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1 Chapter 10 Quality Control

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2 Phases of Quality Assurance Acceptance sampling Process control Continuous improvement Inspection before/after production Corrective action during production Quality built into the process The least progressive The most progressive

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3 Inspection: Appraisal of good/service quality How Much (sample size) /How Often (hourly, daily) Cost Optimal Amount of Inspection Cost of inspection (appraisal and Prevention cost) Cost of passing defectives (failure cost) Total Cost

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4 Inspection Where/When Raw materials Finished products Before a costly operation, PhD comp. exam before candidacy Before an irreversible process, firing pottery Before a covering process, painting, assembly Centralized vs. On-Site, my friend checks quality at cruise lines InputsTransformationOutputs Acceptance sampling Process control Acceptance sampling

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5 Examples of Inspection Points

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6 Statistical Process Control (SPC) SPC: Statistical evaluation of the output of a process during production The Control Process –Define –Measure –Compare to a standard –Evaluate –Take corrective action –Evaluate corrective action

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7 Statistical Process Control Shewhart’s classification of variability: common cause vs. assignable cause Variations and Control –Random variation: Natural variations in the output of process, created by countless minor factors, e.g. temperature, humidity variations. –Assignable variation: A variation whose source can be identified. This source is generally a major factor, e.g. tool failure.

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8 Mean and Variance Given a population of numbers, how to compute the mean and the variance?

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9 Statistical Process Control From a large population of goods or services (random if possible) a sample is drawn. –Example sample: Midterm grades of BA3352 students whose last name starts with letter R {60, 64, 72, 86}, with letter S {54, 60} Sample size= n Sample average or sample mean= Sample range= R Standard deviation of sample means=

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10 Sampling Distribution Sampling distribution Variability of the average scores of people with last name R and S Process distribution Variability of the scores for the entire class Mean Sampling distribution is the distribution of sample means. Grouping reduces the variability.

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11 Normal Distribution Mean 95.44% 99.74% x normdist(x,.,.,0) Probab normdist(x,.,.,1)

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12 Cumulative Normal Density 0 1 x normdist(x,mean,st_dev,1) prob norminv(prob,mean,st_dev)

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13 Normal Probabilities: Example If temperature inside a firing oven has a normal distribution with mean 200 o C and standard deviation of 40 o C, what is the probability that –The temperature is lower than 220 o C =normdist(220,200,40,1) –The temperature is between 190 o C and 220 o C =normdist(220,200,40,1)-normdist(190,200,40,1)

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14 Control Limits Sampling distribution Process distribution Mean LCL Lower control limit UCL Upper control limit Process is in control if sample mean is between control limits. These limits have nothing to do with product specifications!

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15 Setting Control Limits: Hypothesis Testing Framework Null hypothesis: Process is in control Alternative hypothesis: Process is out of control Alpha=P(Type I error)= P(reject the null when it is true)= P(out of control when in control) Beta=P(Type II error)= P(accept the null when it is false) P(in control when out of control) If LCL decreases and UCL increases what happens to –Alpha ? –Beta? Not possible to target alpha and beta simultaneously, control charts target a desired level of Alpha.

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16 Type I Error=Alpha Mean LCLUCL /2 Probability of Type I error

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17 Control Chart UCL LCL Sample number Mean Out of control Normal variation due to chance Abnormal variation due to assignable sources

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18 Observations from Sample Distribution Sample number UCL LCL 1234

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19 Control Charts Control charts for variables (measurable quantities), e.g. length, temperature –Mean control charts To check mean –Range control charts To check variability Control charts for attributes, e.g. fit, defective –p-charts To check proportion of defectives (occurrences) –c-charts To check the number of defectives (occurrences)

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20 Mean control chart Most often z is set to 2 or 3. If the standard deviation of the sample means is not known, use the average of sample ranges to get the limits: Multiplier A_2 depends on n and is available in Table 10-2.

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21 Range Control Chart Multipliers D_4 and D_3 depend on n and are available in Table EX: In the last five years, the range of GMAT scores of incoming PhD class is 88, 64, 102, 70, 74. If each class has 6 students, what are UCL and LCL for GMAT ranges? Are the GMAT ranges in control?

