2. 2.810T.Gutowski1 2.810 Quality and Variation Part Tolerance Process Variation Taguchi “Quality Loss Function” Random Variables and how variation grows with size and complexity Quality Control

3. 2.810T.Gutowski2 References; Kalpakjian pp 982-991 (Control Charts) “Robust Quality” by Genichi Taguchi and Don Clausing A Brief Intro to Designed Experiments Taken from Quality Engineering using Robust Design by Madhav S. Phadke, Prentice Hall, 1989 5 homeworks due Nov 13

4. 2.810T.Gutowski3 Interchangeable Parts; Go, No-Go; Part Tolerance

5. 2.810T.Gutowski4 Product specifications are given as upper and lower limits, for example the dimensional tolerance +0.005 in. Upper Specification Limit Lower Specification Limit Target

6. 2.810T.Gutowski5 Process Variation Process measurement reveals a distribution in output values. Discrete probability distribution based upon measurements Continuous “Normal” distribution In general if the randomness is due to many different factors, the distribution will tend toward a “normal” distribution. (Central Limit Theorem)

7. 2.810T.Gutowski6 Tolerance is the specification given on the part drawing, and variation is the variability in the manufacturing process. This figure confuses the two by showing the process capabilities in terms of tolerance. Never the less, we can see that the general variability (expressed as tolerance over part dimension) one gets from conventional manufacturing processes is on the order of to Homework problem; can you come up with examples of products that have requirements that exceed these capabilities? If so then what?

8. 2.810T.Gutowski7 We can be much more specific about process capability by measuring the process variability and comparing it directly to the required tolerance. Common measures are called Process Capability Indices (PCI’s), such as,

9. 2.810T.Gutowski8 Case 1 In this case the out of specification parts are 4.2% + 0.4% = 4.6% What are the PCI’s? Upper Specification Limit Lower Specification Limit Target

10. 2.810T.Gutowski9 Case 2 However, in general the mean and the target do not have to line up. What are the PCI’s? How many parts are out of spec? Upper Specification Limit Lower Specification Limit Target

11. 2.810T.Gutowski10 Comparison Case 1 C p =  = 2/3 C pk = Min(  )=2/3 Out of Spec = 4.6% Case 2 C p =  = 2/3 C pk = Min(  )=1/3 Out of Spec = 16.1% Note; the out of Spec percentages are off slightly due to round off errors

12. 2.810T.Gutowski11 Why the two different distributions at Sony? 20% Likelihood set will be returned

13. 2.810T.Gutowski12 Quality Loss Deviation,  Goal Post Quality Taguchi Quality Loss Function QL = k  2

14. 2.810T.Gutowski13 Homework Problem Estimate a reasonable factory tolerance if the Quality Loss ($) for a failure in the field is 100 times the cost of fixing a failure in the factory. Say the observed field tolerance level that leads to failure is  field.

15. 2.810T.Gutowski14 Random variables and how variation grows with size and complexity Random variable basics Tolerance stack up Product complexity Mfg System complexity

16. 2.810T.Gutowski15 If the dimension “X” is a random variable, the mean is given by  = E(X)(1) and the variation is given by Var(x) = E[(x -  ) 2 ](2) both of these can be obtained from the probability density function p(x). For a discrete pdf, the expectation operation is: (3)

17. 2.810T.Gutowski16 Properties of the Expectation 1. If Y = aX + b; a, b are constants, E(Y) = aE(X) + b(4) 2. If X 1,…X n are random variables, E(X 1 + … + X n ) = E(X 1 ) +…+ E(X n )(5)

18. 2.810T.Gutowski17 Properties of the Variance 1.For a and b constants Var(aX + b) = a 2 Var(X)(6) 2.If X 1,…..X n are independent random variables Var(X 1 +…+ X n ) = Var(X 1 )+ Var(X 2 )+ + Var(X n ) (7)

19. 2.810T.Gutowski18 If X 1 and X 2 are random variables and not necessarily independent, then Var(X 1 + X 2 ) = Var(X 1 ) + Var(X 2 ) + 2Cov(X 1 Y) (8) this can be written using the standard deviation “  ”, and the correlation “  ” as (9) where L = X 1 + X 2

20. 2.810T.Gutowski19 If X 1 and X 2 are correlated (  = 1), then (14) for X 1 = X 2 = X 0 (15) for N (16) or (17)

21. 2.810T.Gutowski20 Now, if X 1 and X 2 are uncorrelated (  = 0) we get the result as in eq’n (7) or, (10) and for N (11) If X 1 =X =X o (12) Or(13)

22. 2.810T.Gutowski21 Complexity and Variation As the number of variables grow so does the variation in the system; This leads to; more complicated systems may be more likely to fail

23. 2.810T.Gutowski22 Homework; Consider the final dimension and variation of a stack of n blocks. 1, 2 …… n If USL – LSL =  ’, and Cp = 1 a) How many parts are out of compliance? b) Now USL-LSL=  ’, what is Cp? How many parts are out of spec? c) Repeat a) with  ’ Assume that  target.

