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Quality Control

Quality Control. Agenda. - What is quality? - Approaches in quality control - Accept/Reject testing - Sampling (statistical QC) - Control Charts - Robust design methods. What is ‘Quality’. Performance :. - A product that ‘performs better’ than others at same function Example:

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Quality Control

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  1. Quality Control Agenda - What is quality? - Approaches in quality control - Accept/Reject testing - Sampling (statistical QC) - Control Charts - Robust design methods

  2. What is ‘Quality’ Performance: - A product that ‘performs better’ than others at same function Example: Sound quality of Apple iPod vs. iRiver… - Number of features, user interface Examples: Tri-Band mobile phone vs. Dual-Band mobile phone Notebook cursor control (IBM joystick vs. touchpad)

  3. What is ‘Quality’ Reliability: - A product that needs frequent repair has ‘poor quality’ Example: Consumer Reports surveyed the owners of > 1 million vehicles. To calculate predicted reliability for 2006 model-year vehicles, the magazine averaged overall reliability scores for the last three model years (two years for newer models) Best predicted reliability:Sporty cars/Convertibles Coupes Honda S2000 Mazda MX-5 Miata (2005) Lexus SC430 Chevrolet Monte Carlo (2005)

  4. What is ‘Quality’ Durability: - A product that has longer expected service life Nike Air Resolve Plus Mid Men’s Shoe (no warranty) Adidas Barricade 3 Men's Shoe (6-Month outsole warranty)

  5. What is ‘Quality’ Aesthetics: - A product that is ‘better looking’ or ‘more appealing’ Examples? ? or

  6. Defining quality for producers.. Example: [Montgomery] - Real case study performed in ~1980 for a US car manufacturer - Two suppliers of transmissions (gear-box) for same car model Supplier 1: Japanese; Supplier 2: USA - USA transmissions has 4x service/repair costs than Japan transmissions Lower variability  Lower failure rate Distribution of critical dimensions from transmissions

  7. Definitions Quality is inversely proportional to variability Quality improvement is the reduction in variability of products/services. How to reduce in variability of products/services ?

  8. QC Approaches (1) Accept/Reject testing (2) Sampling (statistical QC) (3) Statistical Process Control [Shewhart] (4) Robust design methods (Design Of Experiments) [Taguchi]

  9. Accept/Reject testing - Find the ‘characteristic’ that defines quality - Find a reliable, accurate method to measure it - Measure each item - All items outside the acceptance limits are scrapped Lower Specified Limit Upper Specified Limit target Measured characteristic

  10. Problem with Accept/Reject testing (1) May not be possible to measure all data Examples: Performance of Air-conditioning system, measure temperature of room Pressure in soda can at 10° (2) May be too expensive to measure each sample Examples: Service time for customers at McDonalds Defective surface on small metal screw-heads

  11. Problems with Accept/Reject testing Solution: only measure a subset of all samples This approach is called: Statistical Quality Control What is statistics?

  12. The standard deviation =s= =√( s2) ≈ 0.927. Background: Statistics Average value (mean) and spread (standard deviation) Given a list of n numbers, e.g.: 19, 21, 18, 20, 20, 21, 20, 20. Mean = m =S ai / n = (19+21+18+20+20+21+20+20) / 8 = 19.875 The variance s2 = ≈ 0.8594

  13. Background: Statistics.. Example. Air-conditioning system cools the living room and bedroom to 20; Suppose now I want to know the average temperature in a room: - Measure the temperature at 5 different locations in each room. Living Room: 18, 19, 20, 21, 22. Bedroom: 19, 20, 20, 20, 19. What is the average temperature in the living room? m =Sai / n = (18+19+20+21+22) / 5 = 20. BUT: is m = m ?

  14. Background: Statistics... Example (continued) m =Sai / n = (18+19+20+21+22) / 5 = 20. BUT: is m = m ? If: sample points are selected randomly, thermometer is accurate, … then m is an unbiased estimator of m. - take many samples of 5 data points, - the mean of the set of m-values will approach m - how good is the estimate?

  15. ≈ 1.4142 sn= The unbiased estimator of stdevof a sample = s = Background: Statistics.... Example. Air-conditioning system cools the living room and bedroom to 20; Suppose now I want to know the variation of temperature in a room: - Measure the temperature at 5 different locations in each room. Living Room: 18, 19, 20, 21, 22. BUT: is sn = s? No!

  16. Sampling: Example Soda can production: Design spec: pressure of a sealed can 50PSI at 10C Testing: sample few randomly selected cans each hour Questions: How many should we test? Which cans should we select? To Answer: We need to know the distribution of pressure among all cans Problem: How can we know the distribution of pressure among all cans?

