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Quality and Operations Management Process Control and Capability Analysis

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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)

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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

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Getting Started with SPC X-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 chart X R UCLxLCLx UCLrLCLr

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Basic 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 actions

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

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Type 2 Error Example Suppose: = 10 1 = 10.2 = 4/3 n = 9 thus, x = 4/9 Then, Type 2 Error = Z [( + 3 x - 1 ) / x ] = Z [(10 + 12/9 - 10.2) / (4/9)] = Z [2.55] = 0.9946 if 1 = 11.0, then Type 2 Error = Z[0.75] =.7734 if 1 = 12.0, then Type 2 Error = Z[-1.50] =.0668 Prob.{shift will be detected in 3 rd 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

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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

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

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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

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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 –.1587 4 x (1-.1587) x 5 =.0027 8 in a row on one side of center –.50 8 =.0039

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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

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SPC Quick Reference Card

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P-chart –based on fraction (percentage) of defective units in a varying sample size –UCL/LCL p = p 3 (p)(1-p)/n np-chart –based on number of defective units in a fixed sample size –UCL/LCL np = np 3 (np)(1-p) u-chart –based on the counts of defects in a varying sample size – UCL/LCL u = u 3 u/n c-chart –based on the count of defects found in a fixed sample size –UCL/LCL c = c 3 c X-R chart –variables data –UCL/LCL X = X 3 x = X 3 / n = X A 2 R where R/d 2 –UCL R = D 4 Rand A 2 = 3/d 2 n –LCL R = D 3 R for p, np, u, c and R chart the LCL can not be less than zero.

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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

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How 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 years

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