Control of Manufacturing Processes
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1 Control of Manufacturing Processes Subject Spring 2004 Lecture #8 Hypothesis Testing and Shewhart Charts March 2, /2/04 Lecture 8 D.E. Hardt, all rights reserved 1
2 Applying Statistics to Manufacturing: The Shewhart Approach (circa 1925)* All Physical Processes Have a Degree of Natural Randomness A Manufacturing 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. W.A. Shewhart, The Applications of Statistics as an Aid in Maintaining Quality of a Manufactured Product, Journal of the American Statistical Association, 20, No.. 152, Dec /2/04 Lecture 8 D.E. Hardt, all rights reserved 2
3 In-Control... Time i+1 i+2 Each Sample is from Same Parent i 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 3
4 Not In-Control... Time i i+1 i+2 The Parent Distribution is Not The Same at Each Sample 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 4
5 Not In-Control... Bi-Modal Time Mean Shift + Variance Change Mean Shift 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 5
6 In Other Words A Random - Stationary Process is: In Control Subject to Random Fluctuations Only These Variations are Natural and Irreducible* If the Process Is Not Stationary, the Degree of Variation Can Still Be Reduced Causes can be Assigned * OED:. Incapable of being reduced to a smaller number or amount; the fewest or smallest possible. 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 6
7 Hypothesis Testing and Shewhart Charts Run charts, xbar, R and S charts Estimates for S w/o bias Detecting Trends and Mean Shifts Bands and rules Average run length 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 7
8 A Process that is In Control Shewhart Hypothesis Stationary process with a fixed P and V H 0 : xbar = P (Null Hypothesis) H 1 : xbar P (Alternative Hypothesis) Follows a Normal Distribution Data falls into expected confidence intervals rv rv rv 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 8
9 Hypothesis Testing Given the hypothesis for the statistic I(e.g. the mean) No single value of will= I 0 How do we test the hypothesis given Î? p( Î) H 0 : I I 0 H 1 : I z I 0 Î What is? How well do we want to test H o? 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 9
10 The Test Assume p( Î) is Normal Region of Rejection Region of 0.2 Acceptance area = D area = 1-D P I 0 Region of Rejection D is the significance of the test 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 10
11 Errors H o is rejected when it is in fact true (Type I) p =? p=d for two sided and D/2 for one sided tests H o is accepted when it is in fact false (Type II) p=? = E What is E" 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 11
12 Type II Errors Assume a shift in the true distribution of ±d± p( Î) Assess the probability that we fall in the acceptance region after a shift of ± d occurred 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 12
13 Type II Errors Region of Acceptance P I 0 -d P I 0 area = E P I 0 +d 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 13
14 Other Hypotheses? 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 14
15 Summary Pick Significance Level D Determine an acceptable E (1-E) ) is call the power of the test Î If the test statistic is the sample mean, what is the effect of increasing N?? 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 15
16 Outline Hypothesis Testing and Shewhart Charts Run charts, xbar, R and S charts Estimates for S w/o bias Detecting Trends and Mean Shifts Bands and rules Average run length 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 16
17 Shewhart: Xbar and S Charts Plot sequential average of process Xbar chart Distribution? Plot sequential sample standard deviation S chart Distribution? 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 17
18 Data Sampling and Sequential Averages Given a sequence of process outputs x i : j j+1 j+2... sample interval 'T A sequential sample of size n n measurements at sample j Take at intervals 'T Sample index j 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 18
19 j j+1 j+2... Data Sampling sample interval 'T n measurements at sample j x j 1 n 2 1 S j n 1 j't n x i i ( j 1) 'T 1 sample j mean j't n (x i x j ) 2 sample j variance i ( j 1)'T 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 19
20 Subgroups j j+1 j+2... n measurements at sample j Within Subgroups Statistics xbar j, S j Between Subgroup Statistics Average of xbar j Variance of xbar(j) 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 20
21 xplot of xbar and S Random Data n=5 xbar S Sample Number Run number 20 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 21
22 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 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 22
23 Setting Chart Limits x Expected Ranges x Confidence Intervals x Intervals of + n Standard Deviations 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 23
24 If we knew V x then: V x 1 n V x Chart Limits - Xbar But Since we Estimate the Sample Standard Deviation, then E(S j ) C 4 V x (i.e. it is biased by C 4 ) 2 where C 4 n 1 ¹ 1/ 2 *(n /2) *((n 1) / 2) 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 24
25 Chart Limits xbar chart Since the estimate of True Sample Mean Variance (variance of the mean) is biased To remove this bias for the xbar ± 3s limits we use: UCL x 3 C 4 S n LCL x 3 C 4 S n For the example n=5 C / 2 *(2.5) *(2) /2/04 Lecture 8 D.E. Hardt, all rights reserved 25
26 Chart Limits S The variance of the estimate of S can be shown to be: V S V 1 C 4 2 So we get the chart limits: UCL S 3 S 2 1 C C 4 4 LCL S 3 S 2 1 C 4 C 4 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 26
27 Example xbar UCL Grand Mean LCL sample number 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 27
28 Example S UCL Grand S LCL /2/04 Lecture 8 D.E. Hardt, all rights reserved 28
29 Detecting Problems from Running Data x Appearance of data x Confidence Intervals x Frequency of extremes x Trends 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 29
30 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 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 30
31 Extreme Points Outside ±3V Improbable Points Test for Out of Control 2 of 3 >±2V 4 of 5!r1V All points inside ±1V 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 31
32 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 ±1V 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 32
33 Detecting Mean Shifts: Consider a real shift of 'P x : Chart Sensitivity Sample Number How many samples before we can expect to detect the shift on the xbar chart? 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 33
34 Average Run Length How often will the data exceed the ±3V limits if 'P x = 0? Prob(x! P x 3V x ) Pr ob( x P x 3V x ) 3 / V P V 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 34
35 Average Run Length How often will the data exceed the ±3V limits if 'P x = +1V? Assumed Distribution Prob( x! P x 2V x ) Prob(x P x 4V x ) / V P 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 'P Actual Distribution V p e
36 Definition Average Run Length (arl( arl): Number of runs (or samples) before we can expect a limit to be exceeded = 1/p e for 'P = 0 arl = 3/1000 = 333 samples for 'P = 1V1 arl = 24/1000 = 42 samples Even with a mean shift as large as 1V, it could take 42 samples before we know it!!! 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 36
37 Effect of Sample Size n on arl Assume the same 'P = 1V1 Note that 'Pis an absolute value If we increase n, the Variance of xbar decreases: V V x x n So our ± 3V3 limits move closer together 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 37
38 ARL Example Original Distribution V V P V new limits same absolute shift New Distribution V As n increases p e increases so arl decreases 'P p e 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 38
39 Applying Shewhart Charting x Find a run of points that are in- control x Compute chart centerlines and limits x Begin Plotting subsequent xbar j and S j x Apply the 8 rules, or look for trends, improbable events or extremes. x If these occur, process is out of control 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 39
40 Out of Control Data is not Stationary P or V are not constant) 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 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 40
41 Sample size n Central Limit theorem ARL effects? Selection of Reference Data Is S at a minimum? Sample time 'T Cost of sampling production without data Rapid phenomena j j+1j+2... Design of the Chart Sample size and filtering versus response time to changes sample interval 'T 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 41
42 Limits and Extensions Need for averaging Assumptions of Normality Assumption of independence What are alternatives? 3/2/04 Lecture 8 D.E. Hardt, all rights reserved 42
2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303)
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Control of Manufacturing Processes
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2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008
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2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008
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