  17. Sampling: Example.. How can we know the distribution of pressure among all cans? Plot a histogram showing %-cans with pressure in different ranges

  18. 30 40 35 45 55 70 60 65 50 pressure (psi) Sampling: Example… Limit (as histogram step-size)  0: probability density function why? pdf is (almost) the familiar bell-shaped Gaussian curve! True Gaussian curve: [-∞ , ∞]; pressure: [0, 95psi]

  19. Why is everything normal? pdf of many natural random variables ~ normal distribution WHY ? Central Limit Theorem Let X random variable, any pdf, mean, m, and variance, s2 Let Sn = sum of n randomly selected values of X; As n  ∞ Sn approaches normal distribution with mean = nm, and variance = ns2.

  20. -1, with probability 1/3 0, with probability 1/3 1, with probability 1/3 p(S1) X1 = S1 1 0 -1 -2, with probability 1/9 -1, with probability 2/9 0, with probability 3/9 1, with probability 2/9 2, with probability 1/9 X1 X2 X1 + X2 -1 -1 -2 -1 0 -1 -1 1 0 0 -1 -1 0 0 0 0 1 1 1 -1 0 1 0 1 1 1 2 X1 + X2 = p(S2) S2 1 2 0 -2 -1 -3, with probability 1/27 -2, with probability 3/27 -1, with probability 6/27 0, with probability 7/27 1, with probability 6/27 2, with probability 3/27 3, with probability 1/27 Gaussian curve Curve joining p(S3) X1 + X2 + X3 = p(S3) 3 1 2 S3 0 -2 -1 -3 Central limit theorem.. Example

  21. (Weaker) Central Limit Theorem... Let Sn = X1 + X2 + … + Xn Different pdf, same m and s normalized Sn is ~ normally distributed Another Weak CLT: Under some constraints, even if Xi are from different pdf’s, with different m and s, the normalized sum is nearly normal!

  22. Central Limit Therem.... Observation: For many physical processes/objects variation is f( many independent factors) effect of each individual factor is relatively small Observation + CLT  The variation of parameter(s) measuring the physical phenomenon will follow Gaussian pdf

  23. Sampling for QC Soda Can Problem, recalled: How can we know the distribution of pressure among all cans? Answer: We can assume it is normally distributed Problem: But what is the m, s ? Answer: We will estimate these values  Samples

  24. Background: Scaling of Normal Distribution If x is N(m, s), then z = (x – m)/s is N( 0, 1)  Standard Normal distribution tables

  25. Normal Distribution scaling: example A manufacturer of long life milk estimates that the life of a carton of milk (i.e. before it goes bad) is normally distributed with a mean = 150 days, with a stdev = 14 days. What fraction of milk cartons would be expected to still be ok after 180 days? Z = 180 days (Z - m)/s = (180 - 150)/14 ≈ 2.14 Use tables: Z = 2.14  area = 0.9838 Fraction of milk cartons that are ok Z ≥ 180 days or Z = m + 2.14s, is 1 - 0.9838 = 0.0162

  26. Samples taken from a Normally Distributed Variable Central Limit Theorem Let X random variable, any pdf, mean, m, and variance, s2 Let Sn = sum of n randomly selected values of X; As n  ∞ Sn approaches normal distribution with mean = nm, and variance = ns2. + Scaling  Mean of the sample, mestimates mean of distribution Stdev of sample = s /√n. Estimates reliability of m as an estimate of m  Standard error

  27. Example: QC for raw materials A logistics company buys Shell-C brand diesel for its trucks. Full tank of fuel  average truck travel ~ 510 Km, stdev 31 Km. New seller provides a cheaper fuel, Caltex-B, Claim that it will give similar mileage as the Shell-C. (i) What is the probability that the mean distance traveled over 40 full-tank journeys of Shell-C is between 500 Km and 520 Km? (ii) Mean distance covered by 40 full-tank journeys using Caltex-B ~ 495 Km. What is the probability that Caltex-B is equivalent to Shell-C?

  28. Example: QC for raw materials.. (i) Shell-C: Full tank of fuel  m ~ 510 Km, s ~ 31 Km. P( mean distance)40 is in [500 Km, 520 Km] ? Mean distance ≈ N( 510, s/√40 ) = N( 510, 31/√40 ) ≈ N( 510, 4.9) Use tables, Area between: z= (500 -510)/4.9 ≈ -2.04 and z = (520 - 510)/4.9 ≈ 2.04 Area = 1 - (( 1 - 0.9793) + (1 - 0.9793)) = 0.9586 P( mean distance)40 [500 Km, 520 Km] = 95.86%

