1. OPSM301 Operations Management Spring 2012
Class 23:
Statistical Quality Control

2. Statistical Quality Control
The quantitative aspects of quality management
Processes usually exhibit some variation in their output
Assignable variation: variation that is caused by factors that can be identified and managed
Common variation: variation that is inherent in the process itself

3. Natural Variations Also called common causes
Affect virtually all production processes
Expected amount of variation, inherent due to:
- the nature of the system
- the way the system is managed
- the way the process is organised and operated
can only be removed by
- making modifications to the process
- changing the process
Output measures follow a probability distribution
For any distribution there is a measure of central tendency and dispersion

4. Assignable Variations
Also called special causes of variation
Exceptions to the system
Generally this is some change in the process
Variations that can be traced to a specific reason
considered abnormalities
often specific to a
certain operator
certain machine
certain batch of material, etc.
The objective is to discover when assignable causes are present
Eliminate the bad causes
Incorporate the good causes

5. Statistical Quality Control Objectives
1.Detect and eliminate assignable variation (statistical process control)
If there is no assignable variation, Process is in control
We use Process Control charts to maintain this
2.Reduce normal variation (process capability)
If normal variation is as small as desired, Process is capable
We use capability index to check for this

6. 1. Statistical Process Control: Control Charts
Can be used to monitor ongoing production process quality
970
980
990
1000
1010
1020 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
LCL
UCL

7. Process Control Chart
Information: Monitor process variability over time
Control Limits: Average + z Normal Variability
Decision Rule: Ignore variability within limits as “normal” Investigate variation outside as “abnormal”
Errors: Type I - False alarm (unnecessary investigation) Type II - Missed signal (to identify and correct)

9. Shows sample means over time Monitors process mean
X-bar – Chart
Shows sample means over time
Means of the values in a sample
Monitors process mean

10. Each of these represents one sample of five boxes of cereal
Samples
To measure the process, we take samples and analyze the sample statistics following these 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 #

11. The solid line represents the distribution
Samples
To measure the process, we take samples and analyze the sample statistics following these steps
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

12. 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

13. 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.

14. Samples
To measure the process, we take samples and analyze the sample statistics following these steps
(c) If only natural 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

15. Samples
To measure the process, we take samples and analyze the sample statistics following these steps
Prediction ?
(d) If assignable causes are present, the process output is not stable over time and is not predicable
Weight
Time
Frequency

16. For x-Charts when we know s
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 (=3)
sx = standard deviation of the sample means
= s/ n
s = population standard deviation
n = sample size

17. Setting Control Limits
Hour Mean Hour Mean
Hour 1
sample item 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
n = 9
Sample
size
For 99.73% control limits, z = 3

18. 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
LCLx = x - zsx = (1/3) = 15

19. 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

20. Type of variables control chart Shows sample ranges over time
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

21. Range (R) Chart Average Range R = 10.1 psi
UCL
LCL
Average Range R = 10.1 psi
Standard Deviation of Range = 3.5 psi
Control Limits: (3)(3.5) = [0, 20.6]
Process Is “In Control” (i.e., variation is stable)
Copyright © 2013 Pearson Education Inc. publishing as Prentice Hall

22. 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 mean)
UCL
LCL

23. 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 dispersion)
UCL
LCL
R-chart
(R-chart detects increase in dispersion)
UCL
LCL

24. Process Control and Improvement
Out of Control In Control Improved
UCL 
LCL

25. Important points to remember
Control charts are used to differentiate normal variability from assignable/abnormal variability
X-bar chart monitors control of process mean
R-chart monitors control of process variability
An improvement in the process implies lower normal variability

26. 2. Process Capability Example:Producing bearings for a rotating shaft
Specification Limits
Design requirements:
Diameter: 1.25 inch ±0.005 inch
Lower specification Limit:LSL= =1.245
Upper Specification Limit:USL= =1.255

27. Relating Specs to Process Limits
Process performance (Diameter of the products produced=D):
Average 1.25 inch
Std. Dev: inch
Question:What is the probability
That a bearing does not meet specifications?
(i.e. diameter is outside (1.245,1.255) )
Frequency
Diameter
1.25
P(defect)= = or 1.2% This is not good enough!!

28. Histogram Specs Mean = 82.5 psi, Standard Deviation = 4.2 psi
Fraction Defective = 26% (Theoretical = 30.1%)

29. Process capability
If P(defect)> then the process is not capable of producing
according to specifications.
To have this quality level (3 sigma quality), we need to have:
Lower Spec: mean-3
Upper Spec:mean+3
If we want to have P(defect)0, we aim for 6 sigma quality, then, we need:
Lower Spec: mean-6
Upper Spec:mean+6
What can we do to improve capability of our process? What should  be to have Six-Sigma quality?
We want to have: ( )/ = 6  = inch
We need to reduce variability of the process. We cannot change specifications easily, since they are given by customers or design requirements.

30. Process Capability Index Cpk
Shows how well the parts being produced fit into the range specified by the design specifications
Want Cpk larger than one
When two numbers below are not the same, indicates mean has shifted
For our example:
Cpk tells how many standard deviations can fit between the mean and
the specification limits.
Ideally we want to fit more, so that probability of defect is smaller

31. Next Time
Assignment is due on May 8
Quiz on Quality