1. Operations Management
Supplement 6 – Statistical Process Control
PowerPoint presentation to accompany
Heizer/Render
Principles of Operations Management, 6e
Operations Management, 8e
© 2006 Prentice Hall, Inc.

2. Outline Statistical Process Control (SPC) Process Capability
Control Charts for Variables
Setting Mean Chart Limits (x-Charts)
Setting Range Chart Limits (R-Charts)
Process Capability
Process Capability Ratio (Cp)
Process Capability Index (Cpk )

3. Statistical Process Control (SPC)
Variability is inherent in every process
Natural or common causes
Special or assignable causes
SPC charts provide statistical signals when assignable causes are present
SPC approach supports the detection and elimination of assignable causes of variation
Points which might be emphasized include:
- Statistical process control measures the performance of a process, it does not help to identify a particular specimen produced as being “good” or “bad,” in or out of tolerance.
- Statistical process control requires the collection and analysis of data - therefore it is not helpful when total production consists of a small number of units
- While statistical process control can not help identify a “good” or “bad” unit, it can enable one to decide whether or not to accept an entire production lot. If a sample of a production lot contains more than a specified number of defective items, statistical process control can give us a basis for rejecting the entire lot. The issue of rejecting a lot which was actually good can be raised here, but is probably better left to later.

4. Inspection
Inspection is the activity that is done to ensure that an operation is producing the results expected (post production activity).
Where to inspect
At point of product design
At point of product production (source)
At point of product assembly
At point of product dispatch to customer
At point of product reception by customer

5. What to Inspect Variables of an entity Attributes of an entity
Degree of deviation from a target (continuum scale)
Lifespan of device
Reliability or accuracy of device
Attributes of an entity
Classifies attributes into discrete classes such as good versus bad, or pass or fail
Maximum weight of bag at airport
Minimum height for exit seat

6. Data Used for Quality Judgments
Variables
Attributes
Characteristics that can take any real value
May be in whole or in fractional numbers
Continuous random variables, e.g. weight, length, duration, etc.
Defect-related characteristics
Classify products as either good or bad or count defects
Categorical or discrete random variables
Once the categories are outlined, students may be asked to provide examples of items for which variable or attribute inspection might be appropriate. They might also be asked to provide examples of products for which both characteristics might be important at different stages of the production process.

7. How to Inspect: Methods
Visual inspection
Manual inspection (weigh, count )
Mechanical inspection (machine-based)
Testing of device

8. Disadvantages of Inspection
Most inspections are not done at the source
Errors are discovered after its too late
No link between error and the cause of errors
Errors are often too costly to correct
As products grow in number more staff are needed for inspection
As products grow in number, more time is needed for inspection
Most inspections involve the inspection of good parts as well as bad ones
CHALLENGE: How could one achieve high product quality without having to inspect all goods produced?

9. Statistical Process Control
A statistics-based approach for monitoring and inspecting results of a process, through the gathering, structuring, and analyses of product variables/attributes, as well as the taking of corrective action at the source, during the production process.

10. Class Example
What can we learn from the results of the bodyguards?

11. Common Cause Variations
Also called natural causes
Affects virtually all production processes
Generally this requires some change at the systemic level of an organization
Managerial action is often necessary
The objective is to discover avoidable common causes present in processes
Eliminate (when possible) the root causes of the common variations, e.g. different arrival times of suppliers, different arrival times of customers, weight of products poured in box by machine

12. Assignable Variations
Also called special causes of variation
Generally this is caused by some change in the local activity or process
Variations that can be traced to a specific reason at a localized activity
The objective is to discover special causes that are present
Eliminate the root causes of the special variations, e.g. machine wear, material quality, fatigued workers, misadjusted equipment
Incorporate the good process control

13. SPC and Statistical Samples
To conduct inspection using the SPC approach, one has to compute averages for several small samples instead of using data from individual items: Steps
Each of these represents one sample of five boxes of cereal
(a) Samples of the product, say five boxes of cereal taken off the filling machine line, vary from each other in weight
Frequency
Weight #
Figure S6.1

14. The solid line represents the distribution
SPC Sampling
To measure the process, we take samples of same size at different times. We plot the mean of each sample for each point in time
The solid line represents the distribution
(b) After enough samples are taken from a stable process, they form a pattern called a distribution
Frequency
Weight
Figure S6.1

15. Attributes of Distributions
(c) There are many types of distributions, including the normal (bell-shaped) distribution, but distributions do differ in terms of central tendency (mean), standard deviation or variance, and shape
Weight
Central tendency
Variation
Shape
Frequency
Figure S6.1

16. Identifying Presence of Common Sources of Variation
(d) If only common causes of variation are present, the output of a process forms a distribution that is stable over time and is predictable
Prediction
Weight
Time
Frequency
Figure S6.1