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22 Mean and Range Charts: Which? UCL LCL UCL LCL R-chart x-Chart Detects shift Does not detect shift (process mean is shifting upward) Sampling Distribution

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23 Mean and Range Charts: Which? UCL LC L R-chart Reveals increase x-Chart UCL Does not reveal increase (process variability is increasing) Sampling Distribution

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24 Use of p-Charts p=proportion defective, assumed to be known When observations can be placed into two categories. –Good or bad –Pass or fail –Operate or don’t operate –Go or no-go gauge

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25 Use of c-Charts c=number of occurrences per unit Use only when the number of occurrences per unit can be counted. Scratches, chips, dents, or errors per item Cracks or faults per unit of distance Breaks or Tears per unit of area Bacteria or pollutants per unit of volume Calls, complaints, failures per unit of time

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26 C-chart Example While the nuclear submarine Kursk was being raised in the Barents sea (between Svalbard, No and Novaya Zemlya, Ru), which took 15 hours, engineers took a reading of number of Geiger counts per hour to detect any increase in radiation levels. Should they have stopped before 5 th or 10 th hour given 3-sigma control and the readings data: 42, 48, 50, 45, 52, 66, 64, 84, 92, 76. At the 5 th hour, average number of counts=47.4, stdev of counts=6.88, UCL=47.4+3*6.88=68.05, LCL=47.4-3*6.88= Do not stop. At the 10 th hour, average number of counts=61.9, stdev of counts=7.87, UCL=61.9+3*7.87=85.51, LCL=61.9-3*7.87= Stop, 9 th reading is out of control.

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27 Up and Down Run Charts If all readings are in control, is the process really in control? There could be trends in readings even when they are in control. Counting Up/Down Runs(r=8 runs) U U D U D U D U U D

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28 Up and Down Run Charts EX: What are 3-sigma UCL and LCL for the number of runs in 50 samples?

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29 Tolerances/Specifications –Requirements of the design or customers Process variability –Natural variability in a process –Variance of the measurements coming from the process Process capability –Process variability relative to specification –Capability=Process specifications / Process variability Process Capability

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30 Process Capability: Specification limits are not control chart limits Lower Specification Upper Specification Process variability matches specifications Lower Specification Upper Specification Process variability well within specifications Lower Specification Upper Specification Process variability exceeds specifications Sampling Distribution is used

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31 Process Capability Ratio When the process is centered, process capability ratio Upper specification – lower specification 6 Cp = A capable process has large Cp. Example: The standard deviation, of sample averages of the midterm 1scores obtained by students whose last names start with R, has been 7. The SOM management requires the scores not to differ by more than 50% in an exam. That is the highest score can be at most 50 points above the lowest score. Suppose that the scores are centered, what is the process capability ratio? Answer: 50/42

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32 Process Capability Ratio When the process is not centered, process capability ratio Min{Process mean - lower spec, Upper spec - Process mean} 3 Cpk= When the process is not centered, the closest spec to mean determines the capability of the process because that spec is likely to be more of a limiting factor than the other. Example: Suppose that the process is not centered in the previous example and the SOM wants all the scores to fall within 50% and 100%. What is the Capability ratio if the average score was 70? Answer: From the lower limit, we have (70-50)/21 From the upper limit, we have (100-70)/21 Then the ratio is 20/21

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33 Process mean Lower specification Upper specification +/- 3 Sigma +/- 6 Sigma 3 Sigma and 6 Sigma Quality

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34 Chapter 10 Supplement Acceptance Sampling

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35 Acceptance Sampling Acceptance sampling: Is a lot of N products good if a random sample of n (n

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36 Why not to emphasize Acceptance Sampling (AS) AS plans have no clearly stated economic objective. They target some levels of type I and II errors. AS incorporate an attitude of punishment by rejecting entire lots after examining small samples. This feeds the mistrust between supplier and the customer. AS does not attempt to find the root cause of defectives. It merely detects defectives. Real problem is actually finding the root cause. Some people say that: –“AS provides elegant solutions to balance type I and II errors by making a type III error: solving the wrong problem”.

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