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Quality Control

Quality Control. Dr. Everette S. Gardner, Jr. Correlation:. x. Strong positive. Positive. x. x. x. Negative. x. x. Strong negative. *. Competitive evaluation. Engineering characteristics.

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Quality Control

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  1. Quality Control Dr. Everette S. Gardner, Jr.

  2. Correlation: x Strong positive Positive x x x Negative x x Strong negative * Competitive evaluation Engineering characteristics Source: Based on John R. Hauser and Don Clausing, “The House of Quality,” Harvard Business Review, May-June 1988. Acoustic trans., window Energy needed to open door Check force on level ground Energy needed to close door x Water resistance = Us Door seal resistance Importance to customer = Comp. A A = Comp. B B Customer requirements (5 is best) 1 2 3 4 5 x Easy to close 7 AB Stays open on a hill x AB 5 Easy to open 3 AB x x Doesn’t leak in rain 3 B A x No road noise 2 B A Importance weighting 3 2 10 9 6 6 Relationships: Strong = 9 Medium = 3 Target values Reduce energy level to 7.5 ft/lb Reduce energy to 7.5 ft/lb Small = 1 Maintain current level Maintain current level Maintain current level Reduce force to 9 lb. 5 BA B BA x x A 4 B B B x Technical evaluation (5 is best) A x 3 A x 2 A x 1 Quality

  3. Taguchi analysis Loss function L(x) = k(x-T)2 where x = any individual value of the quality characteristic T = target quality value k = constant = L(x) / (x-T)2 Average or expected loss, variance known E[L(x)] = k(σ2 + D2) where σ2 = Variance of quality characteristic D2 = ( x – T)2 Note: x is the mean quality characteristic. D2 is zero if the mean equals the target. Quality

  4. Taguchi analysis (cont.) Average or expected loss, variance unkown E[L(x)] = k[Σ ( x – T)2 / n] When smaller is better (e.g., percent of impurities) L(x) = kx2 When larger is better (e.g., product life) L(x) = k (1/x2) Quality

  5. Introduction to quality control charts Definitions • Variables Measurements on a continuous scale, such as length or weight • Attributes Integer counts of quality characteristics, such as nbr. good or bad • Defect A single non-conforming quality characteristic, such as a blemish • Defective A physical unit that contains one or more defects Types of control charts Data monitored Chart name Sample size • Mean, range of sample variables MR-CHART 2 to 5 units • Individual variables I-CHART 1 unit • % of defective units in a sample P-CHART at least 100 units • Number of defects per unit C/U-CHART 1 or more units Quality

  6. Sample mean value 0.13% Upper control limit Normal tolerance of process 99.74% Process mean Lower control limit 0.13% 7 6 8 1 3 4 5 2 0 Sample number Quality

  7. Reference guide to control factors n A A2 D3 D4 d2 d3 2 2.121 1.880 0 3.267 1.128 0.853 3 1.732 1.023 0 2.574 1.693 0.888 4 1.500 0.729 0 2.282 2.059 0.880 5 1.342 0.577 0 2.114 2.316 0.864 • Control factors are used to convert the mean of sample ranges ( R ) to: (1) standard deviation estimates for individual observations, and (2) standard error estimates for means and ranges of samples For example, an estimate of the population standard deviation of individual observations (σx) is: σx = R / d2 Quality

  8. Reference guide to control factors (cont.) • Note that control factors depend on the sample size n. • Relationships amongst control factors: A2 = 3 / (d2 x n1/2) D4 = 1 + 3 x d3/d2 D3 = 1 – 3 x d3/d2, unless the result is negative, then D3 = 0 A = 3 / n1/2 D2 = d2 + 3d3 D1 = d2 – 3d3, unless the result is negative, then D1 = 0 Quality

  9. Process capability analysis 1. Compute the mean of sample means ( X ). 2. Compute the mean of sample ranges ( R ). 3. Estimate the population standard deviation (σx): σx = R / d2 4. Estimate the natural tolerance of the process: Natural tolerance = 6σx 5. Determine the specification limits: USL = Upper specification limit LSL = Lower specification limit Quality