17. Identifying Presence of Special Sources of Variation
Prediction ?
(e) If assignable causes are present, the process output is not stable over time and is not predicable
Weight
Time
Frequency
Figure S6.1

18. Central Limit Theorem
Regardless of the distribution of the population, the distribution of sample means drawn from the population will tend to follow a normal curve
The mean of the sampling distribution (x) will be the same as the population mean m
x = m
This slide introduces the difference between “natural” and “assignable” causes.
The next several slides expand the discussion and introduce some of the statistical issues.
The standard deviation of the sampling distribution (sx) will equal the population standard deviation (s) divided by the square root of the sample size, n s n
sx =

19. Interpreting SPC Charts
(a) In statistical control and capable of producing within control limits
Frequency
Lower Control Limit
Upper Control Limit
(b) In statistical control but not capable of producing within control limits
This slide helps introduce different process outputs.
It can also be used to illustrate natural and assignable variation.
(c) Out of statistical control and incapable of producing within limits
(weight, length, speed, etc.)
Size
Figure S6.2

20. Population and Sampling Distributions
Three population distributions
Beta
Normal
Uniform
Distribution of sample means
Standard deviation of the sample means
= sx = s n
Mean of sample means = x
| | | | | | |
-3sx -2sx -1sx x +1sx +2sx +3sx
99.73% of all x
fall within ± 3sx
95.45% fall within ± 2sx
Figure S6.3

21. Sampling Distribution
Sampling distribution of means
Process distribution of means
x = m
(mean)
It may be useful to spend some time explicitly discussing the difference between the sampling distribution of the means and the mean of the process population.
Figure S6.4

22. Steps In Creating Control Charts
Take representative sample from output of a process over a long period of time, e.g. 10 units every hour for 24 hours.
Compute means and ranges for the variables and calculate the control limits
Draw control limits on the control chart
Plot a chart for the means and another for the mean of ranges on the control chart
Determine state of process (in or out of control)
Investigate possible reasons for out of control events and take corrective action
Continue sampling of process output and reset the control limits when necessary

23. In-Class Exercise : Control Charts
6/15
6/16
8AM
10AM
12AM
2PM 5 6 8
6.5
5.5 7
7.5
Calculate X bar and R’s for new data
Calculate X double bar and R bar figures for new data
Draw X bar chart Calculate LCL and UCL for X bar chart Draw lines for LCL and UCL and for X double bar in chart

24. Control Charts for Variables
For variables that have continuous dimensions
Weight, speed, length, strength, etc.
x-charts are to control the central tendency of the process
R-charts are to control the dispersion of the process
These two charts must be used together

25. Setting Chart Limits For x-Charts when we know s
Upper control limit (UCL) = x + zsx
Lower control limit (LCL) = x - zsx
where x = mean of the sample means or a target value set for the process
z = number of normal standard deviations
sx = standard deviation of the sample means
= s/ n
s = population standard deviation
n = sample size

26. Setting Control Limits
Hour 1
Sample Weight of Number Oat Flakes
1 17
2 13
3 16
4 18
5 17
6 16
7 15
8 17
9 16
Mean 16.1
s = 1
Hour Mean Hour Mean
n = 9
For 99.73% control limits, z = 3
UCLx = x + zsx = (1/3) = 17 ozs
LCLx = x - zsx = (1/3) = 15 ozs

27. Setting Control Limits
Control Chart for sample of 9 boxes
Variation due to assignable causes
Out of control
Sample number
| | | | | | | | | | | |
17 = UCL
15 = LCL
16 = Mean
Variation due to natural causes
Out of control

28. Setting Chart Limits For x-Charts when we don’t know s
Upper control limit (UCL) = x + A2R
Lower control limit (LCL) = x - A2R
where R = average range of the samples
A2 = control chart factor found in Table S6.1
x = mean of the sample means

29. Control Chart Factors Sample Size Mean Factor Upper Range Lower Range
n A2 D4 D3
Table S6.1

30. Setting Control Limits
Process average x = ounces
Average range R = .25
Sample size n = 5

31. Setting Control Limits
Process average x = ounces
Average range R = .25
Sample size n = 5
UCLx = x + A2R
= (.577)(.25) =
= ounces
From Table S6.1

32. Setting Control Limits
Process average x = ounces
Average range R = .25
Sample size n = 5
UCL =
Mean = 16.01
LCL =
UCLx = x + A2R
= (.577)(.25) =
= ounces
LCLx = x - A2R =
= ounces

33. R – Chart Type of variables control chart
Shows sample ranges over time
Difference between smallest and largest values in sample
Monitors process variability
Independent from process mean

34. Setting Chart Limits For R-Charts Upper control limit (UCLR) = D4R
Lower control limit (LCLR) = D3R
where
R = average range of the samples
D3 and D4 = control chart factors from Table S6.1

