1. Chapter 9A. Process Capability & Statistical Quality Control
Outline:
Basic Statistics
Process Variation
Process Capability
Process Control Procedures
Variable data
X-bar chart and R-chart
Attribute data
p-chart
Acceptance Sampling
Operating Characteristic Curve

2. Focus
This technical note on statistical quality control (SQC) covers the quantitative aspects of quality management
SQC is a number of different techniques designed to evaluate quality from a conformance view
How are we doing in meeting specifications?
SQC can be applied to both manufacturing and service processes
SQC techniques usually involve periodic sampling of the process and analysis of data
Sample size
Number of samples
SQC techniques are looking for variance
Most processes produce variance in output
we need to monitor the variance (and the mean also) and correct processes when they get out of range

3. Basic Statistics Mean Standard Deviation
99.7% μ
Normal Distributions have a mean (μ) and a standard deviation (σ)
For a sample of N observations:
Mean
where:
xi = Observed value
N = Total number of observed values
Standard Deviation 3

4. Statistics and Probability
SQC relies on central limit theorem and normal dist.
We establish the Upper Control Limits (UCL) and the Lower Control Limits (LCL) with plus or minus 3 standard deviations. Based on this we can expect 99.7% of our sample observations to fall within these limits.
Acceptance sampling relies on Binomial and Hyper geometric probability concepts
-3s
-2s
+2s
+3s
99.7%
LCL
UCL
a/2
a = Prob. of Type I error

5. Basic Stats
Using SQC,
samples of a process output are taken, and
sample statistics are calculated
The purpose of sampling is to find when the process has changed in some nonrandom way
The reason for the change can then be quickly determined and corrected
This allows us to detect changes in the actual distribution process 3

6. Variation
Random (common) variation is inherent in the production process.
Assignable variation is caused by factors that can be clearly identified and possibly managed
Using a saw to cut 2.1 meter long boards as a sample process
Discuss random vs. assignable variation
Generally, when variation is reduced, quality improves.
It is impossible to have zero variability. T or F ?

7. Taguchi’s View of Variation
Traditional view is that quality within the LS and US is good and that the cost of quality outside this range is constant, where Taguchi views costs as increasing as variability increases, so seek to achieve zero defects and that will truly minimize quality costs.
Incremental
Cost of
Variability
High
Zero
Lower
Spec
Target
Upper
Traditional View
Incremental
Cost of
Variability
High
Zero
Lower
Spec
Target
Upper
Taguchi’s View
Exhibits TN8.1 & TN8.2 30

8. Process Capability Tolerance (specification, design) Limits
Bearing diameter inches
LTL = inches UTL = inches
Process Limits
The actual distribution from the process
Run the process to make 100 bearings, compute the mean and std. dev. (and plot/graph the complete results)
Suppose, mean = 1.250, std. dev = 0.002
How do they relate to one another? 28

9. Tolerance Limits vs. Process Capability
Specification Width
Actual Process Width
Specification Width
Actual Process Width

10. Process Capability Example
Design Specs: Bearing diameter inches
LTL = inches UTL = inches
The actual distribution from the process  mean = 1.250, s = 0.002
+- 3s limits  (0.002)  [1.244, 1.256]
Anew process,  std. dev. =

11. Process Capability Index, Cpk
Capability Index shows how well parts being produced fit into design limit specifications
Compute the Cpk for the bearing example.
Old process, mean = 1.250, s = 0.002
What is the probability of producing defective bearings?
New process, mean = 1.250, s= , re-compute the Cpk
When the computed (sample) mean = design (target) mean, what does that imply?

12. The Cereal Box Example
Recall the cereal example. Consumer Reports has just published an article that shows that we frequently have less than 15 ounces of cereal in a box.
Let’s assume that the government says that we must be within ± 5 percent of the weight advertised on the box.
Upper Tolerance Limit = (16) = 16.8 ounces
Lower Tolerance Limit = 16 – 0.05(16) = 15.2 ounces
We go out and buy 1,000 boxes of cereal and find that they weight an average of ounces with a standard deviation of ounces.

13. Cereal Box Process Capability
Specification or Tolerance Limits
Upper Spec = 16.8 oz,
Lower Spec = 15.2 oz
Observed Weight
Mean = oz, Std Dev = oz
What does a Cpk of mean?
Many companies look for a Cpk of 1.3 or better… 6-Sigma company wants 2.0!

