Transcript Production and Operations Management: Manufacturing and

1
CHAPTER 8TN
Process Capability and Statistical Quality
Control


Process Variation
Process Capability
Process Control Procedures
–
Acceptance Sampling

2
Types of Statistical
Quality Control
Statistical
Quality Control
Process
Control
Variables
Charts
Attributes
Charts
Acceptance
Sampling
Variables
Attributes
3
Basic Forms of Variation

Assignable variation

Common variation
4
Taguchi’s View of Variation
High
High
Incremental
Cost of
Variability
Incremental
Cost of
Variability
Zero
Zero
Lower Target
Spec
Spec
Upper
Spec
Traditional View
Lower
Spec
Target
Spec
Upper
Spec
Taguchi’s View
5
Process Capability

Process limits

Tolerance limits

How do the limits relate to one another?
6
Process Capability Index, Cpk
Capability Index shows
how well parts being
produced fit into design
limit specifications.
As a production
process produces
items small shifts in
equipment or systems
can cause differences
in production
performance from
differing samples.
 X  LTL UTL - X 

Cpk = min
or
 3
3 


Shifts in Process Mean
7
Types of Statistical Sampling

Attribute (Go or no-go information)
–
–

Defectives refers to the acceptability of product
across a range of characteristics.
Defects refers to the number of defects per unit
which may be higher than the number of
defectives.
Variable (Continuous)
–
Usually measured by the mean and the standard
deviation.
8
Statistical Process Control
(SPC) ChartsNormal Behavior
UCL
LCL
1
2
3
4
5
6
Samples
over time
UCL
Possible problem, investigate
LCL
1
2
3
4
5
6
Samples
over time
UCL
Possible problem, investigate
LCL
1
2
3
4
5
6
Samples
over time
9
Control Limits are based on the Normal
Curve
x
m
-3
-2
-1
Standard
deviation
units or “z”
units.
0
1
2
3
z
10
Control Limits
We establish the Upper Control Limits (UCL)
and the Lower Control Limits (LCL) with plus
or minus 3 standard deviations. Based on
this we can expect 99.7% of our sample
observations to fall within these limits.
99.7%
LCL
x
UCL
11
STANDARD DEVIATION
TABLE
% Data Points
------------------68
95
95.5
99
99.7
# of Std Dev From Mean
---------------------------------
12
Example of Constructing a p-Chart:
Required Data
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
4
2
5
3
6
4
3
7
1
2
3
2
2
8
3
13
Statistical Process Control Formulas:
Attribute Measurements (p-Chart)
Given:
T o tal N u m b er o f D efe ctiv es
p =
T o tal N u m b er o f O b se rv atio n s
sp =
p (1 - p )
n
Compute control limits:
UCL = p + z s p
LCL = p - z s p
14
Example of Constructing a p-chart:
Step 1
1. Calculate the
sample proportions, p
(these are what can
be plotted on the pchart) for each
sample.
Sample
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
n
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
Defectives
4
2
5
3
6
4
3
7
1
2
3
2
2
8
3
p
0.04
0.03
0.07
0.01
0.02
0.03
0.02
0.02
0.08
0.03
15
Example of Constructing a p-chart:
Steps 2&3
2. Calculate the average of the sample proportions.
p =
3. Calculate the standard deviation of the sample
proportion
sp =
16
Example of Constructing a p-chart:
Step 4
4. Calculate the control limits.
UCL
=
LCL
=
17
Example of Constructing a p-Chart: Step 5
5. Plot the individual sample proportions, the average
0.1
of the proportions, and the control limits
0.09
0.08
0.07
0.06
p 0.05
0.04
0.03
0.02
0.01
0
1
2
3
4
5
6
7
8
9
Observation
10
11
12
13
14
15
18
R Chart

Type of variables control chart
– Interval or ratio scaled numerical data

Shows sample ranges over time
– Difference between smallest & largest values in
inspection sample