35. Setting Control Limits
Average range R = 5.3 pounds
Sample size n = 5
From Table S6.1 D4 = 2.115, D3 = 0
UCL = 11.2
Mean = 5.3
LCL = 0
UCLR = D4R
= (2.115)(5.3)
= 11.2 pounds
LCLR = D3R
= (0)(5.3)
= 0 pounds

36. Mean and Range Charts (a)
These sampling distributions result in the charts below
(Sampling mean is shifting upward but range is consistent)
x-chart
(x-chart detects shift in central tendency)
UCL
LCL
R-chart
(R-chart does not detect change in spread)
UCL
LCL
Figure S6.5

37. Mean and Range Charts (b)
These sampling distributions result in the charts below
(Sampling mean is constant but dispersion is increasing)
x-chart
(x-chart does not detect the increase in mean)
UCL
LCL
R-chart
(R-chart detects increase in dispersion)
UCL
LCL
Figure S6.5

38. Automated Control Charts

39. Control Charts for Attributes
For variables that are categorical
Good/bad, yes/no, acceptable/unacceptable
Measurement is typically counting defectives
Charts may measure
Percent defective (p-chart)
Number of defects (c-chart)

40. Patterns in Control Charts
Upper control limit
Target
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
Normal behavior. Process is “in control.”
Figure S6.7

41. Patterns in Control Charts
Upper control limit
Target
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
One plot out above (or below). Investigate for cause. Process is “out of control.”
Figure S6.7

42. Patterns in Control Charts
Upper control limit
Target
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
Trends in either direction, 5 plots. Investigate for cause of progressive change.
Figure S6.7

43. Patterns in Control Charts
Upper control limit
Target
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
Run of 5 above (or below) central line. Investigate for cause.
Figure S6.7

44. Process Capability
The natural variation of a process should be small enough to produce products that meet the standards required
A process in statistical control does not necessarily meet the design specifications
Process capability is a measure of the relationship between the natural variation of the process and the design specifications

45. Process Capability Ratio
Cp =
Upper Specification - Lower Specification 6s
A capable process must have a Cp of at least 1.0
Does not look at how well the process is centered in the specification range
Often a target value of Cp = 1.33 is used to allow for off-center processes
Six Sigma quality requires a Cp = 2.0

46. Process Capability Ratio
Insurance claims process
Process mean x = minutes
Process standard deviation s = .516 minutes
Design specification = 210 ± 3 minutes
Cp =
Upper Specification - Lower Specification 6s

47. Process Capability Ratio
Insurance claims process
Process mean x = minutes
Process standard deviation s = .516 minutes
Design specification = 210 ± 3 minutes
Cp =
Upper Specification - Lower Specification 6s
= = 1.938
6(.516)

48. Process Capability Ratio
Insurance claims process
Process mean x = minutes
Process standard deviation s = .516 minutes
Design specification = 210 ± 3 minutes
Cp =
Upper Specification - Lower Specification 6s
= = 1.938
6(.516)
Process is capable

49. Process Capability Index
Cpk = minimum of ,
Upper Specification - x Limit 3s
Lower x - Specification Limit
A capable process must have a Cpk of at least 1.0
A capable process is not necessarily in the center of the specification, but it falls within the specification limit at both extremes

50. Process Capability Index
New Cutting Machine
New process mean x = .250 inches
Process standard deviation s = inches
Upper Specification Limit = .251 inches
Lower Specification Limit = .249 inches

51. Process Capability Index
New Cutting Machine
New process mean x = .250 inches
Process standard deviation s = inches
Upper Specification Limit = .251 inches
Lower Specification Limit = .249 inches
Cpk = minimum of ,
(.251)
(3).0005

52. Process Capability Index
New Cutting Machine
New process mean x = .250 inches
Process standard deviation s = inches
Upper Specification Limit = .251 inches
Lower Specification Limit = .249 inches
Cpk = minimum of ,
(.251)
(3).0005
(.249)
Both calculations result in
New machine is NOT capable
Cpk = = 0.67
.001
.0015

53. Process Capability Comparison
New Cutting Machine
New process mean x = .250 inches
Process standard deviation s = inches
Upper Specification Limit = .251 inches
Lower Specification Limit = .249 inches
Upper Specification - Lower Specification 6s
Cp =
New machine is NOT capable
.0030
Cp = =

54. Interpreting Cpk Cpk = negative number Cpk = zero
Cpk = between 0 and 1
Cpk = 1
Cpk > 1
Figure S6.8

55. Acceptance Sampling
Form of quality testing used for incoming materials or finished goods
Take samples at random from a lot (shipment) of items
Inspect each of the items in the sample
Decide whether to reject the whole lot based on the inspection results
Only screens lots; does not drive quality improvement efforts
Here again it is useful to stress that acceptance sampling relates to the aggregate, not the individual unit. You might also discuss the decision as to whether one should take only a single sample, or whether multiple samples are required.