Control of Manufacturing Processes

Size: px

Start display at page:

Download "Control of Manufacturing Processes"

Melvin Dalton
5 years ago
Views:

1 Control of Processes David Hardt

2 Topics for Today! Physical Origins of Variation!Process Sensitivities! Statistical Models and Interpretation!Process as a Random Variable(s)!Diagnosis of Problems! Shewhart Charts! Process Capability! Next Steps: Optimization and Control Process Control 9/25/17 2

3 Learning Expectations! Processes Vary in Two Key Ways!Deterministic!Random! We Seek to Minimize or Eliminate! Control Chart are used to Monitor the State of Randomness! Statistical Control = Purely Random! Achieved by Minimizing any Deterministic Effects Process Control 9/25/17 3

4 ! Quality! Rate! Cost! Flexibility Process Objectives? Process Control 9/25/17 4

5 Process Control Objectives?! Quality!Conformance to Specifications wrt! Geometry! Properties! Rate! Cost! Flexibility Process Control 9/25/17 5

6 Lab Processes CNC Turning Critical Dimension: Shaft Diameter Process Control 9/25/17 6

7 Lab Processes Brake Bending Critical Dimension: Part Angle Process Control 9/25/17 7

8 Lab Processes Injection Molding Critical Dimension:? Process Control 9/25/17 8

9 Lab Processes Thermoforming Critical Dimension: Process Control 9/25/17 9

10 Origins of Variation Process Control 9/25/17 10

11 The Process Components Operator Equipment Workpiece Material Part! Etch bath! Injection Molder! Lathe! Draw Press!...!Coated Silicon!Plastic Pellets! Bar stock! Sheet Metal!...! Semiconductor!Connector Body!Shaft!Hood! Process Control 9/25/17 11

12 What Causes Variation in the Process Output?! Material Variations! Intrinsic Properties, Initial Geometry! Equipment Variations! Non-repeatable, long term wear, deflections! Operator Variations! Inconsistent control, excessive tweaking! Environment Variations! Temperature and Handling inconsistencies Process Control 9/25/17 12

13 Can We Rank These?! Likelihood of Variation?! Frequency of Variation?! Magnitude of Variation?! Sensitivity of Output to Variation? Process Control 9/25/17 13

14 ! Equipment Can We Rank These?!Fixed Iron!Can be Automated (Controlled) to Keep Energy States as Desired! Material! Flows Through the Process! Constantly Changing!Energy Transfer from Equipment! Variable Cycle to Cycle? Process Control 9/25/17 14

15 Process Control Hierarchy! Identify and Reduce Disturbances! Good Housekeeping (Ops Management)! Standard Operations (SOP s)! Statistical Analysis and Identification of Sources! Feedback Control of Machines! Reduce Sensitivity (Process Optimization or Robustness)! Measure Sensitivities via Designed Experiments! Adjust free parameters to minimize! Measure Output and Manipulate Inputs! Feedback control of Output(s)! Control What you Measure Continuously Process Control 9/25/17 15

16 Why not Always Use Process Output Control?! Lack of Measurements!Shape not accessible! Lack of Spatial Resolution!Complex shape, simple control u! Cost/Benefit vs. Other Methods! Sufficiency of Equipment Control!e.g. numerical control Process Control 9/25/17 16

17 Modeling Variation! Deterministic (Know Cause)!1 st Principles!Numerical!Empirical! Random (Unknown Casue)!Theory of Random Processes Process Control 9/25/17 17

18 Feedback The General Process s Control Problem Desired Product CONTROLLER EQUIPMENT MATERIAL Product Equipment loop Material loop Process output loop Control of Equipment: Forces, Velocities Temperatures Control of Material: Strains Stresses Temperatures Pressure,.. Control of Product: Geometry and Properties Process Control 9/25/17 18

Feedback The General Process s Control Problem Desired Product CONTROLLER EQUIPMENT MATERIAL Product Equipment loop Material loop Process output loop Control of Equipment:

19 Feedback The General Process s Control Problem Desired Product CONTROLLER EQUIPMENT MATERIAL Product Equipment loop Material loop Process output loop Control of Equipment: Forces, Velocities Temperatures Control of Material: Strains Stresses Temperatures Pressure,.. Control of Product: Geometry and Properties Process Control 9/25/17 19

