Quality Control
Dr. Everette S. Gardner, Jr.
Correlation:
Strong positive
Positive
x Negative
* Strong negative
Competitive
evaluation
x = Us
A = Comp. A
B = Comp. B
(5 is best)
1 2 3 4 5
x
AB
x AB
x AB
A xB
x A
B
x
x
x
Reduce energy
to 7.5 ft/lb
Acoustic trans.,
window
6
Energy needed
to open door
6
9
2
3
B xA
BA
x
7
5
3
3
2
Importance weighting
10
Target values
Technical evaluation
(5 is best)
Check force on
level ground
Easy to close
Stays open on a hill
Easy to open
Doesn’t leak in rain
No road noise
Reduce force
to 9 lb.
Customer
requirements
Door seal
resistance
Engineering
characteristics
x
Maintain
current level
x
5
4
3
2
1
B
A
x
BA
x
B
A
x
Quality
B
x
A
Relationships:
Strong = 9
Medium = 3
Small = 1
Source: Based on John R. Hauser
and Don Clausing, “The House of
Quality,” Harvard Business Review,
May-June 1988.
2
Taguchi analysis
Loss function
L(x) = k(x-T)2
where
x = any individual value of the quality characteristic
T = target quality value
k = constant = L(x) / (x-T)2
Average or expected loss, variance known
E[L(x)] = k(σ2 + D2)
where
σ2 = Variance of quality characteristic
D2 = ( x – T)2
Note: x is the mean quality characteristic. D2 is zero if the mean
equals the target.
Quality
3
Taguchi analysis (cont.)
Average or expected loss, variance unkown
E[L(x)] = k[Σ ( x – T)2 / n]
When smaller is better (e.g., percent of impurities)
L(x) = kx2
When larger is better (e.g., product life)
L(x) = k (1/x2)
Quality
4
Introduction to quality control charts
Definitions
•
Variables
•
Attributes
•
Defect
•
Defective
Measurements on a continuous scale, such as length or
weight
Integer counts of quality characteristics, such as nbr. good or
bad
A single non-conforming quality characteristic, such as a
blemish
A physical unit that contains one or more defects
Types of control charts
Data monitored
• Mean, range of sample variables
• Individual variables
• % of defective units in a sample
• Number of defects per unit
Chart name
MR-CHART
I-CHART
P-CHART
C/U-CHART
Quality
Sample size
2 to 5 units
1 unit
at least 100 units
1 or more units
5
Sample mean
value
0.13%
Upper control limit
99.74%
Normal
tolerance
of
process
Process mean
Lower control limit
0.13%
0
1
2
3
4
5
6
7
8
Sample number
Quality
6
Reference guide to control factors
n
2
3
4
5
A
2.121
1.732
1.500
1.342
A2
1.880
1.023
0.729
0.577
D3
0
0
0
0
D4
3.267
2.574
2.282
2.114
d2
1.128
1.693
2.059
2.316
d3
0.853
0.888
0.880
0.864
• Control factors are used to convert the mean of sample ranges
( R ) to:
(1) standard deviation estimates for individual observations,
and
(2) standard error estimates for means and ranges of samples
For example, an estimate of the population standard deviation
of individual observations (σx) is:
σx = R / d2
Quality
7
Reference guide to control factors
(cont.)
• Note that control factors depend on the sample size n.
• Relationships amongst control factors:
A2 = 3 / (d2 x n1/2)
D4 = 1 + 3 x d3/d2
D3 = 1 – 3 x d3/d2, unless the result is negative, then D3 = 0
A = 3 / n1/2
D2 = d2 + 3d3
D1 = d2 – 3d3, unless the result is negative, then D1 = 0
Quality
8
Process capability analysis
1. Compute the mean of sample means ( X ).
2. Compute the mean of sample ranges ( R ).
3. Estimate the population standard deviation (σx):
σx = R / d2
4. Estimate the natural tolerance of the process:
Natural tolerance = 6σx
5. Determine the specification limits:
USL = Upper specification limit
LSL = Lower specification limit
Quality
9
Process capability analysis (cont.)
6. Compute capability indices:
Process capability potential
Cp = (USL – LSL) / 6σx
Upper capability index
CpU = (USL – X ) / 3σx
Lower capability index
CpL = ( X – LSL) / 3σx
Process capability index
Cpk = Minimum (CpU, CpL)
Quality
10
Mean-Range control chart
MR-CHART
1. Compute the mean of sample means ( X ).
2. Compute the mean of sample ranges ( R ).
3. Set 3-std.-dev. control limits for the sample means:
UCL = X + A2R
LCL = X – A2R
4. Set 3-std.-dev. control limits for the sample ranges:
UCL = D4R
LCL = D3R
Quality
11
Control chart for percentage defective
in a sample — P-CHART
1. Compute the mean percentage defective ( P ) for all samples:
P = Total nbr. of units defective / Total nbr. of units sampled
2. Compute an individual standard error (SP ) for each sample:
SP = [( P (1-P ))/n]1/2
Note: n is the sample size, not the total units sampled.
If n is constant, each sample has the same standard error.