  29. Example: QC for raw materials... (ii) Shell-C: Full tank of fuel  m ~ 510 Km, s ~ 31 Km. Mean distance covered by 40 full-tank journeys using Caltex-B ~ 495 Km. What is the probability that Caltex-B is equivalent to Shell-C? P(mean distance over 40 journeys) ≤ 495 ? m= 495  z = (495 - 510)/4.9 ≈ -3.06  P( m40 using Shell-C or similar ≥ 495) = 0.9989  P(Caltex-B is equivalent to Shell-C) = (1 - 0.9989) = 0.0011 This method of reasoning is related to Hypothesis Testing

  30. Summary/Comments on Sampling - Statistics provides basis for reasoning; - Sampling is economical and more efficient than accept/reject - We may not know the population m and/or s  more complex reasoning (not covered in this course)

  31. Control Charts in QC 1. Use sampling of product/process 2. Repeat sampling at regular intervals 3. Plot the time series data 4. Look for any ‘patterns’ that may indicate ‘out-of-control’ process 4.1. Look for problem 4.2. Solve problem  bring process back to ‘under-control’

  32. Measure random sample of 5 rings in each hour Record mean value of the inside diameter Plot Process Control Charts: example Piston rings manufacturing Critical dimension: inside diameter Mfg process designed for: mean diameter = 74mm, s = 0.01 mm

  33. Process Control Charts example: X-bar charts Mfg process designed for: mean diameter = 74mm, s = 0.01 mm [source: Montgomery]

  34. is normally distributed with s = 0.01/√5 = 0.0045 m lies in acceptance interval m lies in the rejection interval No error Type II error Accept the claim Type I error No error Reject the claim X-bar charts – UCL and LCL s = 0.01, and n = 5; Process is in-control  We should avoid a “False rejection” a = P( Type I error)

  35. m lies in acceptance interval m lies in the rejection interval No error Type II error Accept the claim is N( 74, 0.0045) Type I error No error Reject the claim X-bar charts – UCL and LCL.. Process is in-control  We should avoid a “False rejection” If we never reject the claim  never commit Type I error 100(1 - a)% of the sample m must lie in [ 74 - Za/2(0.0045), 74 + Za/2(0.0045)] a = P( Type I error) Typical: P( Type I error) < 0.0027  Za/2 = 3

  36. X-bar charts – UCL and LCL... Avoid “False rejection”  P( Type I error) < 0.0027  Za/2 = 3 3-sigma control limits Piston Rings: Control limits = 74 ± 3(0.0045)  UCL = 74.0135, LCL = 73.9865 [source: Montgomery]

  37. X-bar charts: relationship between sample and x-bar [source: Montgomery]

  38. Points of interest -- larger sample size  control limit lines move close together -- Larger sample size  control chart can identify smaller shifts in the process -- ±2s warning lines [source: Montgomery+]

  39. Using Control Charts

  40. Using Control Charts..

  41. Process Control Charts… - Great practical use in factories - First introduced by Walter A. Shewhart - Help to reduce variability - Monitor performance over time - Trends and out-of-control are immediately detected - Other common control charts: Range-charts (R-charts), …

  42. Robust Design and Taguchi Methods Example: The INA Tile Company - Tiles made in Kiln - Variability in size too high - Variation due to baking process - Accept/Reject is expensive!

  43. Ina Tile Example.. Cause: Different temperature profile in different regions SPC approach: Eliminate cause  redesign Kiln

  44. Ina Tile Example... Cause: Different temperature profile in different regions SPC approach: Eliminate cause  reduce Temp variation  How ? redesign Kiln  Expensive!

  45. Ina Tile example: Taguchi Method Response: Tile dimension Control Parameters (tile design): Amount of Limestone Fineness of additive Amount of Agalmatolite Type of Agalmatolite Raw material Charging Quantity Amount of Waste Return Amount of Feldspar Noise parameter was the temperature gradient. Taguchi: Experiment with different values of Control Parameters!

  46. Ina Tile example: Taguchi Method.. Experiment with different values of Control Parameters Higher Limestone content  desensitize design to noise

  47. Robust Design definition A method of designing a process or product aimed at reducing the variability (deviations from target performance) by lowering sensitivity to noise. HOW ?

  48. Design of Experiments

  49. Typical Objectives of DOE (i) Determine which input variables have the most influence on the output; (ii) Determine what value of xi’s will lead us closest to our desired value of y; (iii) Determine where to set the most influential xi’s so as to reduce the variability of y; (iv) Determine where to set the most influential xi’s such that the effects of the uncontrollable variables (zi’s) are minimized. Tool used: ANalysis Of VAriance  ANOVA

  50. Concluding Remarks Statistical Tools are critical to QC QC is critical to all productive activities next topic: review for exam!

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