20 !"#$%&'(()*+$%,%$-'*").#($ CNC Mill with Position and Velocity Servos Process Control 9/25/17 20

21 /#'&0!)1#$234').$%5.345&$ Die Maximum Minimum Strain Gages Locations A Minimum B Maximum C Representative! o e! X o X! Feedback Law Hydraulic Servo-system Sheet Clamp End Representative B CL Free Length Strain Gage l C l A CL Shape Area Grip Area Figure 6-6 Strain gages locations on sheet Process Control 9/25/17 21

22 6)4#*3$%5.345&$57$8#&9$:#51#34;$ Process Control 9/25/17 22

23 Why not Always Output Feedback! Difficult if not Impossible to Measure Output During Process! Measurements will be Noisy!Material is flowing through the process! Process May not Have Sufficient Control Authority!Time and Space!Serial vs. Parallel Process Control 9/25/17 23

24 Solution: Focus on Randomness! Assume Deterministic Behavior can be Control/Eliminated! Address the Random Part of the Response! Seek a State of Pure Randomness Process Control 9/25/17 24

25 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. 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 Process Control 9/25/17 25

26 Shewhart Applied to! Measure and Plot the Process Output! Look for Any Sign of Non-Random (Deterministic) Behavior!Not in Statistical Control! Identify the Cause of that Behavior and Reduce or Eliminate it! Verify That the Process is Now Purely Random!In Statistical Control Process Control 9/25/17 26

27 Statistical Models for Process Control 9/25/17 27

28 Consider: Turning Process Process Control 9/25/17 28

29 Observations from Experiments! Randomness + Deterministic Changes CNC Turning random or unknown!" Shift changes Run Number Process Control 9/25/17 29

30 Brake Bending of Sheet Output: Angle M M! " Process Control 9/25/17 30

31 Bending Process Springback Process Control 9/25/17 31

32 Observations from Bending Process! Clear Input-Output Effects (Deterministic)! Also Randomness as well Aluminum 0.3 In Steel 0.3 In Aluminum 0.6 In Steel 0.6 In Angle changes with depth!y ->!u Process Control 9/25/17 32

Observations from Injection Molding 41.00 40.

33 Observations from Injection Molding Holding Time = 5 sec Injection Press = 40% Run Chart for Injection Molded Part Holding Time = 10 sec Injection Press = 40% Holding Time = 5 sec Injection Press = 60% Holding Time = 10 sec Injection Press = 60% 2/3/ Width (mm) Average Number of Run Process Control 9/25/17 33

34 How To Model to Distinguish these Effects? Irreducible Disturbances (Common) Inputs Process Ouputs + "Noise" Disturbances (Reducible) (Assignable)! A Random Process + A Deterministic Process Process Control 9/25/17 34

35 Random Processes! Consider the Output-only, Black Box view of the Run Chart Process Y(t)! Implies no Intentional Deterministic Chagnes! How do We Characterize The Process?! Using Y(t) only! WHY do we Characterize the Process! Using Y(t) only? Process Control 9/25/17 35

36 How to Describe Randomness?! Look at a Frequency Histogram of the Data! Estimates likelihood of certain ranges occurring:!pr(y 1 < Y <y 2 )! Probability that a random variable Y falls between the limits y 1 and y Process Control 9/25/17 36

37 Histogram for Bending (MIT 2002 data) 150 Bending Data Sorted by Depth Angle (Deg.) Aluminum 0.6 in depth Steel 0.6 in depth Aluminum 0.3in depth Dashed Lines are at group changes Steel 0.3 in depth 0.3 in depth only Process Control 9/25/17 37

38 Injection Molding Frequency Process Control 9/25/17 38

How do we use knowledge of this pattern?!predict behavior!