  10. Process capability analysis (cont.) 6. Compute capability indices: Process capability potential Cp = (USL – LSL) / 6σx Upper capability index CpU = (USL – X ) / 3σx Lower capability index CpL = ( X – LSL) / 3σx Process capability index Cpk = Minimum (CpU, CpL) Quality

  11. Mean-Range control chartMR-CHART 1. Compute the mean of sample means ( X ). 2. Compute the mean of sample ranges ( R ). 3. Set 3-std.-dev. control limits for the sample means: UCL = X + A2R LCL = X – A2R 4. Set 3-std.-dev. control limits for the sample ranges: UCL = D4R LCL = D3R Quality

  12. Control chart for percentage defective in a sample — P-CHART 1. Compute the mean percentage defective ( P ) for all samples: P = Total nbr. of units defective / Total nbr. of units sampled 2. Compute an individual standard error (SP ) for each sample: SP = [( P (1-P ))/n]1/2 Note: n is the sample size, not the total units sampled. If n is constant, each sample has the same standard error. 3. Set 3-std.-dev. control limits: UCL = P + 3SP LCL = P – 3SP Quality

  13. Control chart for individual observations — I-CHART 1. Compute the mean observation value ( X ) X = Sum of observation values / N where N is the number of observations 2. Compute moving range absolute values, starting at obs. nbr. 2: Moving range for obs. 2 = obs. 2 – obs. 1 Moving range for obs. 3 = obs. 3 – obs. 2 … Moving range for obs. N = obs. N – obs. N – 1 3. Compute the mean of the moving ranges ( R ): R = Sum of the moving ranges / N – 1 Quality

  14. Control chart for individual observations — I-CHART (cont.) 4. Estimate the population standard deviation (σX): σX = R / d2 Note: Sample size is always 2, so d2 = 1.128. 5. Set 3-std.-dev. control limits: UCL = X + 3σX LCL = X – 3σX Quality

  15. Control chart for number of defects per unit — C/U-CHART 1. Compute the mean nbr. of defects per unit ( C ) for all samples: C = Total nbr. of defects observed / Total nbr. of units sampled 2. Compute an individual standard error for each sample: SC = ( C / n)1/2 Note: n is the sample size, not the total units sampled. If n is constant, each sample has the same standard error. 3. Set 3-std.-dev. control limits: UCL = C + 3SC LCL = C – 3SC Notes: ● If the sample size is constant, the chart is a C-CHART. ● If the sample size varies, the chart is a U-CHART. ● Computations are the same in either case. Quality

  16. Seasonal adjustment of quality observations 1. Compute a 4-quarter or 12-month moving average. Position the first average as follows: a. Quarterly: Place the first average opposite the 3rd quarter. The first 2 quarters and the last quarter have no moving average. b. Monthly: Place the first average opposite the 7th month. The first 6 months and the last 5 months have no moving average. 2. Divide each data observation by the corresponding moving average. 3. Compute a mean ratio for each quarter or month. 4. Compute a normalization factor to adjust the mean ratios so that they sum to 4 (quarterly) or 12 (monthly): a. Quarterly: Normalization factor = 4 / Sum of mean ratios b. Monthly: Normalization factor = 12 / Sum of mean ratios Quality

  17. Seasonal adjustment of quality observations (cont.) 5. Multiply each mean ratio by the normalization factor to get a set of final seasonal indices. Each quarter or month has an individual index. 6. Deseasonalize each data observation by dividing by the appropriate seasonal index. 7. Develop a control chart for the deseasonalized (seasonally-adjusted) data. Quality

  18. Seasonal adjustment illustrated: 3 years of quarterly sales of Wolfpack Red Soda Step 1. Moving averages t Qtr. Xt4-Qtr. moving average 1 1 53 NA 2 2 83 NA 3 3 95 (53 + 83 + 95 + 72) / 4 = 75.75 4 4 72 (83 + 95 + 72 + 50) / 4 = 75.00 5 1 50 (95 + 72 + 50 + 75) / 4 = 73.00 6 2 75 (72 + 50 + 75 + 102) / 4 = 74.75 7 3 102 (50 + 75 + 102 + 66) / 4 = 73.25 8 4 66 (75 + 102 + 66 + 55) / 4 = 74.50 9 1 55 (102 + 66 + 55 + 81) / 4 = 76.00 10 2 81 (66 + 55 + 81 + 93) / 4 = 73.75 11 3 93 (55 + 81 + 93 + 76) / 4 = 76.25 12 4 76 NA Quality

