1. Statistical Process Control (SPC)
Stephen R. Lawrence
Assoc. Prof. of Operations Mgmt
Leeds.colorado.edu/faculty/lawrence

2. Process Control Tools

3. Process Control Tools
Process tools assess conditions in existing processes to detect problems that require intervention in order to regain lost control.
Check sheets Pareto analysis
Scatter Plots Histograms
Run Charts Control charts
Cause & effect diagrams

4. Check Sheets 27 15 19 20 28 Check sheets explore what and where
an event of interest is occurring.
Attribute Check Sheet
Order Types am-9am 9am-11am 11am-1pm pm-3pm 3pm-5-pm
Emergency
Nonemergency
Rework
Safety Stock
Prototype Order
Other

5. Run Charts
measurement
time
Look for patterns and trends…

6. SCATTERPLOTS Variable A Variable B
x x x
x x x xx
x x x
x xx x x x
x x xx x
xx x x x
x xx x x x
x x xx x x
x xx xxx x x x
x xx xx x x
x x xx x x
x xxx xx x x
x x xxx x x xx
x x xx xx x x
x x x xx x
x xxx xx xx xxx x
x xx xxx x x x
x x x xx x x x
x x x xx
x x xx x x
xx x x x
Variable A
Larger values of variable A appear to be associated with larger values of variable B.
Variable B

7. Frequency of Occurrences
HISTOGRAMS
A statistical tool used to show the extent
and type of variance within the system.
Frequency of Occurrences
Outcome

8. A B C D E F G PARETO ANALYSIS H Percentage of Occurrences
A method for identifying and separating
the vital few from the trivial many. A B
Percentage of Occurrences C D E F G H I J
Factor

9. CAUSE & EFFECT DIAGRAMS
Classification
Error
Inspection
Pins not
Assigned
Defective
Pins
Received
Damaged
in storage
CPU Chip
BAD
CPU
Maintenance
Equipment
Condition
Employees
Procedures
and Methods
Training
Speed

10. Example: Rogue River Adventures

11. Process Variation

12. Deming’s Theory of Variance
Variation causes many problems for most processes
Causes of variation are either “common” or “special”
Variation can be either “controlled” or “uncontrolled”
Management is responsible for most variation
Categories of Variation
Management
Employee
Controlled Variation
Uncontrolled Variation
Common Cause
Special Cause

13. Causes of Variation What prevents perfection? Process variation...
Natural Causes
Assignable Causes
Inherent to process
Random
Cannot be controlled
Cannot be prevented
Examples
weather
accuracy of measurements
capability of machine
Exogenous to process
Not random
Controllable
Preventable
Examples
tool wear
“Monday” effect
poor maintenance
Common causes: * Inherit to the process
* Random
* Not controllable by operators
* Management is responsible
(e.g., a filling cereal machine may be replaced by a better one)
Assignable causes: * Not part of the process
* Not random
* Operators have control
* Management is responsible for training operators.
(e.g., a machine is not properly set)

14. Specification vs. Variation
Product specification
desired range of product attribute
part of product design
length, weight, thickness, color, ...
nominal specification
upper and lower specification limits
Process variability
inherent variation in processes
limits what can actually be achieved
defines and limits process capability
Process may not be capable of meeting specification!

15. Process Capability

16. Process Capability Capable process (Very) capable process
Process variation
LSL
Spec
USL
Capable process
(Very) capable process
Out of control - The process is out of control because the distribution is not centered around the target value. The assignable cause may be a wrong setting for the particularned.
Not capable - The process is in control but is not capable. This is determined by the large probability of producing parts that do not meet the engineering index (called Cp) is less than 1. Cp has a value of one when the range of the distribution (measure by standard deviations from the mean) equals the range of the tolerances. Capable process have a Cp value of at least 1.3.
Capable - The process is now capable because the entire distribution falls within the tolerance limits. Improvements occur when the range of the distribution is reduce, increasing the probability for producing on target.
Capability - It is the ability to meet customer’s specifications.
In control - A process that does not show signs of assignable causes of variation.
Process not capable