39 Conclusion?! When there are no input change (e.g. using SOP s) a consistent histogram pattern can emerge! How do we use knowledge of this pattern?!predict behavior!set limits on normal behavior! Define analytical probability density functions Process Control 9/25/17 39

40 Analysis of Histograms! Is there a consistent pattern?! Is an underlying parent distribution suggested? Process Control 9/25/17 40

41 Underlying or Parent Probability! A model of the true, continuous behavior of the random process! Can be thought of as a continuous random variable obeying a set of rules (the probability function)! We can only glimpse into these rules by sampling the random variable (i.e. the process output)! Underlying process can have! Continuous Values (e.g. geometry)! Discrete Values (e.g. defect occurrence) Process Control 9/25/17 41

42 Process Outputs as a Random Variable! The Histogram suggests a pdf!parent or underlying behavior sampled by the process! Standard Forms (There are many)!e.g. The Uniform and Normal pdf s Normal Probability Density Function for! 2 = 1, µ= p(x) x Process Control 9/25/17 42

43 Standard Normal Distribution Normal Probability Density Function for! 2 = 1, µ=0 p(x) = 1 1! 2" e# 2 $ & % x # µ ' )! ( p(x) z = x! µ " Process Control 9/25/17 zx 43

44 Properties of the Normal pdf! Symmetric about mean! Only two parameters: µ and # 2 p(x) = 1 1! 2" e# 2 $ & % x # µ ' )! ( 2! Good Model for Additive Effects!Central Limit Theorem Process Control 9/25/17 44

45 Interpretation of the PDF: Confidence Intervals 0.4! Probability that x >µ+3! 0.3 = 3/ % 95% % ±! z ±2! ±3! Process Control 9/25/17 45

46 Model Calibration: Sample Statistics! For the Normal PDF, we need two parameters: µ and!, which are unknown! We must estimate µ and! using sample statistic based on samples of the output (i.e. measurements) Process Control 9/25/17 46

47 x = 1 n n Estimators: Sample Statistics x( j) = samples of x(t) taken n times! x( j) : Average or Sample Mean j=1 S 2 = 1 n! 1 n "(x( j)! x) 2 : Sample Variance j =1 S = n 1 "(x( j)! x) 2 : Sample Std.Dev. n! 1 j = Process Control 9/25/17 47

48 Statistical Modeling of Processes! All Physical Processes Have a Degree of Natural Randomness! We can Model this Behavior using Probability Distribution Functions! We can Calibrate and Evaluate the Quality of this Model from Measurement Data using appropriate Sample Statistics Process Control 9/25/17 48

49 In-Control... i i+1 i+2 The Parent Distribution Does Not Change with Time Process Control 9/25/17 49

50 Not In-Control... i i+1 i+2 The Parent Distribution Changes with Time Process Control 9/25/17 50

51 In-Control (Almost) Frequency Process Control 9/25/17 51

52 Not In-Control CNC Turning Diameter (in) Shift changes Run Number Process Control 9/25/17 52

53 Not In-Control (How do we know?) Process Control 9/25/17 53

54 Not In-Control... Variance Increase Mean Shift + Variance Change Mean Shift Process Control 9/25/17 54

55 Applying in Real-Time: Xbar and S Charts! Shewhart: Plot the Evolving Sample Statistics ( x & s) µ!!these are the estimated & for the Normal process model!plot sequential average* of process! Xbar chart! Distribution?!Plot sequential sample standard deviation! S chart! Distribution? *aka Sample Mean Process Control 9/25/17 55

56 xbar Chart as Hypothesis Test Hypothesis H o : mean =µ and SD = # Samples Unlikely to Fall Here given H 0 x Samples Likely to Fall Here given H 0 ±3 # Samples Likely to Fall Here given H 0 Samples Unlikely to Fall Here given H Process Control 9/25/17 56

57 Xbar Chart: Confidence Intervals x j UCL x 34% 34% 2.35% 13.5% 3 ˆ! X! X! ˆ Confidence X 2 ˆ Interval's p(x) 13.5% 2.35% LCL Process Control 9/25/17 57

58 Data Sampling j j+1 j+2... sample interval "T n measurements at sample j x j = 1 n 2 1 S j = n "1 j#t +n $ sample j mean x i i =( j" 1)#T +1 j#t +n $ (x i " x j ) 2 sample j variance i =(j "1)#T Process Control 9/25/17 58

59 !Plot of xbar and S Random Data n=5 xbar S Sample Number Run number Process Control 9/25/17 59

60 Setting Chart Limits! Expected Ranges! Confidence Intervals! Intervals of + n Standard Deviations! Most Typical is + 3# Process Control 9/25/17 60

