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

2. 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

3. 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.

4. 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 )

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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 )

11. 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

12. 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

13. 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

14. 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

15. 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.

16. Quality16 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 3 rd quarter. The first 2 quarters and the last quarter have no moving average. b.Monthly: Place the first average opposite the 7 th 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

17. Quality17 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.

18. Quality18 Seasonal adjustment illustrated: 3 years of quarterly sales of Wolfpack Red Soda Step 1. Moving averages t Qtr. X t 4-Qtr. moving average 1 1 53NA 2 2 83NA 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 3102(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 76NA

19. Quality19 Seasonal adjustment illustrated: 3 years of quarterly sales of Wolfpack Red Soda Step 2. Ratios Ratio = X t / Average 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

20. Quality20 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

21. Quality21 Seasonal adjustment illustrated: 3 years of quarterly sales of Wolfpack Red Soda Step 5. Final seasonal indices Qtr.Mean ratio x Factor = Index 10.7043 x 1.0083 = 0.7101 21.0508 x 1.0083 = 1.0595 31.2888 x 1.0083 = 1.2995 40.9230 x 1.0083 = 0.9307 Sum of indices = 3.9998

22. Quality22 Seasonal adjustment illustrated: 3 years of quarterly sales of Wolfpack Red Soda Step 6. Deseasonalize data t Qtr. X t / Index= Des. X t 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 3102 / 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

23. Quality23 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.

24. Quality24 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.

25. Quality25 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 )

26. Quality26 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