14. Types of Statistical Sampling
Sampling to accept or reject the immediate lot of product at hand (Acceptance Sampling).
Attribute (Binary; Yes/No; Go/No-go information)
Defectives refers to the acceptability of product across a range of characteristics.
Defects refers to the number of defects per unit which may be higher than the number of defectives.
p-chart application
Sampling to determine if the process is within acceptable limits (Statistical Process Control)
Variable (Continuous)
Usually measured by the mean and the standard deviation.
X-bar and R chart applications

15. Evidence for Investigation…

16. Control Limits
If we establish control limits at +/- 3 standard deviations, then we would expect 99.7% of our observations to fall within these limits x
LCL
UCL

17. Attribute Measurements (p-Chart)
Item is “good” or “bad”
Collect data, compute average fraction bad (defective) and std. dev. using:
The, UCL, LCL using:
Excel time!

18. Variable Measurements (x-Bar and R Charts)
A variable of the item is measured (e.g., weight, length, salt content in a bag of chips)
Note that the item (sample) is not declared good or bad
Since the actual the standard deviation of the process is not known (and it may indeed fluctuate also) we use the sample data to compute the UCL & LCL
For 3-sigma limits, factors A2 , D3 , and D4 and are given in Exhibit 9A.6, p. 341
Excel time!

19. Acceptance Sampling vs. SPC
Sampling to accept or reject the immediate lot of product at hand (Acceptance Sampling).
Determine quality level
Ensure quality is within predetermined (agreed) level
Sampling to determine if the process is within acceptable limits - Statistical Process Control (SPC) 3 3

20. Acceptance Sampling Advantages Economy Less handling damage
Fewer inspectors
Upgrading of the inspection job
Applicability to destructive testing
Entire lot rejection (motivation for improvement)
Disadvantages
Risks of accepting “bad” lots and rejecting “good” lots
Added planning and documentation
Sample provides less information than 100-percent inspection

21. n the sample size (how many units to sample from a lot)
A Single Sampling Plan
A Single Sampling Plan simply requires two parameters to be determined:
n the sample size (how many units to sample from a lot)
c the maximum number of defective items that can be found in the sample before the lot is rejected.

22. RISK RISKS for the producer and consumer in sampling plans:
Acceptable Quality Level (AQL)
Max. acceptable percentage of defectives defined by producer.
a (Producer’s risk)
The probability of rejecting a good lot.
Lot Tolerance Percent Defective (LTPD)
Percentage of defectives that defines consumer’s rejection point.
 (Consumer’s risk)
The probability of accepting a bad lot.

23. Operating Characteristic Curve
The OCC brings the concepts of producer’s risk, consumer’s risk, sample size, and maximum defects allowed together
n = 99
c = 4
AQL
LTPD
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 2 3 4 5 6 7 8 9 10 11 12
Percent defective
Probability of acceptance
B =.10
(consumer’s risk)
a = .05 (producer’s risk)
The shape or slope of the curve is dependent on a particular combination of the four parameters 9 9

24. Example: Acceptance Sampling
Zypercom, a manufacturer of video interfaces, purchases printed wiring boards from an outside vender, Procard. Procard has set an acceptable quality level of 1% and accepts a 5% risk of rejecting lots at or below this level. Zypercom considers lots with 3% defectives to be unacceptable and will assume a 10% risk of accepting a defective lot.
Develop a sampling plan for Zypercom and determine a rule to be followed by the receiving inspection personnel. 10 10

25. Developing A Single Sampling Plan
Determine:
AQL? a?
LTPD? ?
Divide LTPD by AQL  0.03/0.01 = 3
Then find the value for “c” by selecting the value in the TN8.10 “n(AQL)”column that is equal to or just greater than the ratio above (3).
Thus, c = 6
From the row with c=6, get  nAQL = and divide it by AQL
3.286/0.01 = 328.6, round up to 329,  n = 329
Problems: 3, 4, 6, 7, 8, 9, 11, 13 c
LTPD/AQL
n AQL
44.890
0.052 5
3.549
2.613 1
10.946
0.355 6
3.206
3.286 2
6.509
0.818 7
2.957
3.981 3
4.890
1.366 8
2.768
4.695 4
4.057
1.970 9
2.618
5.426
Sampling Plan:
Take a random sample of 329 units from a lot.
Reject the lot if more than 6 units are defective.