61 Superposition of Random Variables If we define a variable y = c 1 x 1 + c 2 x 2 + c 3 x 3 + c 4 x ! c i are constants! x i are independent random variables Then µ y = c 1 µ 1 + c 2 µ 2 + c 3 µ 3 + c 4 µ ! y 2 = c 12! # 2 + c 22! c 32! c 42! 4 2 For example y = x = 1! x i c j = 1 n n n i=1! 2 x = 1 n! x Process Control 9/25/17 61

62 Chart Limits - Xbar! If we knew! x then by superposition: " x = 1 n " x! But Since we Estimate the Sample Standard Deviation, then E(S j ) = C 4! x (S j is a biased estimator) where C 4 = # $ 2 % n "1& 1/2 '(n / 2) '((n "1) / 2) Process Control 9/25/17 62

63 Chart Limits xbar chart The estimate of True Sample Mean Variance (variance of the mean) is biased To remove this bias for the xbar ± 3# limits we use: 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!(2.5)!(2) = = Process Control 9/25/17 63

64 Chart Limits S The variance of the estimate of S can be shown to be:! S =! 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 4 C Process Control 9/25/17 64

Example xbar +3! x 1 0.9 0.8 UCL 0.7 x 0.6 0.5 0.4 Grand Mean!3! x 0.3 0.2 0.

65 Example xbar +3! x UCL 0.7 x Grand Mean!3! x LCL sample number Process Control 9/25/17 65

Example S 0.7 0.6 UCL 0.5 0.4 0.3 Grand S 0.2 0.

66 Example S UCL Grand S LCL Process Control 9/25/17 66

67 Detecting Problems from Running Data! Appearance of data!confidence Intervals!Frequency of extremes!trends Process Control 9/25/17 67

68 Western Electric Rules! 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 Process Control 9/25/17 68

69 Test for Out of Control! Extreme Points!Outside ±3#! Improbable Points!2 of 3 >±2#!4 of 5 >± 1#!All points inside ±1# Process Control 9/25/17 69

70 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 ±1# Process Control 9/25/17 70

71 Applying Shewhart Charting! Find a run of points that are incontrol, and get a grand mean and standard deviation ( x and S )! Compute xbar and S chart limits using the appropriate +- 3 sigma limits! Begin Plotting subsequent xbar j and S j! Apply rules, or look for trends, improbable events or extremes.! If these occur, process is out of control Process Control 9/25/17 71

72 Summary of xbar Chart x UCL = x + 3 C 4 S n x NB: n is the number of samples used to find xbar LCL = x!3 C 4 S n Process Control 9/25/17 72

73 Summary of S Chart S UCL = S + 3 S 2 1! C 4 C 4 S LCL = S! 3 S 2 1! C 4 C Process Control 9/25/17 73

74 The True Issue: xbar Chart in Real-Time xbar Sample Number Process Control 9/25/17 74

75 Average Run Length! How often will the data exceed the ±3# limits if "µ x = 1s? Prob(x > µ x + 3! x ) + Pr ob(x < µ x " 3! x ) = 3 / # µ +3# Process Control 9/25/17 75

76 Average Run Length! How often will the data exceed the ±3# limits if "µ x = +1!? Assumed Distribution "µ # µ +3# Actual Distribution Prob(x > µ x + 2! x ) + Prob(x < µ x " 4! x ) = = 24 / 1000 p e Lecture /6.780J 3/1/

77 Out of Control! Data is not Stationary (µ or! 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 Process Control 9/25/17 77

78 Out of Control! Data is not Stationary (µ or! 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 Process Control 9/25/17 78

79 In-Control... i+1 i+2 What will chart look like? i Process Control 9/25/17 79

80 Not In-Control... i+1 i+2 What will chart look like? i Process Control 9/25/17 80

81 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? # µ +3# Process Control 9/25/17 81

82 Process Capability! Assume Process is In-control! Described fully by xbar and s! Compare to Design Specifications!Tolerances!Quality Loss Process Control 9/25/17 82

83 Design Specifications! Tolerances: Upper and Lower Limits Lower Specification Limit LSL Target x* Characteristic Dimension Upper Specification Limit USL Process Control 9/25/17 83

84 LSL x* USL # Use of Tolerances: Process Capability! Define Process using a Normal Distribution! Superimpose x*, LSL and USL! Evaluate Expected Performance µ +3# Process Control 9/25/17 84