17. Process Capability 6 99.7% 3 LSL Spec USL
Measure of capability of process to meet (fall within) specification limits
Take “width” of process variation as 6
If 6 < (USL - LSL), then at least 99.7% of output of process will fall within specification limits
LSL
Spec
USL 6
99.7% 3

18. Variation -- RazorBlade

19. Process Capability Ratio
Define Process Capability Ratio Cp as
If Cp > 1.0, process is... capable
If Cp < 1.0, process is... not capable

20. Process Capability -- Example
A manufacturer of granola bars has a weight specification
2 ounces plus or minus 0.05 ounces. If the standard deviation
of the bar-making machine is 0.02 ounces, is the process capable?
USL = = 2.05 ounces
LSL = = 1.95 ounces
Cp = (USL - LSL) / 6
= ( ) / 6(0.02)
= / 0.12 =
Therefore, the process is not capable!

21. Process Centering Capable and centered Capable, but not centered
LSL
Spec
USL
Capable and centered
Capable, but not centered
Out of control - The process is out of control because the distribution is not centered around the target value. The assignable cause may be a wrong setting for the particularned.
Not capable - The process is in control but is not capable. This is determined by the large probability of producing parts that do not meet the engineering index (called Cp) is less than 1. Cp has a value of one when the range of the distribution (measure by standard deviations from the mean) equals the range of the tolerances. Capable process have a Cp value of at least 1.3.
Capable - The process is now capable because the entire distribution falls within the tolerance limits. Improvements occur when the range of the distribution is reduce, increasing the probability for producing on target.
Capability - It is the ability to meet customer’s specifications.
In control - A process that does not show signs of assignable causes of variation.
Not capable, and
not centered

22. Process Centering -- Example
For the granola bar manufacturer, if the process is
incorrectly centered at 2.05 instead of 2.00 ounces, what
fraction of bars will be out of specification?
2.0
LSL=1.95
USL=2.05
Out of spec!
50% of production will be out of specification!

23. Process Capability Index Cpk
Std dev 
Mean m
If Cpk > 1.0, process is... Centered & capable
If Cpk < 1.0, process is... Not centered &/or not capable

24. Cpk Example 1
A manufacturer of granola bars has a weight specification
2 ounces plus or minus 0.05 ounces. If the standard deviation
of the bar-making machine is s = 0.02 ounces and the process mean is m = 2.01, what is the process capability index?
USL = 2.05 oz LSL = 1.95 ounces
Cpk = min[(m -LSL) / 3 , (USL- m) / 3 ]
= min[(2.01–1.95) / 0.06 , (2.05 – 2.01) / 0.06 ]
= min[1.0 , 0.67 ]
= 0.67
Therefore, the process is not capable and/or not centered !

25. Cpk Example 2
Venture Electronics manufactures a line of MP3 audio players. One of the components manufactured by Venture and used in its players has a nominal output voltage of 8.0 volts. Specifications allow for a variation of plus or minus 0.6 volts. An analysis of current production shows that mean output voltage for the component is volts with a standard deviation of volts. Is the process "capable: of producing components that meet specification? What fraction of components will fall outside of specification? What can management do to improve this fraction?

26. Process Control Charts

27. Process Control Charts
Statistical technique for tracking a process and
determining if it is going “out to control”
Establish capability of process under normal conditions
Use normal process as benchmark to statistically identify abnormal process behavior
Correct process when signs of abnormal performance first begin to appear
Control the process rather than inspect the product!

