.0 2% % This is what the Taguchi approach analyzed!!! Rev: 3/25/03 16

9 Radiator-Core: ANOVA Approach Step 4. Graph the results of the analysis What graph(s) should we make? Full Model Source MS *VC % Fr % Pos % Fr*Pos % Pos*Core % Error % Rest 0.0 2% % Rev: 3/25/03 17 Radiator-Core: Interaction Plot Can t we do better? Make graph connect more to physical reality 0.50 Pos Mean Fr Rev: 3/25/03 18

10 Radiator-Core: Interaction Plot Fr/Pos Rev: 3/25/03 19 Radiator-Core: One More Plot Fr/Pos Full Model Source MS *VC % Fr % Pos % Fr*Pos % Pos*Core % Error % Rest 0.0 2% % Rev: 3/25/03 20

11 Radiator-Core: One More Plot Frame Rev: 3/25/03 21 Radiator-Core: Lessons Learned Push the Taguchi button? Run ANOVA that reflects your design! Pull apart those degrees of freedom! Push the Interaction-Graph button? Make up problem-specific graphs! Fr/Pos Rev: 3/25/03 22

12 Case Study 2 Unbalance: 2D Gauge R&R Studies Rev: 3/25/03 23 Unbalance Rev: 3/25/03 24

13 Unbalance Unbalance * Units 3.00 oz, 0.5 in 0.75 oz, 2.0 in 1.00 oz, 1.5 in oz-in (1.5 oz-in) g-mm Coordinates Polar (r, θ) Cartesian (X,) Rev: 3/25/03 25 Another measure Unbalance Unbalance= 1.5 oz-in Take weight of impeller into account Center of Axis of Rotation Light impeller Heavy impeller Eccentricity e=u / m units: in or mm Center of Mass of Rotating Body Rev: 3/25/03 26

14 Unbalance: 2D Gauge R&R 20 repeat measurements of unbalance 1 impeller, appraiser, machine How would you summarize the variability in these measurements? Rev: 3/25/03 27 Unbalance: 2D Gauge R&R One approach that s been used Measure r from origin for each point Use regular gauge R&R measures Rev: 3/25/03 28

15 1D Analysis of 2D Data Now estimate range where 99% of values would be Find s = sample standard deviation = 0.42 Find 5.15s = 2.17 = EV Rev: 3/25/ D Analysis of 2D Data But consider (X,) (X+1,+1) 90 rotation * All have same variation Rev: 3/25/03 30

16 1D Analysis of 2D Data 90 deg rotation. EV= (X+1,+1). EV=1.40 (X,). EV= r What is a better way? Rev: 3/25/03 31 Extending 1D Summaries to 2D Fundamental point: engineering tolerance is a circle The proposed twodimensional method repeatability (equipment variation) EV EV diameter of circle capturing p = 99% of such readings Rev: 3/25/03 32

17 Comparison of 2D to 1D Summary EV = 2.17, 1.40, 3.67? EV = 3.73 Rev: 3/25/03 33 Example 3 appraisers measured each of 10 parts twice ( = 60). For more validity, they did this at each of two time periods (60 2 = 120) Will sometimes act here as if 6 appraisers instead ( = 120). This is (pretty much) OK. We will see how this technique can summarize variation, just as in the 1D case Rev: 3/25/03 34

18 Example. Step 1: Set up ESD Appraiser(3) Parts(10) Time(2) Meas(2) Rev: 3/25/03 35 Example: All 3 Appraisers 6 appraiser s 10 impellers 2 trials = Adjust data (graphs) to mean of (0,0) for each impeller. 8.00E E E E E E E E E-04 E E-04 E-04 E-04 E E E X -8.00E-04 X Rev: 3/25/03 36

19 Example. Step 2: ANOVA-based Diameter Summaries 6 appraiser s 10 impellers 2 trials = 120 %D.tol %Total D.tol 100% 114% R&R 58% 66% Reprocibility 56% 63% Oper/Time Oper/Time*Imp Repeatability 17% 20% Impeller 66% 75% Total 88% 100% Data adjusted to a mean of (0,0) for each impeller X Rev: 3/25/03 37 ±0.0013" Step 3: Analysis. Matt Only %D.tol %Total D.tol 100% 137% R&R 37% 51% Reprocibility 34% 46% Oper/Time Oper/Time*Imp Repeatability 16% 22% Impeller 64% 87% Total 73% 100% X Rev: 3/25/03 38

20 Step 3: Analysis. Tom Only %D.tol %Total D.tol 100% 118% R&R 40% 48% Reprocibility 32% 38% Oper/Time Oper/Time*Imp Repeatability 24% 29% Impeller 75% 88% Total 85% 100% X Rev: 3/25/03 39 Step 3: Analysis. Ben Only %D.tol %Total D.tol 100% 147% R&R 14% 20% Reprocibility 10% 15% Oper/Time Oper/Time*Imp Repeatability 9% 13% Impeller 67% 99% Total 68% 100% X Rev: 3/25/03 40

21 Step 4: Graphical Analysis. Tom Data = Averages (N=2) of each Part-Time combination Tom R&R = 40% AV = 32% EV = 24% Time 1 Time 2 EV Rev: 3/25/03 41 Step 4: Graphical Analysis. Ben Data = Averages (N=2) of each Part-Time combination Ben R&R = 14% AV = 10% EV = 9% Time 1 Time 2 EV Rev: 3/25/03 42

22 Results Standardization was done based on Ben s techniques. Measurement variation greatly reduced Good methods provide insight, often visual quantify the extent of the problem may predict results of improvements Rev: 3/25/03 43 Case Study 3: Multivariate Methods: Experimental Design Many experiments: Multiple Responses Multiple Responses=Multivariate Case Example: L18 Film Experiment 4 Factors 24 Responses Many are MD/TD types Emphasis in this talk: Responses only See how the responses are associated How many different features exist among the responses? Rev: 3/25/03 44

23 Multivariate Methods: Exp l Design The Responses Microsoft Excel Worksheet How are the responses associated? How would you try to find this out? Minitab 1. Correlations 2. Matrix Plot 3. Add Col # s 4. CorrCode 5. PC s 6. Plots Good methods provide insight, often visual quantify the extent of the problem Minitab Worksheet may predict results of improvements Rev: 3/25/03 45 Case Study 4: SPC with a Twist Consider this Process data 40 subgroups Dimensional measurement Subgroup size 6. Microsoft Excel Worksheet Six parts in a row sampled from a machine How would you look at these data? Minitab Minitab Worksheet Rev: 3/25/03 46

24 Case Study 4: SPC with a Twist But here is the real situation 40 subgroups, Subgroup size 6. Six parts sampled from a Six-Cavity machine Now, how would you analyze these data? Minitab Minitab Project Rev: 3/25/03 47 The Actual Mold Good methods provide insight, often visual quantify the extent of the problem may predict results of improvements Rev: 3/25/03 48