24. 2.810T.Gutowski23 Homework Problem: Experience shows that when composites are cured by autoclave processing on one sided tools the variation in thickness is about 7%. After careful measurements of the prepreg thickness it is determined that their variation is about 7%. What can you tell about the source of variation?

25. 2.810T.Gutowski24 Complexity and Reliability ref. Augustine’s Laws

26. 2.810T.Gutowski25 Quality and System Design Data from D. Cochran

27. 2.810T.Gutowski26 Quality Control Inputs “I”; Mat’l, Energy, Info Operator inputs,”u”; initial settings, feedback, action? Disturbances, “d”; temperature, humidity, vibrations, dust, sunlight Outputs, “X” Machine “M”

28. 2.810T.Gutowski27 Who controls what? X = f (M, I, u, d) Equipment Purchase Q.C., Utilities, etc Operator, Real Time Control Physical Plant, etc So who is in charge of quality?

29. 2.810T.Gutowski28 How do you know there is a quality problem? 1. Detection 2. Measurement 3. Source Identification 4. Action 5. Goal should be prevention

30. 2.810T.Gutowski29 Detection Make problems obvious Poke yoke at the process level Clear flow paths and responsibility Andon board Simplify the system Stop operations to attend to quality problems Stop line Direct attention to problem Involve Team

31. 2.810T.Gutowski30 Measurement Statistical Process Control Average value x Sampling period Upper Control Limit Lower Control Limit Centerline

32. 2.810T.Gutowski31 Statistical Process Control Issues Sampling Period Establish Limits Sensitivity to Change

33. 2.810T.Gutowski32 Source Identification; Ishikawa Cause and Effect Diagram ManMachine MaterialMethod Effect Finding the cause of a disturbance is the most difficult part of quality control. There are only aids to help you with this problem solving exercise like the Ishikawa Diagrams which helps you cover all categories, and the “5 Whys” which helps you go to the root cause.

34. 2.810T.Gutowski33 Truck front suspension assembly Problem; warranty rates excessive

35. 2.810T.Gutowski34 Setting the best initial parameters Tables and Handbooks E.g. Feeds and speeds Models E.g. Moldflow for injection molding Designed Experiments E.g. Orthogonal Arrays

36. 2.810T.Gutowski35 Designed Experiments 1. Temp “T” (3 settings) 2. Pressure “P” (3 settings) 3. Time “t” (3 values) 4. Cleaning Methods “K” (3 types) How Many Experiments? One at a time gives 3 4 = 81

37. 2.810T.Gutowski36 But what if we varied all of the factors at once? Our strategy would be to measure one of the factors, say temperature, while “randomizing” the other factors. For example measure T2 with all combinations of the other factors e.g. (P,t,K) = (123), (231), (312). Notice that all levels are obtained for each factor.

38. 2.810T.Gutowski37 “Orthogonal Array” for 4 factors at 3 levels. Only 9 experiments are needed Exp temppressuretimeclean 11111 21222 31333 42123 52231 62312 73132 83213 93321

39. 2.810T.Gutowski38 Homework Can you design an orthogonal array for 3 factors at 2 levels?

40. 2.810T.Gutowski39 Summary – the best ways to reduce variation Simplify design Simplify the manufacturing system Plan on variation and put in place a system to address it

41. 2.810T.Gutowski40 Aircraft engine case study

42. 2.810T.Gutowski41 Engine Data engine A1 engine A2 engine B1 engine B2 engine C1 engine C2 number of part numbers ~2,000 ~1,400~1,3004,4653,485 total number of parts ~15,000~19,000~7,000 26,07323,580 weight [lb] 2.3k- 3.5k 9k-10k1.5k- 1.6k 2.3k- 3.5k 1.5k- 1.6k thrust [lb] unless otherwise noted 14k-21k40k-50k4k-5k hp 7k-9k14k-21k7k-9k by-pass ratio 0.36:14.9:1-5.15:10.34:16.2:1 engine A1 engine A2 engine B1 engine B2 engine C1 engine C2 annual production 150 110150 286 planned through- put time [days] 15208102321 approx. takt time [shifts/engine] 7.30 6.644.87 2.55 Engine “complexity”

43. 2.810T.Gutowski42 Scheduled build times Vs part count Scheduled build times

44. 2.810T.Gutowski43 Engine Delivery Late Times

45. 2.810T.Gutowski44 Late times compared to scheduled times

46. 2.810T.Gutowski45 Reasons for delay at site A

47. 2.810T.Gutowski46 Reasons for delay at site B (Guesses)

48. 2.810T.Gutowski47 Reasons for delay at site A (data)

49. 2.810T.Gutowski48 Engines shipped over a 3 month period at aircraft engine factory “B”

50. 2.810T.Gutowski49 Engines shipped over a 3 month period at aircraft engine factory “C”