2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303)

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1 MIT OpenCourseWare J / 6.780J / ESD.63J Control of Processes (SMA 6303) Spring 2008 For information about citing these materials or our Terms of Use, visit:

2 Control of Processes Subject 2.830/6.780/ESD.63 Spring 2008 Lecture #7 Shewhart SPC & Process Capability February 28,

3 Applying Statistics to : The Shewhart Approach Text removed due to copyright restrictions. Please see the Abstract of Shewhart, W. A. The Applications of Statistics as an Aid in Maintaining Quality of a Manufactured Product. Journal of the American Statistical Association 20 (December 1925):

4 Applying Statistics to : The Shewhart Approach Text removed due to copyright restrictions. Please see the Abstract of Shewhart, W. A. The Applications of Statistics as an Aid in Maintaining Quality of a Manufactured Product. Journal of the American Statistical Association 20 (December 1925):

5 Applying Statistics to : The Shewhart Approach (circa 1925)* All Physical Processes Have a Degree of Natural Randomness A Process is a Random Process if all Assignable Causes (identifiable disturbances) are eliminated A Process is In Statistical Control if only Common Causes (Purely Random Effects) are present. 4

6 In-Control... Time i+1 i+2 Each Sample is from Same Parent i 5

7 Not In-Control... Time i i+1 i+2 The Parent Distribution is Not The Same at Each Sample 6

8 Not In-Control... Bi-Modal Time Mean Shift + Variance Change Mean Shift What will appear in Samples? 7

9 Shewhart: Xbar and S Charts Plot sequential average of process Xbar chart Distribution? Plot sequential sample standard deviation S chart Distribution? 8

10 Data Sampling and Sequential Averages Given a sequence of process outputs x i : j j+1 j+2... sample interval ΔT A sequential sample of size n n measurements at sample j Take at intervals ΔT Sample index j 9

11 Data Sampling j j+1 j+2... sample interval ΔT n measurements at sample j x j = 1 n S j 2 = 1 n 1 jδt + n x i i = ( j 1) ΔT +1 jδt +n i = (j 1)ΔT (x i sample j mean x j ) 2 sample j variance 10

12 Subgroups j j+1 j+2... Within Subgroup Statistics n measurements at sample j xbar j, S j Between Subgroup Statistics Average of xbar j Variance of xbar(j) 11

13 Plot of xbar and S Random Data n=5 xbar S Sample Number Run number 12

14 Overall Statistics x = 1 N N x j i =1 Grand Mean Sample Number S = 1 N N S j i =1 Grand Standard Deviation Run number 13

15 Setting Chart Limits Expected Ranges Grand mean and Variance (based on what data and how many data points?) Confidence Intervals Intervals of + n Standard Deviations Most Typical is + 3σ (US) or 0.1% (Europe) 14

16 If we knew σ x then: Chart Limits - Xbar σ x = 1 n σ x But Since we Estimate the Sample Standard Deviation, then E(S j ) = C 4 σ x (S j is a biased estimator) where C 4 = 2 n 1 1/2 Γ(n /2) Γ((n 1) / 2) 15

17 Chart Limits xbar chart With this correction we can set limit at ±3σ xbar Or set a confidence interval of 99.7% Or a test significance of 0.3% UCL = x + 3 C 4 S n LCL = x 3 C 4 S n For the example n=5 C 4 = ( 0.5) 1/2 Γ(2.5) Γ(2) = =

18 Chart Limits S The variance of the estimate of S can be shown to be: σ S = σ 1 C 4 2 So we get the chart limits: UCL = S + 3 S 2 1 C 4 C 4 LCL = S 3 S 2 1 C C

19 Example xbar UCL Grand Mean LCL sample number 18

20 Example S UCL Grand S LCL

21 Detecting Problems from Running Data Appearance of data Confidence Intervals Frequency of extremes Trends 20

22 The 8 rules from Devor et al (Based on Confidence Intervals) Prob. of data in a band Based on Periodicity Based on Linear Trends Based on Mean Shift 21

23 Extreme Points Outside ±3σ Improbable Points Test for Out of Control 2 of 3 >±2σ 4 of 5>± 1σ All points inside ±1σ 22

24 Tests for Out of Control Consistently above or below centerline Runs of 8 or more Linear Trends 6 or more points in consistent direction Bi-Modal Data 8 successive points outside ±1σ 23

25 Applying Shewhart Charting Find a run of points that are in-control Compute chart centerlines and limits Begin Plotting subsequent xbar j and S j Apply the 8 rules, or look for trends, improbable events or extremes. If these occur, process is out of control 24

26 Data is not Stationary (μ or σ are not constant) Out of Control Process Output is being caused by a disturbance (common cause) This disturbance can be identified and eliminated Trends indicate certain types Correlation with know events shift changes material changes 25

27 Western Electric Rules (See Table 4-1) Points outside limits 2-3 consecutive points outside 2 sigma Four of five consecutive points beyond 1 sigma Run of 8 consecutive points on one side of center 26

28 In-Control... Time i+1 i+2 What will chart look like? i 27

29 Not In-Control... Time i+1 i+2 What will chart look like? i 28

30 Detecting Mean Shifts: Chart Sensitivity Consider a real shift of Δμ x : Sample Number How many samples before we can expect to detect the shift on the xbar chart? 29

31 Average Run Length How often will the data exceed the ±3σ limits if Δμ x = 0? Prob(x > μ x + 3σ x ) + Prob(x < μ x 3σ x ) = 3 / σ μ +3σ 30