  19. Seasonal adjustment illustrated: 3 years of quarterly sales of Wolfpack Red Soda Step 2. Ratios Ratio = Xt / Average NA NA 95 / 75.75 = 1.2541 72 / 75.00 = 0.9600 50 / 73.00 = 0.6849 75 / 74.75 = 1.0033 102 / 73.25 = 1.3925 66 / 74.50 = 0.8859 55 / 76.00 = 0.7237 81 / 73.75 = 1.0983 93 / 76.25 = 1.2197 NA Quality

  20. Seasonal adjustment illustrated: 3 years of quarterly sales of Wolfpack Red Soda Step 3. Mean ratios Qtr.Sum of ratios for each qtr. / Nbr. 1 (0.6849 + 0.7237) / 2 = 0.7043 2 (1.0033 + 1.0983) / 2 = 1.0508 3 (1.2542 + 1.3925 + 1.2197) / 3 = 1.2888 4 (0.9600 + 0.8859) / 2 = 0.9230 Sum of mean ratios = 3.9669 Step 4. Normalization Factor Factor = 4 / (Sum of mean ratios) Factor = 4 / 3.9669 = 1.0083 Quality

  21. Seasonal adjustment illustrated: 3 years of quarterly sales of Wolfpack Red Soda Step 5. Final seasonal indices Qtr.Mean ratio x Factor = Index 1 0.7043 x 1.0083 = 0.7101 2 1.0508 x 1.0083 = 1.0595 3 1.2888 x 1.0083 = 1.2995 4 0.9230 x 1.0083 = 0.9307 Sum of indices = 3.9998 Quality

  22. Seasonal adjustment illustrated: 3 years of quarterly sales of Wolfpack Red Soda Step 6. Deseasonalize data t Qtr. Xt / Index = Des. Xt 1 1 53 / 0.7101 = 74.6 2 2 83 / 1.0595 = 78.3 3 3 95 / 1.2995 = 73.1 4 4 72 / 0.9307 = 77.4 5 1 50 / 0.7101 = 70.4 6 2 75 / 1.0595 = 70.8 7 3 102 / 1.2995 = 78.5 8 4 66 / 0.9307 = 70.9 9 1 55 / 0.7101 = 77.5 10 2 81 / 1.0595 = 76.5 11 3 93 / 1.2995 = 71.6 12 4 76 / 0.9307 = 81.7 Quality

  23. How to start up a control chart system 1. Identify quality characteristics. 2. Choose a quality indicator. 3. Choose the type of chart. 4. Decide when to sample. 5. Choose a sample size. 6. Collect representative data. 7. If data are seasonal, perform seasonal adjustment. 8. Graph the data and adjust for outliers. Quality

  24. How to start up a control chart system (cont.) 9. Compute control limits 10. Investigate and adjust special-cause variation. 11. Divide data into two samples and test stability of limits. 12. If data are variables, perform a process capability study: a. Estimate the population standard deviation. b. Estimate natural tolerance. c. Compute process capability indices. d. Check individual observations against specifications. 13. Return to step 1. Quality

  25. Quick reference to quality formulas • Control factors n A A2 D3 D4 d2 d3 2 2.121 1.880 0 3.267 1.128 0.853 3 1.732 1.023 0 2.574 1.693 0.888 4 1.500 0.729 0 2.282 2.059 0.880 5 1.342 0.577 0 2.114 2.316 0.864 • Process capability analysis σx = R / d2 Cp = (USL – LSL) / 6σx CpU = (USL – X ) / 3σx CpL = ( X – LSL) / 3σx Cpk = Minimum (CpU, CpL) Quality

  26. Quick reference to quality formulas (cont.) • Means and ranges UCL = X + A2R UCL = D4R LCL = X – A2R LCL = D3R • Percentage defective in a sample SP = [( P (1-P ))/n]1/2 UCL = P + 3SP LCL = P – 3SP • Individual quality observations σx = R / d2 UCL = X + 3σX LCL = X – 3σX • Number of defects per unit SC = ( C / n)1/2 UCL = C + 3SC LCL = C – 3SC Quality

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