1. Chapter 10
Quality Control

2. 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

3. 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

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
Inputs
Transformation
Outputs
Acceptance
sampling
Process
control

5. Examples of Inspection Points

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

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.

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

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=

10. 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.

11. Normal Distribution normdist(x,.,.,1) normdist(x,.,.,0) x Probab Mean




95.44%
99.74%

12. Cumulative Normal Density1
prob
normdist(x,mean,st_dev,1) x
norminv(prob,mean,st_dev)

13. 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)

14. 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

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.

16. Type I Error=Alpha
Mean
LCL
UCL
/2
Probability of Type I error

17. 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

18. Observations from Sample Distribution
Sample number
UCL
LCL 1 2 3 4

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)

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.

21. 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?

22. 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

23. 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

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

25. 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

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 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.

27. 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

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

29. 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

30. 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

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

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

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

34. Chapter 10 Supplement
Acceptance Sampling

35. 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.

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”.