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Chapter 10 Quality Control

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**Phases of Quality Assurance**

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

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**Inspection: Appraisal of good/service quality**

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

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**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 Inputs Transformation Outputs Acceptance sampling Process control

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**Examples of Inspection Points**

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

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**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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**Sampling Distribution**

Sampling distribution is the distribution of sample means. 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 Grouping reduces the variability.

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**Normal Distribution normdist(x,.,.,1) normdist(x,.,.,0) x Probab Mean**

95.44% 99.74%

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**Cumulative Normal Density**

1 prob normdist(x,mean,st_dev,1) x norminv(prob,mean,st_dev)

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**Normal Probabilities: Example**

If temperature inside a firing oven has a normal distribution with mean 200 oC and standard deviation of 40 oC, what is the probability that The temperature is lower than 220 oC =normdist(220,200,40,1) The temperature is between 190 oC and 220oC =normdist(220,200,40,1)-normdist(190,200,40,1)

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

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**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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Type I Error=Alpha Mean LCL UCL /2 Probability of Type I error

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Control Chart 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 UCL LCL Sample number Mean Out of control Normal variation due to chance Abnormal variation due to assignable sources

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**Observations from Sample Distribution**

Sample number UCL LCL 1 2 3 4

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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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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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Range Control Chart Multipliers D_4 and D_3 depend on n and are available in Table 10-2. 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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**Mean and Range Charts: Which?**

(process mean is shifting upward) Sampling Distribution UCL x-Chart Detects shift LCL UCL LCL Does not detect shift R-chart

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**Mean and Range Charts: Which?**

Sampling Distribution (process variability is increasing) UCL LCL Does not reveal increase x-Chart UCL R-chart Reveals increase LCL

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**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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**c=number of occurrences per unit **

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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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 5th or 10th hour given 3-sigma control and the readings data: 42, 48, 50, 45, 52, 66, 64, 84, 92, 76. At the 5th 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 10th 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, 9th reading is out of control.

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**If all readings are in control, is the process really in control? **

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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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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**Process Capability Tolerances/Specifications Process variability**

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

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**Process Capability: Specification limits are not control chart limits**

Lower Specification Upper Specification Process variability matches specifications Process variability well within specifications Sampling Distribution is used Process variability exceeds specifications

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**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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**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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**3 Sigma and 6 Sigma Quality**

Lower specification Upper specification Process mean +/- 3 Sigma +/- 6 Sigma

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Chapter 10 Supplement Acceptance Sampling

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Acceptance Sampling Acceptance sampling: Is a lot of N products good if a random sample of n (n<N) products contain only c defects? For example take a sample of 10(=n) milk bottles out of every 100(=N). If 1(=c) or more bottles do not fit specifications, reject the entire lot of 100 bottles. c is determined to balance type I and type II errors. This is a smart compromise between 100% inspection and no inspection. Generally used for input/output inspection.

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**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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