Monitors variability in process
Example: Weigh samples of coffee &
compute ranges of samples; Plot
19
R Chart
Control Limits
UCL
 D4 R
R
From Table
LCL R  D 3  R
k
R 
R
i 1
k
i
Sample Range
at Time i
# Samples
20
R Chart
Example
You’re manager of a 500room hotel. You want to
analyze the time it takes to
deliver luggage to the
room. For 7 days, you
collect data on 5 deliveries
per day. Is the process in
control?
21
R Chart
Hotel Data
Day
Delivery Time
1 7.30 4.20 6.10 3.45 5.55
2 4.60 8.70 7.60 4.43 7.62
3
5.98 2.92 6.20 4.20 5.10
4 7.20 5.10
5 4.00 4.50
6 10.10 8.10
7 6.77 5.08
5.19
5.50
6.50
5.90
6.80
1.89
5.06
6.90
4.21
4.46
6.94
9.30
Sample
Mean Range
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R Chart
Hotel Data
Sample
Day
Delivery Time
Mean Range
1 7.30 4.20 6.10 3.45 5.55 5.32 3.85
Largest
Smallest
Sample Range = 7.30 - 3.45
23
R Chart
Hotel Data
Sample
Day
Delivery Time
Mean Range
1 7.30 4.20 6.10 3.45 5.55 5.32
7.30 + 4.20 + 6.10 + 3.45 + 5.55
Sample Mean =
5
24
R Chart
Hotel Data
Day
Sample
Mean Range
Delivery Time
1 7.30 4.20
2 4.60 8.70
3 5.98 2.92
4 7.20 5.10
5 4.00 4.50
6 10.10 8.10
7 6.77 5.08
6.10
7.60
6.20
5.19
5.50
6.50
5.90
3.45
4.43
4.20
6.80
1.89
5.06
6.90
5.55
7.62
5.10
4.21
4.46
6.94
9.30
5.32
6.59
4.88
5.70
4.07
7.34
6.79
3.85
4.27
3.28
2.99
3.61
5.04
4.22
25
Example of R charts:
R Chart Control Limits
UCL = D 4 R
LCL = D 3 R
n
2
3
4
5
6
7
8
9
10
11
A2
1.88
1.02
0.73
0.58
0.48
0.42
0.37
0.34
0.31
0.29
D3
0
0
0
0
0
0.08
0.14
0.18
0.22
0.26
D4
3.27
2.57
2.28
2.11
2.00
1.92
1.86
1.82
1.78
1.74
26
Example of R charts:
R Chart Control Limits
UCL = D 4 R
From Exhibit TN7.7
LCL = D 3 R
27
R Chart
Control Chart Solution
R, Minutes
8
6
4
2
0
1
2
UCL
3
4
Day
5
6
7
28
`X Chart

Type of variables control chart

Shows sample means over time
Monitors process average
Example:


29
`X Chart
Control Limits
From Exhibit
7.7TN(n = 5)
UCL X  X  A 2  R
Sample
Mean at
Time i
LCLX  X  A 2  R
k
X
X
i 1
k
k
i
R
R
i 1
k
i
Sample
Range at
Time i
# Samples
30
`X Chart Example
You’re manager of a 500room hotel. You want to
analyze the time it takes to
deliver luggage to the
room. For 7 days, you
collect data on 5 deliveries
per day. Is the process in
control?
Alone
Group Class
31
X Chart
Hotel Data
Day
1 7.30
2 4.60
3 5.98
4 7.20
5 4.00
6 10.10
7 6.77
Delivery Time
4.20 6.10 3.45
8.70 7.60 4.43
2.92 6.20 4.20
5.10 5.19 6.80
4.50 5.50 1.89
8.10 6.50 5.06
5.08 5.90 6.90
5.55
7.62
5.10
4.21
4.46
6.94
9.30
Sample
Mean Range
5.32 3.85
6.59 4.27
4.88 3.28
5.70 2.99
4.07 3.61
7.34 5.04
6.79 4.22
32
Example of x-bar charts: Tabled Values
From Exhibit TN7.7
x Chart Control Limits
UCL = x + A 2 R
LCL = x - A 2 R
n
2
3
4
5
6
7
8
9
10
11
A2
1.88
1.02
0.73
0.58
0.48
0.42
0.37
0.34
0.31
0.29
D3
0
0
0
0
0
0.08
0.14
0.18
0.22
0.26
D4
3.27
2.57
2.28
2.11
2.00
1.92
1.86
1.82
1.78
1.74
33
Example of x-bar charts: Tabled Values
x Chart
UCL
Control
= x + A
2
R
From Exhibit TN7.7
LCL
= x - A
2
R
Limits
34
`X Chart
Control Chart Solution*
X, M in u tes
8
UCL
6
4
LCL
2
0
1
2
3
4
Day
5
6
7
35
Process Capability
36
Process Capability

