1. QUALITY CONTROL AND SPC
CHAPTER 16
DAVID A. COLLIER AND JAMES R. EVANS

2. Statistical Process Control and Variation
Statistical process control (SPC) is a methodology for monitoring quality of manufacturing and service delivery processes to help identify and eliminate unwanted causes of variation.

3. Statistical Process Control and Variation
Common cause variation is the result of complex interactions of variations in materials, tools, machines, information, workers, and the environment.
Common cause variation accounts for 80 to 95 percent of the observed variation in a process.
Only management has the power to change systems and infrastructure that cause common cause variation.

4. Statistical Process Control and Variation
Special (assignable) cause variation arises from external sources that are not inherent in the process, appear sporadically, and disrupt the random pattern of common causes.
Special cause variation accounts for 15 to 20 percent of observed variation.
Front-line employees and supervisors have the power to identify and solve special causes of variation.

5. Statistical Process Control and Variation
If no special causes affect the output of a process, we say that the process is in control.
When special causes are present in a process, the process is said to be out of control.

6. Constructing Control Charts
Preparation
Choose the metric to be monitored.
Determine the basis, size, and frequency of sampling.
Set up the control chart.
Data collection
Record the data.
Calculate relevant statistics: averages, ranges, proportions, and so on.
Plot the statistics on the chart.

7. Constructing Control Charts
Determination of trial control limits
Draw the center line (process average) on the chart.
Compute the upper and lower control limits.
Analysis and interpretation
Investigate the chart for lack of control.
Eliminate out-of-control points.
Recompute control limits if necessary.

8. Constructing Control Charts
Use as a problem-solving tool
Continue data collection and plotting.
Identify out-of-control situations and take corrective action.
Determination of process capability using the control chart data

9. Constructing Control Charts
A continuous metric is one that is calculated from data that are measured as the degree of conformance to a specification on a continuous scale of measurement.
A discrete metric is one that is calculated from data that are counted.

10. Constructing x- and R-Charts
Collect k = samples of n = 3-10 observations each.
Compute the mean and range of each sample.
Compute the overall mean and average range:
Compute the control limits:
[16.1]
[16.2]
[16.3]

11. Solved Problem
Goodman Tire periodically tests its tires for tread wear under simulated road conditions. The company collects twenty samples, each containing three radial tires from different shifts over several days of operations. The overall mean is computed as 31.88, and the average range is 10.8.

12. Exhibit 16.1 Excel Template for Goodman Tire x-bar and R-Charts

13. Solved Problem
Control limits are:

14. Exhibit 16.2 R-Chart for Goodman Tire Example

15. Exhibit 16.3 x-Chart for Goodman Tire Example

16. Interpreting Patterns in Control Charts
A process is said to be “in control” when the control chart has the following characteristics:
No points are outside the control limits (the traditional and most popular SPC chart guideline).
The number of points above and below the center line is about the same.
The points seem to fall randomly above and below the center line.
Most points, but not all, are near the center line, and only a few are close to the control limits.

17. Interpreting Patterns in Control Charts
Rules for identifying a shift in the process:
8 points in a row above or below the center line.
10 of 11 consecutive points above or below the center line.
12 of 14 consecutive points above or below the center line.
2 of 3 consecutive points in the outer one-third region between the center line and one of the control limits.
4 of 5 consecutive points in the outer two-thirds region between the center line and one of the control limits.

18. Exhibit Extra Illustration of Some Rules for Identifying Out-of-Control Conditions

19. Constructing p-Charts
A p-chart monitors the proportion of nonconforming items (also called a fraction nonconforming or fraction defective chart).
pi is the fraction nonconforming in sample i.
Average fraction nonconforming:
Standard deviation:
[16.4]
[16.5]

20. Constructing p-Charts
Control limits:
[16.6]

21. Solved Problem
The operators of automated sorting machines in a post office must read the ZIP code on letters and divert the letters to the proper carrier routes. Over a month’s time, 25 samples of 100 letters were chosen, and the number of errors was recorded.
The average proportion defective, p is computed as The standard deviation is:
UCL = (.01467) = .066, and LCL = (.01467) =
Since the LCL is negative and the actual proportion nonconforming cannot be less than zero, the LCL is set equal to zero.