85 ! Definitions Process Capability C p = (USL! LSL) 6" = tolerance range 99.97% confidence range! Compares ranges only! No effect of a mean shift: Process Control 9/25/17 85

86 Process Capability: C pk $ C pk = min % (USL " µ) 3#, (LSL " µ) 3# & ' = Minimum of the normalized deviation from the mean! Compares effect of offsets Process Control 9/25/17 86

87 Cp = 1; Cpk = Process Control 9/25/17 87

88 Cp = 1; Cpk = Process Control 9/25/17 88

89 Cp = 2; Cpk = Process Control 9/25/17 89

90 Cp = 2; Cpk = Process Control 9/25/17 90

91 Effect of Changes! In Design Specs! In Process Mean! In Process Variance! What are good values of Cp and Cpk? Process Control 9/25/17 91

Cpk Table Cpk z P<LS or P>USL 1 3 1E-03 1.

92 Cpk Table Cpk z P<LS or P>USL 1 3 1E E E E Process Control 9/25/17 92

93 The 6 Sigma problem P(x > 6#) = 18.8x10-10 C p =2 LSL # +3# C pk =2 USL Process Control 9/25/17 6# 93

94 ! The 6 # problem: Mean Shifts P(x>4#) = 31.6x10-6 Even with a mean shift of 2# we have only 32 ppm out of spec USL LSL 4# C p =2 C pk =4/ Process Control 9/25/17 94

95 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)}? Process Control 9/25/17 95

96 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 Process Control 9/25/17 96

97 Process Control Hierarchy! Identify and Reduce Causal Disturbances! Good Housekeeping! Standard Operations (SOP s)! Feedback Control of Machines! Eliminate Equipment Variations! Statistical Analysis and Identification of Sources (SPC)! Eliminate Assignable Causes Process Control 9/25/17 97

98 Process Control Hierarchy! NEXT: Reduce Sensitivity to Disturbance! Measure Sensitivities via Designed Experiments (DOE)! Adjust free parameters to minimize variations Process Control 9/25/17 98

99 Example: Bending Sensitivity to Yield Stress Simple Example: Die Width for Air Bending (An adjustable equipment property):! Wide Die:!!! Low force, high spring back, high sensitivity to variations in yield stress! Narrow Die:!!!! High force, Higher material stress, Lower spring back, Lower sensitivity to variations in yield stress, Process Control 9/25/17 99

100 Final Step : Output Control Desired Output C Equipment E(t) Material Geometry & Properties Examples:! Web Thickness in Milling! Sheet Thickness in Rolling! Sheet Angle in Bending critical dimensions Process 9/25/17Control 100

101 Implementing Product Feedback Control! Continuous In-Process Measurements!Regulate Process States In-Process! Sampling and Monitoring (SPC)!Measure After-Process and Diagnose! Part to Part Sampling and Control!Cycle to Cycle Control: Measure After each Cycle and Improve Process Capability Process 9/25/17Control 101

102 Conclusions: Single Variable Case! Cycle to Cycle Control!Obeys Root Locus Prediction wrt Dynamics!Amplifies White Noise Disturbance Attenuates Colored Noise Disturbance!Can Reduce Mean Error (Zero if I-control)!Can Reduce Open Loop Expected Loss Lower Spec T Upper Spec Process 9/25/17Control 102

103 Feedback Control Objectives Steady-State Error from Target Disturbance Rejection Target Process 9/25/17Control 103

104 It Works!: Bending Step Disturbance! Effect of Material Change! Switch to a Stiffer Material more springback Material Shift Run number Process 9/25/17Control 104

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0-3 -2.8-2.6-2.4-2.2-2 -1.8-1.6-1.4-1.2-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.

005 1 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 Process Output Design Goal LSL T USL! Method?:!

105 Objective Width 2 In) Process Process Output Design Goal LSL T USL! Method?:!!! SPC Optimization Output Feedback Process 9/25/17Control 105

106 ! Shewhart Charts Conclusions! Application of Statistics to Production! Plot Evolution of Sample Statistics and S! Look for Deviations from Model! Process Capability! A measure of the process to meet a requirement! Includes variance and bias! Gets design and manufacturing talking! If That s Not Good Enough! DOE/Optimization! Feedback Control!! x Process Control 9/25/17 106

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

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.