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Quality Control Dr. Everette S. Gardner, Jr. Correlation: Strong positive Positive x Negative * Strong negative Competitive evaluation x = Us A = Comp. A B = Comp. B (5 is best) 1 2 3 4 5 x AB x AB x AB A xB x A B x x x Reduce energy to 7.5 ft/lb Acoustic trans., window 6 Energy needed to open door 6 9 2 3 B xA BA x 7 5 3 3 2 Importance weighting 10 Target values Technical evaluation (5 is best) Check force on level ground Easy to close Stays open on a hill Easy to open Doesn’t leak in rain No road noise Reduce force to 9 lb. Customer requirements Door seal resistance Engineering characteristics x Maintain current level x 5 4 3 2 1 B A x BA x B A x Quality B x A Relationships: Strong = 9 Medium = 3 Small = 1 Source: Based on John R. Hauser and Don Clausing, “The House of Quality,” Harvard Business Review, May-June 1988. 2 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 3 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 4 Introduction to quality control charts Definitions • Variables • Attributes • Defect • Defective Measurements on a continuous scale, such as length or weight Integer counts of quality characteristics, such as nbr. good or bad A single non-conforming quality characteristic, such as a blemish A physical unit that contains one or more defects Types of control charts Data monitored • Mean, range of sample variables • Individual variables • % of defective units in a sample • Number of defects per unit Chart name MR-CHART I-CHART P-CHART C/U-CHART Quality Sample size 2 to 5 units 1 unit at least 100 units 1 or more units 5 Sample mean value 0.13% Upper control limit 99.74% Normal tolerance of process Process mean Lower control limit 0.13% 0 1 2 3 4 5 6 7 8 Sample number Quality 6 Reference guide to control factors n 2 3 4 5 A 2.121 1.732 1.500 1.342 A2 1.880 1.023 0.729 0.577 D3 0 0 0 0 D4 3.267 2.574 2.282 2.114 d2 1.128 1.693 2.059 2.316 d3 0.853 0.888 0.880 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 7 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 8 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 9 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 10 Mean-Range control chart MR-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 11 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 12 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 13 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 14 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 15 Quick reference to quality formulas • Control factors n A A2 2 2.121 1.880 3 1.732 1.023 4 1.500 0.729 5 1.342 0.577 D3 0 0 0 0 D4 3.267 2.574 2.282 2.114 • Process capability analysis σx = R / d2 Cp = (USL – LSL) / 6σx CpL = ( X – LSL) / 3σx d2 1.128 1.693 2.059 2.316 d3 0.853 0.888 0.880 0.864 CpU = (USL – X ) / 3σx Cpk = Minimum (CpU, CpL) Quality 16 Quick reference to quality formulas (cont.) • Means and ranges UCL = X + A2R LCL = X – A2R UCL = D4R 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 17 Multiplicative seasonality The seasonal index is the expected ratio of actual data to the average for the year. Actual data / Index = Seasonally adjusted data Seasonally adjusted data x Index = Actual data Quality 18 Multiplicative seasonal adjustment 1. Compute moving average based on length of seasonality (4 quarters or 12 months). 2. Divide actual data by corresponding moving average. 3. Average ratios to eliminate randomness. 4. Compute normalization factor to adjust mean ratios so they sum to 4 (quarterly data) or 12 (monthly data). 5. Multiply mean ratios by normalization factor to get final seasonal indexes. 6. Deseasonalize data by dividing by the seasonal index. 7. Forecast deseasonalized data. 8. Seasonalize forecasts from step 7 to get final forecasts. Quality 19 Additive seasonality The seasonal index is the expected difference between actual data and the average for the year. Actual data - Index = Seasonally adjusted data Seasonally adjusted data + Index = Actual data Quality 20 Additive seasonal adjustment 1. Compute moving average based on length of seasonality (4 quarters or 12 months). 2. Compute differences: Actual data - moving average. 3. Average differences to eliminate randomness. 4. Compute normalization factor to adjust mean differences so they sum to zero. 5. Compute final indexes: Mean difference – normalization factor. 6. Deseasonalize data: Actual data – seasonal index. 7. Forecast deseasonalized data. 8. Seasonalize forecasts from step 7 to get final forecasts. Quality 21 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 22 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 23