Chapter 9
Other Univariate Statistical Process
Monitoring and Control Techniques
Introduction to Statistical Quality Control,
4th Edition
9-1. Statistical Process Control for
Short Production Runs
9-1.1 x and R Charts for Short Production Runs
•
Deviation from nominal (DNOM) control chart can be
used for short production runs. The procedure is:
1. Calculate xi = Mi – Tk, the deviation from nominal for
each observation. (Where Mi is the ith actual
measurement and Tk is the target value for a particular
group or part.)
2. Calculate x and R for each sample using the deviations.
3. Construct standard x and R charts for the deviations
calculated.
Introduction to Statistical Quality Control,
4th Edition
9-1. Statistical Process Control for
Short Production Runs
9-1.1 x and R Charts for Short Production Runs
•
Using data in Table 9-1, the control charts are
Deviation from Nominal X-bar and R Charts
Part A
Part B
Sample Mean
3
3.0SL=2.929
2
1
CL=0.1667
0
-1
-2
-3.0SL=-2.596
-3
Subgroup
0
1
2
Sample Range
Part A
8
7
6
5
4
3
2
1
0
3
4
5
6
7
8
9
10
Part B
3.0SL=6.950
R=2.700
-3.0SL=0.00E+00
Introduction to Statistical Quality Control,
4th Edition
9-1. Statistical Process Control for
Short Production Runs
9-1.1 x and R Charts for Short Production Runs
Important points about relative to the DNOM approach:
1. An assumption is that the process standard deviation is
approximately the same for all parts. If this assumption
is invalid, use standardized x and R charts.
2. This procedure works best when the sample size is
constant for all part numbers
3. Deviation from nominal control charts have intuitive
appeal when the nominal specification is the desired
target value for the process.
Introduction to Statistical Quality Control,
4th Edition
9-1. Statistical Process Control for
Short Production Runs
9-1.1 Standardized x and R Charts
• If process standard deviations are different
for different part numbers, the deviation
from nominal control charts will not work
effectively.
• Standardized control charts can effectively
handle the situation of different process
standard deviations.
Introduction to Statistical Quality Control,
4th Edition
9-1. Statistical Process Control for
Short Production Runs
9-1.1 Standardized x and R Charts
• R i - average range for ith part
• Ti – nominal value of x for the ith part
• For all samples from this part number
R
R 
Ri
s
is plotted on a standardized R chart with LCL =
D3 and UCL = D4
Introduction to Statistical Quality Control,
4th Edition
9-1. Statistical Process Control for
Short Production Runs
9-1.1 Standardized x and R Charts
• R i - average range for ith part
• Ti – nominal value of x for the ith part
• For all samples from this part number
x  Ti
x 
Ri
s
is plotted on a standardized x chart with
LCL = -A2 and UCL = A2
Introduction to Statistical Quality Control,
4th Edition
9-1. Statistical Process Control for
Short Production Runs
9-1.2 Attribute Control Charts for Short
Production Runs
• Standardized control charts for attribute data can
be used for short production runs.
• These charts are discussed in Chapter 6.
• Table 9-2 provides a summary of the
standardized attribute control charts for short
production runs.
Introduction to Statistical Quality Control,
4th Edition
9-1. Statistical Process Control for
Short Production Runs
9-1.2 Attribute Control Charts for Short Production Runs
Attribute
p̂ i
Target
Value
p
Standard
Deviation
p (1  p )
n
np̂ i
np
n p (1  p )
ci
c
c
ui
u
u
n
Statistic to Plot on
the control chart
p̂ i  p
Zi 
p (1  p )
n
np̂ i  np
Zi 
np (1  p )
ci  c
c
u u
Zi  i
u
n
Zi 
Introduction to Statistical Quality Control,
4th Edition
9-2. Modified and Acceptance
Control Charts
•
•
When a high level of process capability
has been achieved, it may be useful to
relax the level of surveillance provided by
standard control charts.
Modified control limits for x charts or
acceptance control charts can be employed
for this situation.
Introduction to Statistical Quality Control,
4th Edition
9-2. Modified and Acceptance
Control Charts
x
9-2.1 Modified Control Limits for the Chart
• Modified control limits commonly used in the
case Cp or Cpk > 1.
• The modified x control chart is concerned only
with detecting whether the true process mean 
is located such that the process is producing a
fraction nonconforming in excess of some
specified value, .
–
 can vary over the range L    U
Introduction to Statistical Quality Control,
4th Edition
9-2. Modified and Acceptance
Control Charts
x
9-2.1 Modified Control Limits for the Chart
• The upper and lower control limits for the
modified control chart are
3 

UCL  USL   Z 

n

3 

LCL  LSL   Z 

n

Introduction to Statistical Quality Control,
4th Edition
9-2. Modified and Acceptance
Control Charts
9-2.2 Acceptance Control Charts
• Acceptance control charts take into account
both the -risk and the -risk.
• Two approaches to designing the acceptance
control chart
1. Design the chart based on a specified n and process
nonconforming  that we would like to reject with
probability 1 - .
2. Choose a sample size for an acceptance control chart
so that specified values of   , and  are obtained.
Introduction to Statistical Quality Control,
4th Edition
9-2. Modified and Acceptance
Control Charts
9-2.2 Acceptance Control Charts
Approach 1: The control limits are
Z 


UCL  USL   Z  
n

Z 


LCL  LSL   Z  
n

Introduction to Statistical Quality Control,
4th Edition
9-2. Modified and Acceptance
Control Charts
9-2.2 Acceptance Control Charts
Approach 2: A sample size of
 Z   Z 

n
 Z Z 
 
 
2
will yield the required values of   , and .
Introduction to Statistical Quality Control,
4th Edition