TQM’s emphasis on “making it right the first
time” has resulted in organizations
emphasizing the ability of a production
system to meet design specifications rather
than evaluating the quality of outputs after the
fact with acceptance sampling.

Process Capability -
37
Process Capability

Process limits –

Tolerance limits -
38
Process Capability

How do the limits relate to one another?
You want: tolerance range > process range
Two methods of
accomplishing
this:
1. Make bigger
2. Make smaller
39
Process Capability Measurement
Cp index = Tolerance range / Process range
What value(s) would you like for Cp?

Larger Cp –

The Cp index
– Assumes
»
40
Process Capability Depends On:




Location of the process mean.
Natural variability inherent in the process.
Stability of the process.
Product’s design requirements.
41
Natural Variation Versus Product Design
Specifications
42
Process Capability Index
Cp 
product' s design specification range
UTL - LTL

6 standard deviations of the production system
6
Cp < 1:
Cp > 1:
As rule of thumb, many organizations desire a Cp index
of at least 1.5.
Six sigma quality (fewer than 3.4 defective parts per
million) corresponds to a Cp index of 2.
43
(UTL LTL)
Cp 
6
6
Cp   1
6
LTL
UTL

-4

X
44
12
Cp 
2
6
UTL
LTL

-4

X
45
Process Capability
Light-bulb Production
Upper specification = 120 hours
Lower specification = 80 hours
Average life = 90 hours s = 4.8 hours
CP =
UTL - LTL
6s
Process Capability Ratio
46
Process Capability
Light-bulb Production
CP =
Process Capability Ratio
47
Cpk Index
 X  LTL UTL- X 

Cpk = min
or
 3

3



X
= estimate of the process mean
 = estimate of the standard deviation
Together, these process capability Indices
show how well parts being produced conform
to design specifications.
48
Light-bulb Production
C pk
 X  LTL
UTL - X

= min
or

3

3






49
Another example of the use of process
capability indices
The design specifications for a machined slot is 0.5± .003
inches. Samples have been taken and the process mean is
estimated to be .501. The process standard deviation is
estimated to be .001.
What can you say about the capability of this process to
produce this dimension?
UTL  LTL
Cp 
6
 X  LTL UTL- X 

Cpk = min
or
 3

3



50
Basic Forms of Statistical Sampling
for Quality Control

Sampling to accept or reject the immediate
lot of product at hand (Acceptance
Sampling).

Sampling to determine if the process is
within acceptable limits (Statistical Process
Control)
51
Acceptance Sampling

Purposes
–
–

Determine quality level
Ensure quality is within predetermined level
Advantages
–
–
–
–
–
–
Economy
Less handling damage
Fewer inspectors
Upgrading of the inspection job
Applicability to destructive testing
Entire lot rejection (motivation for improvement)
52
Acceptance Sampling

Disadvantages
–
–
–
Risks of accepting “bad” lots and rejecting
“good” lots
Added planning and documentation
Sample provides less information than 100percent inspection
53
Risk

Acceptable Quality Level (AQL)

Lot Tolerance Percent Defective (LTPD)