Quality and Operations Management: Process Control and Capability Analysis

1. Quality and OperationsManagement Process Control and Capability Analysis

2. Process Control • Recognizes that variance exists in all processes • Sources of variation • systematic • assignable • Purpose • to detect and eliminate ‘out-of-control’ conditions • to return a process to an ‘in-control’ state • Basic tool -- the SPC chart(s)

3. Measuring A Process • Types of measurements • variables data • length, weight, speed, output, etc • discrete values • attributes data • good vs bad, pass vs fail, etc • binary values • Types of charts • variables -- X-R chart • attributes -- p, np, c and u • Basic assumption -- sample means are normally distributed

4. Getting Started with SPCX-R Charts X R • Determine sample size and frequency of data collection • Collect sufficient historical data • Ensure normality of distribution • Calculate factors for control charts • Construct control chart • Plot data points • Determine outliers and eliminate assignable causes • Recalculate control limits with reduced data set • Implement new process control chart UCLx LCLx LCLr UCLr

5. Basic Properties • sx = std dev of sample mean = s/n (where s = process standard deviation) • conventional approach uses  3 s/n • limitations of control charts • Type I Error: probability that an in-control value would appear as out-of-control • Type II Error: probability that a shift causing an out-of-control situation would be mis-reported as in-control • delays due to sampling interval • charting without taking action on assignable causes • over control actions

6. Type 1 and Type 2 Error Alarm No Alarm Type 1 error No error In Control Out of Control Type 2 error No error Suppose m1 > m, then Type 2 Error = Z [(m + 3 sx - m1) / sx] Type 1 Error = 0.0027 for 3s charts

7. Type 2 Error Example Suppose: m = 10 m1= 10.2 s = 4/3 n = 9 thus, sx = 4/9 Then, Type 2 Error = Z [(m + 3 sx - m1) / sx ] = Z [(10 + 12/9 - 10.2) / (4/9)] = Z [2.55] = 0.9946 if m1= 11.0, then Type 2 Error = Z[0.75] = .7734 if m1= 12.0, then Type 2 Error = Z[-1.50] = .0668 Prob.{shift will be detected in 3rd sample after shift occurs} = 0.0668*0.0668*(1-0.0668) = 0.0042 Average number of samples taken before shift is detected = 1/(1-0.0668) = 1.0716 Prob.{no false alarms first 32 runs, but false alarm on 33rd} = (0.9973)32*(0.0027) = .0025 Average number of samples taken before a false alarm = 1/0.0027 = 370

8. Tests for Unnatural Patterns • Probability that “odd” patterns observed are not “natural” variability are calculated by using the probabilities associated with each zone of the control chart • Use the assumption that the population is normally distributed • Probabilities for X-chart are shown on next slide

9. Normal Distribution Applied to X-R Control Charts X Probability = .00135 UCLx +3s A Probability = .02135 Outer 3rd +2s B Probability = .1360 Middle 3rd C +1s Probability = .3413 Inner 3rd C -1s Inner 3rd Probability = .3413 B -2s Middle 3rd Probability = .1360 -3s Outer 3rd A Probability = .02135 LCLx Probability = .00135

10. A Few Standard Tests • 1 point outside Zone A • 2 out of 3 in Zone A or above (below) • 4 out of 5 in Zone B or above (below) • 8 in a row in Zone C or above (below) • 10 out of 11 on one side of center

11. Tests for Unnatural Patterns • 2 out of 3 in A or beyond • .0227 x .0227 x (1-.0227) x 3 = .0015 • 4 out of 5 in B or beyond • .15874 x (1-.1587) x 5 = .0027 • 8 in a row on one side of center • .508 = .0039

12. Other Charts • P-chart • based on fraction (percentage) of defective units in a varying sample size • np-chart • based on number of defective units in a fixed sample size • u-chart • based on the counts of defects in a varying sample size • c-chart • based on the count of defects found in a fixed sample size

13. SPC Quick Reference Card

14. P-chart • based on fraction (percentage) of defective units in a varying sample size • UCL/LCLp = p  3(p)(1-p)/n • np-chart • based on number of defective units in a fixed sample size • UCL/LCLnp = np  3(np)(1-p) • u-chart • based on the counts of defects in a varying sample size • UCL/LCLu = u  3u/n • c-chart • based on the count of defects found in a fixed sample size • UCL/LCLc = c  3c • X-R chart • variables data • UCL/LCLX = X  3 sx = X  3 s/  n = X  A2R where s = R/d2 • UCLR = D4R and A2 = 3/d2n • LCLR = D3R • for p, np, u, c and R chart the LCL can not be less than zero.

15. Process Capability • Cp: process capability ratio • a measure of how the distribution compares to the width of the specification • not a measure of conformance • a measure of capability, if distribution center were to match center of specification range • Cpk: process capability index • a measure of conformance (capability) to specification • biased towards “worst case” • compares sample mean to nearest spec. against distribution width

16. How Good is Good Enough? • Cp = 1.0 =>  3s => 99.73% (in acceptance) => .9973 => 2700 ppm out of tolerance • PG&E operates non-stop • 23.65 hours per year without electricity • average car driven 15,000 miles per year • 41 breakdowns or problems per year • Cp = 2.0 =>  6s => 99.99983% (in acceptance) => .9999983 => 3.4 ppm out of tolerance • PG&E operates non-stop • 1.78 minutes per year without electricity • average car driven 15,000 miles per year • 0.051 breakdowns or problems per year or one every 20 years