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Quality Control Dr. Everette S. Gardner, Jr.

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Quality2 Energy needed to close door Door seal resistance Check force on level ground Energy needed to open door Acoustic trans., window Water resistance Maintain current level Reduce energy level to 7.5 ft/lb Reduce force to 9 lb. Reduce energy to 7.5 ft/lb Maintain current level Engineering characteristics Customer requirements Importance to customer 5 4 3 2 1 Easy to close Stays open on a hill Easy to open Doesn’t leak in rain No road noise Importance weighting 7 5 3 3 2 10 66 9 Source: Based on John R. Hauser and Don Clausing, “The House of Quality,” Harvard Business Review, May-June 1988. 2 3 x x x x x x * Competitive evaluation x A B (5 is best) 1 2 3 4 5 = Us = Comp. A = Comp. B Target values Technical evaluation (5 is best) Correlation: Strong positive Positive Negative Strong negative x x x x x x x x x x x AB BA A A A A A A B B B B B B Relationships: Strong = 9 Medium = 3 Small = 1

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Quality3 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(σ 2 + D 2 ) where σ 2 = Variance of quality characteristic D 2 = ( x – T) 2 Note: x is the mean quality characteristic. D 2 is zero if the mean equals the target.

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Quality4 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) = kx 2 When larger is better (e.g., product life) L(x) = k (1/x 2 )

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Quality5 Introduction to quality control charts Definitions VariablesMeasurements on a continuous scale, such as length or weight AttributesInteger counts of quality characteristics, such as nbr. good or bad DefectA single non-conforming quality characteristic, such as a blemish DefectiveA 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

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Quality6 Sample mean value Sample number 99.74% 0.13% Upper control limit Lower control limit Process mean Normal tolerance of process 0 1 2 345 6 7 8

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Quality7 Reference guide to control factors n A A 2 D 3 D 4 d 2 d 3 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 (σ x ) is: σ x = R / d 2

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Quality8 Reference guide to control factors (cont.) Note that control factors depend on the sample size n. Relationships amongst control factors: A 2 = 3 / (d 2 x n 1/2 ) D 4 = 1 + 3 x d 3 /d 2 D 3 = 1 – 3 x d 3 /d 2, unless the result is negative, then D 3 = 0 A = 3 / n 1/2 D 2 = d 2 + 3d 3 D 1 = d 2 – 3d 3, unless the result is negative, then D 1 = 0

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Quality9 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 (σ x ): σ x = R / d 2 4. Estimate the natural tolerance of the process: Natural tolerance = 6σ x 5. Determine the specification limits: USL = Upper specification limit LSL = Lower specification limit

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Quality10 Process capability analysis (cont.) 6. Compute capability indices: Process capability potential C p = (USL – LSL) / 6σ x Upper capability index C pU = (USL – X ) / 3σ x Lower capability index C pL = ( X – LSL) / 3σ x Process capability index C pk = Minimum (C pU, C pL )

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Quality11 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 + A 2 R LCL = X – A 2 R 4. Set 3-std.-dev. control limits for the sample ranges: UCL = D 4 R LCL = D 3 R

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Quality12 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 (S P ) for each sample: S P = [( 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 + 3S P LCL = P – 3S P

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Quality13 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

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Quality14 Control chart for individual observations — I-CHART (cont.) 4. Estimate the population standard deviation (σ X ): σ X = R / d 2 Note: Sample size is always 2, so d 2 = 1.128. 5. Set 3-std.-dev. control limits: UCL = X + 3σ X LCL = X – 3σ X

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Quality15 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: S C = ( 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 + 3S C LCL = C – 3S C 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.

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Quality16 Quick reference to quality formulas Control factors n A A 2 D 3 D 4 d 2 d 3 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 σ x = R / d 2 C p = (USL – LSL) / 6σ x C pU = (USL – X ) / 3σ x C pL = ( X – LSL) / 3σ x C pk = Minimum (C pU, C pL )

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Quality17 Quick reference to quality formulas (cont.) Means and ranges UCL = X + A 2 RUCL = D 4 R LCL = X – A 2 RLCL = D 3 R Percentage defective in a sample S P = [( P (1-P ))/n] 1/2 UCL = P + 3S P LCL = P – 3S P Individual quality observations σ x = R / d 2 UCL = X + 3σ X LCL = X – 3σ X Number of defects per unit S C = ( C / n) 1/2 UCL = C + 3S C LCL = C – 3S C

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Quality18 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

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Quality19 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.

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Quality20 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

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Quality21 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.

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Quality22 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.

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Quality23 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.

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To Accompany Russell and Taylor, Operations Management, 4th Edition, 2003 Prentice-Hall, Inc. All rights reserved. Chapter 15 Statistical Quality Control.

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