Quality and Operations ManagementProcess ControlMeasuring A ProcessGetting Started with SPC X-R ChartsBasic PropertiesType 1 and Type 2 ErrorType 2 Error ExampleTests for Unnatural PatternsNormal Distribution Applied to X-R Control ChartsA Few Standard TestsSlide 11Other ChartsSPC Quick Reference CardSlide 14Process CapabilityHow Good is Good Enough?Quality and OperationsManagementProcess Control and Capability AnalysisProcess 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)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 distributedGetting Started with SPCX-R Charts•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 chartXRUCLx LCLxUCLrLCLrBasic Properties• x = std dev of sample mean = /n (where  = process standard deviation)•conventional approach uses  3 /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 actionsType 1 and Type 2 ErrorType 1errorType 2errorNo errorNo error Alarm No AlarmIn ControlOut of ControlSuppose 1 > , thenType 2 Error = Z [( + 3 x - 1) / x ]Type 1 Error = 0.0027 for 3 chartsType 2 Error ExampleSuppose: = 101= 10.2 = 4/3n = 9thus,x = 4/9Then, Type 2 Error = Z [( + 3 x - 1) / x ]= Z [(10 + 12/9 - 10.2) / (4/9)]= Z [2.55] = 0.9946if 1= 11.0, then Type 2 Error = Z[0.75] = .7734if 1= 12.0, then Type 2 Error = Z[-1.50] = .0668Prob.{shift will be detected in 3rd sample after shift occurs}= 0.0668*0.0668*(1-0.0668) = 0.0042Average number of samples taken before shift is detected= 1/(1-0.0668) = 1.0716Prob.{no false alarms first 32 runs, but false alarm on 33rd}= (0.9973)32*(0.0027) = .0025Average number of samples taken before a false alarm= 1/0.0027 = 370Tests 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 slideNormal Distribution Applied to X-R Control ChartsAABBCC+3+2+1-1-2-3Probability = .00135Probability = .1360Probability = .3413Probability = .3413Probability = .1360Probability = .02135Probability = .00135Probability = .02135UCLxLCLxXOuter 3rdOuter 3rdMiddle 3rdMiddle 3rdInner 3rdInner 3rdA 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 centerTests 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 = .0039Other 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 sizeSPC Quick Reference Card•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 x = X  3 /  n = X  A2R where  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.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 widthHow Good is Good Enough?•Cp = 1.0 =>  3 => 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 =>  6 => 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