28. Process Control Charts
Upper Spec Limit
Upper Control Limit 6
Target Spec 3
Lower Control Limit
Lower Spec Limit

29. Process Control Charts
Look for
special
cause !
In control
Out of control !
Back in control!
UCL
Target
LCL
Time
Samples
Natural variation

30. When to Take Action
A single point goes beyond control limits (above or below)
Two consecutive points are near the same limit (above or below)
A run of 5 points above or below the process mean
Five or more points trending toward either limit
A sharp change in level
Other erratic behavior

31. Samples vs. Population Sample Distribution Population Distribution
Mean

32. Types of Control Charts
Attribute control charts
Monitors frequency (proportion) of defectives
p - charts
Defects control charts
Monitors number (count) of defects per unit
c – charts
Variable control charts
Monitors continuous variables
x-bar and R charts

33. 1. Attribute Control Charts
p - charts
Estimate and control the frequency of defects in a population
Examples
Invoices with error s (accounting)
Incorrect account numbers (banking)
Mal-shaped pretzels (food processing)
Defective components (electronics)
Any product with “good/not good” distinctions

34. Using p-charts
Find long-run proportion defective (p-bar) when the process is in control.
Select a standard sample size n
Determine control limits

35. p-chart Example
Chic Clothing is an upscale mail order clothing company selling merchandise to successful business women. The company sends out thousands of orders five days a week. In order to monitor the accuracy of its order fulfillment process, 200 orders are carefully checked every day for errors. Initial data were collected for 24 days when the order fulfillment process was thought to be "in control." The average percent defective was found to be 5.94%.

36. 2. Defect Control Charts c-charts
Estimate & control the number of defects per unit
Examples
Defects per square yard of fabric
Crimes in a neighborhood
Potholes per mile of road
Bad bytes per packet
Most often used with continuous process (vs. batch)

37. Using c-charts
Find long-run proportion defective (c-bar) when the process is in control.
Determine control limits

38. 2. c-chart Example
Dave's is a restaurant chain that employs independent evaluators to visit its restaurants as secret shoppers to the asses the quality of service. The company evaluates restaurants in two categories, food quality, and service (promptness, order accuracy, courtesy, friendliness, etc.) The evaluator considers not only his/her order experiences, but also evaluations throughout the restaurant. Initial surveys find that the total number of service defects per survey is 7.3 when a restaurant is operating normally.

39. 3. Control Charts for Variables
x-bar and R charts
Monitor the condition or state of continuously variable processes
Use to control continuous variables
Length, weight, hardness, acidity, electrical resistance
Examples
Weight of a box of corn flakes (food processing)
Departmental budget variances (accounting
Length of wait for service (retailing)
Thickness of paper leaving a paper-making machine

40. x-bar and R charts Two things can go wrong Two control solutions
process mean goes out of control
process variability goes out of control
Two control solutions
X-bar charts for mean
R charts for variability

41. Problems with Continuous Variables
“Natural”
Process
Distribution
Mean not
Centered
Increased
Variability
Target

42. Range (R) Chart Choose sample size n
Determine average in-control sample ranges R-bar where R=max-min
Construct R-chart with limits:

43. Mean (x-bar) Chart Choose sample size n (same as for R-charts)
Determine average of in-control sample means (x-double-bar)
x-bar = sample mean
k = number of observations of n samples
Construct x-bar-chart with limits:

44. x & R Chart Parameters

45. R and x-bar Chart Example
Resistors for electronic circuits are being manufactured on a high-speed automated machine. The machine is set up to produce resistors of 1,000 ohms each. Fifteen samples of 4 resistors each were taken over a period of time when the machine was operating normally. The average range of the samples was found to be R-bar=21.7 and the average mean of the samples was x-double-bar=999.1.

46. When to Take Action
A single point goes beyond control limits (above or below)
Two consecutive points are near the same limit (above or below)
A run of 5 points above or below the process mean
Five or more points trending toward either limit
A sharp change in level
Other statistically erratic behavior

47. Control Chart Error Trade-offs
Setting control limits too tight (e.g., m ± 2) means that normal variation will often be mistaken as an out-of-control condition (Type I error).
Setting control limits too loose (e.g., m ± 4) means that an out-of-control condition will be mistaken as normal variation (Type II error).
Using control limits works well to balance Type I and Type II errors in many circumstances.
3s is not sacred -- use judgement.

48. Video: SPC at Frito Lay

49. Statistical Process Control