32 Average Run Length How often will the data exceed the ±3σ limits if Δμ x = +1σ? Assumed Distribution Prob(x > μ x + 2σ x ) + Prob(x < μ x 4σ x ) = = 24 / Δμ Actual Distribution σ μ +3σ p e 31

33 Definition Average Run Length (arl): Number of runs (or samples) before we can expect a limit to be exceeded = 1/p e for Δμ = 0 arl = 3/1000 = 333 samples for Δμ = 1σ arl = 24/1000 = 42 samples Even with a mean shift as large as 1σ, it could take 42 samples before we know it!!! 32

34 Effect of Sample Size n on ARL Assume the same Δμ = 1σ Note that Δμ is an absolute value If we increase n, the Variance of xbar decreases: σ x = σ x So our ± 3σ limits move closer together n 33

35 ARL Example Original Distribution σ Δμ 3σ new limits same absolute shift New Distribution μ +3σ +3σ As n increases p e increases so ARL decreases p e 34

36 Design of the Chart Sample size n Central Limit theorem ARL effects? Selection of Reference Data Is S at a minimum? Sample time ΔT Cost of sampling production without data Rapid phenomena Sample size and filtering versus response time to changes j j+1j+2... sample interval ΔT 35

37 Limits and Extensions Need for averaging Assumptions of Normality Assumption of independence Pitfalls Misinterpretation of Data Improper Sampling What are alternatives? Different Sampling Schemes Different Averaging Schemes Continuous Update to Improve Statistics 36

38 Hypothesis Testing Conclusions Use knowledge of PDFs to evaluate hypotheses Quantify the degree of certainty (a and b) Evaluate effect of sampling and sample size Shewhart Charts Application of Statistics to Production Plot Evolution of Sample Statistics and S Look for Deviations from Model x 37

39 Detection : The SPC Hypothesis Process Y In-Control Sample Number p(y) Not In-Control 38

40 Out of Control Data is not Stationary (μ or σ are not constant) Process Output is being caused by a disturbance (assignable or special cause) This disturbance can be identified and eliminated Trends indicate certain types Correlation with know events shift changes material changes 39

41 Use of the S Chart Plot of sample Variance Variance of the Mean for Shewhart xbar (n>1) What Does it Tell Us about State of Control? It simply plots the other statistic 40

42 Consider this Process Xbar Chart UCL LCL sample number 41

43 And the S Chart 0.6 UCL LCL

44 In Control? 43

45 Same Process Later in Time Xbar UCL LCL sample number 44

46 Later S Chart 0.6 UCL LCL

47 What Changed?? 46

48 A Different Sequence Xbar UCL LCL sample number 47

49 S Chart 0.6 UCL LCL

50 Use of S Chart Detect Changes in Variance of Parent Distribution Distinguish Between Mean and Variance Changes 49

51 Statistical Process Control Model Process as a Normal Independent* Random Variable Completely described by μ and σ Estimate using xbar and s Enforce Stationary Conditions Look for Deviations in Either Statistic If so..? Call an Engineer! 50

52 Another Use of the Statistical Process Model: The -Design Interface We now have an empirical model of the process 0.45 How good is the process? Is it capable of producing what we need? σ μ +3σ 51

53 Process Capability Assume Process is In-control Described fully by xbar and s Compare to Design Specifications Tolerances Quality Loss 52

54 Design Specifications Tolerances: Upper and Lower Limits Lower Specification Limit LSL Target x* Characteristic Dimension Upper Specification Limit USL 53

55 Design Specifications Quality Loss: Penalty for Any Deviation from Target QLF = L*(x-x*) 2 How to Calibrate? x*=target 54

56 Define Process using a Normal Distribution Superimpose x*, LSL and USL Evaluate Expected Performance LSL x* USL σ Use of Tolerances: Process Capability μ +3σ 55

57 Process Capability Definitions C p = (USL LSL) 6σ = tolerance range 99.97% confidence range Compares ranges only No effect of a mean shift: 56

58 Process Capability: C pk C pk = min (USL μ), 3σ (LSL μ) 3σ = Minimum of the normalized deviation from the mean Compares effect of offsets 57

59 Cp = 1; Cpk =

60 Cp = 1; Cpk =

61 Cp = 2; Cpk =

62 Cp = 2; Cpk =

63 In Design Specs In Process Mean In Process Variance Effect of Changes What are good values of Cp and Cpk? 62

64 Cpk Table Cpk z P<LS or P>USL 1 3 1E E E E-09 63

65 The 6 Sigma problem P(x > 6σ) = 18.8x10-10 C p =2 LSL σ +3σ 6σ C pk =2 USL 64

66 The 6 σ problem: Mean Shifts P(x>4σ) = 31.6x10-6 C p =2 Even with a mean shift of 2σ we have only 32 ppm out of spec LSL USL 4σ C pk =4/3 65

67 Capability from the Quality Loss Function QLF = L(x) =k*(x-x*) 2 x* Given L(x) and p(x) what is E{L(x)}? 66

68 Expected Quality Loss E{L(x)}= Ek(x [ x*) 2 ] = ke(x [ 2 ) 2E(xx*) + E(x * 2 )] = kσ x 2 + k(μ x x*) 2 Penalizes Variation Penalizes Deviation 67

69 Process Capability The reality (the process statistics) The requirements (the design specs) Cp - a measure of variance vs. tolerance Cpk - a measure of variance from target Expected Loss- An overall measure of goodness 68

2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008

MIT OpenCourseWare http://ocw.mit.edu 2.830J / 6.780J / ESD.63J Control of Processes (SMA 6303) Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.