22. Exhibit 16.4
Data and Calculations for the p-Chart Solved Problem

23. Exhibit 16.5 p-Chart for Solved Problem

24. √ √ Constructing c-charts UCLc = c + 3 c LCLc = c - 3 c
A c-chart monitors the number of nonconformances (i.e., a count of the number of defects or errors) per unit, when the size of the sampling unit or number of opportunities for errors is constant.
Example: A purchase order may have several errors, such as wrong items, order quantity, or wrong price.
c is the average number of nonconformances/unit. Control limits are given by:
UCLc = c + 3 √ c
[16.7]
LCLc = c - 3 √ c

25. Solved Problem
The total number of machine failures over a 25-day period is 45.
The average number of failures per day, c, is 45/25 = 1.8.
Control limits for the c-chart are:

26. Exhibit 16.6
Machine Failure Data for c-Chart Solved Problem

27. Exhibit 16.7 c-Chart for Machine Failures

28. Practical Issues in SPC Implementation
Sample size: Small sample size keeps costs lower; however, large sample sizes provide greater degrees of statistical accuracy in estimating the true state of control.
In practice, samples of about 5 have been found to work well in detecting process shifts of 2 standard deviations or larger. To detect smaller shifts in the process mean, larger sample sizes of 15 to 25 must be used.

29. Controlling Six Sigma Processes
SPC is a useful methodology for processes that operate at a low sigma level (less than or equal to 3-sigma).
For processes with a high sigma level (greater than 3-sigma), few defects will be discovered even with large sample sizes.
More advanced tools are necessary.

30. Process Capability
Process capability refers to the natural variation in a process that results from common causes.
A process capability study is a carefully planned study designed to yield specific information about the performance of a process under specified operating conditions.
Typical questions:
Where is the process centered?
How much variability exists in the process?
Is the performance relative to specifications acceptable?
What proportion of output will be expected to meet specifications?

31. Process Capability Index
The relationship between the natural variation and specifications is often quantified by a measure known as the process capability index.
Cp = (UTL – LTL) [16.9] 6σ
Where:
UTL = Upper tolerance limit
LTL = Lower tolerance limit
σ = Standard deviation of the process (or an estimate based on the sample standard deviation, s)

32. Process Capability Index
When Cp = 1, the natural variation is the same as the design specification width.
When Cp < 1, a significant percentage of output will not conform to the specifications.
When Cp > 1, nearly all the output will conform to the specifications.

33. Exhibit 16.8 c-Process Capability versus Design Specifications

34. Process Capability Index
Many firms require Cp values of 1.66 or greater from their suppliers, which equates to a tolerance range of about 10 standard deviations.
The value of Cp does not depend on the mean of the process (it depends on the specification range and actual process standard deviation); thus, a process may be off-center, and still show an acceptable value of Cp.

35. Process Capability Index
One-sided process capability indexes consider off-centering in a process:
Cpu = (UTL – µ)/3σ (upper one-sided index) [16.10]
Cpl = (µ – LTL)/3σ (lower one-sided index) [16.11]
Cpk = min (Cpl, Cpu) [16.12]

36. Solved Problem
A controlled process shows an overall mean of 2.50 and an average range of Samples of size 4 were used to construct the control charts. If specifications are 2.60 ± 0.25, how well can this process meet them? What are the process capability indexes?

37. Solution UTL = 2.60 + 0.25 = 2.85 LTL = 2.60 – 0.25 = 2.35
We estimate the standard deviation using equation From Appendix B, d2 = and s = R/d2 = 0.42/2.059 = 0.20.
Actual process performance is 2.50 ± 3(0.20), or ranges from 1.90 to This variation exceeds the specifications by a large amount and is also off-center.

38. Exhibit 16.9 Comparison of Observed Variation and Design Specifications for Solved Problem