3. Set 3-std.-dev. control limits:
UCL = P + 3SP
LCL = P – 3SP
Quality
12
Control chart for individual
observations — I-CHART
1. Compute the mean observation value ( X )
X = Sum of observation values / N
where N is the number of observations
2. Compute moving range absolute values, starting at obs. nbr. 2:
Moving range for obs. 2 = obs. 2 – obs. 1
Moving range for obs. 3 = obs. 3 – obs. 2
…
Moving range for obs. N = obs. N – obs. N – 1
3. Compute the mean of the moving ranges ( R ):
R = Sum of the moving ranges / N – 1
Quality
13
Control chart for individual
observations — I-CHART (cont.)
4. Estimate the population standard deviation (σX):
σX = R / d2
Note: Sample size is always 2, so d2 = 1.128.
5. Set 3-std.-dev. control limits:
UCL = X + 3σX
LCL = X – 3σX
Quality
14
Control chart for number of defects
per unit — C/U-CHART
1. Compute the mean nbr. of defects per unit ( C ) for all samples:
C = Total nbr. of defects observed / Total nbr. of units sampled
2. Compute an individual standard error for each sample:
SC = ( C / n)1/2
Note: n is the sample size, not the total units sampled.
If n is constant, each sample has the same standard error.
3. Set 3-std.-dev. control limits:
UCL = C + 3SC
LCL = C – 3SC
Notes:
● If the sample size is constant, the chart is a C-CHART.
● If the sample size varies, the chart is a U-CHART.
● Computations are the same in either case.
Quality
15
Seasonal adjustment of quality
observations
1. Compute a 4-quarter or 12-month moving average. Position the first
average as follows:
a.
Quarterly: Place the first average opposite the 3rd quarter. The first
2 quarters and the last quarter have no moving average.
b.
Monthly: Place the first average opposite the 7th month. The first 6
months and the last 5 months have no moving average.
2. Divide each data observation by the corresponding moving average.
3. Compute a mean ratio for each quarter or month.
4. Compute a normalization factor to adjust the mean ratios so that they sum
to 4 (quarterly) or 12 (monthly):
a.
Quarterly: Normalization factor = 4 / Sum of mean ratios
b.
Monthly: Normalization factor = 12 / Sum of mean ratios
Quality
16
Seasonal adjustment of quality
observations (cont.)
5. Multiply each mean ratio by the normalization factor to get a set
of final seasonal indices. Each quarter or month has an
individual index.
6. Deseasonalize each data observation by dividing by the
appropriate seasonal index.
7. Develop a control chart for the deseasonalized (seasonallyadjusted) data.
Quality
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Seasonal adjustment illustrated: 3 years
of quarterly sales of Wolfpack Red Soda
Step 1. Moving averages
t
1
2
3
4
Qtr.
1
2
3
4
Xt
53
83
95
72
5
6
7
8
1
2
3
4
50
75
102
66
9
10
11
12
1
2
3
4
55
81
93
76
4-Qtr. moving average
NA
NA
(53 + 83 + 95 + 72) / 4 = 75.75
(83 + 95 + 72 + 50) / 4 = 75.00
(95
(72
(50
(75
+
+
+
+
72 + 50 + 75) / 4
50 + 75 + 102) / 4
75 + 102 + 66) / 4
102 + 66 + 55) / 4
= 73.00
= 74.75
= 73.25
= 74.50
(102 + 66 + 55 + 81) / 4 = 76.00
(66 + 55 + 81 + 93) / 4 = 73.75
(55 + 81 + 93 + 76) / 4 = 76.25
NA
Quality
18
Seasonal adjustment illustrated: 3 years
of quarterly sales of Wolfpack Red Soda
Step 2. Ratios
Ratio = Xt / Average
NA
NA
95 / 75.75 = 1.2541
72 / 75.00 = 0.9600
50 / 73.00
75 / 74.75
102 / 73.25
66 / 74.50
= 0.6849
= 1.0033
= 1.3925
= 0.8859
55 / 76.00 = 0.7237
81 / 73.75 = 1.0983
93 / 76.25 = 1.2197
NA
Quality
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Seasonal adjustment illustrated: 3 years
of quarterly sales of Wolfpack Red Soda
Step 3. Mean ratios
Qtr.
1
2
3
4
Sum of ratios for each qtr. / Nbr.
(0.6849 + 0.7237) / 2
= 0.7043
(1.0033 + 1.0983) / 2
= 1.0508
(1.2542 + 1.3925 + 1.2197) / 3 = 1.2888
(0.9600 + 0.8859) / 2
= 0.9230
Sum of mean ratios
= 3.9669
Step 4. Normalization Factor
Factor = 4 / (Sum of mean ratios)
Factor = 4 / 3.9669 = 1.0083
Quality
20
Seasonal adjustment illustrated: 3 years
of quarterly sales of Wolfpack Red Soda
Step 5. Final seasonal indices
Qtr.
1
2
3
4
Mean ratio
0.7043
1.0508
1.2888
0.9230
x
x
x
x
x
Factor = Index
1.0083 = 0.7101
1.0083 = 1.0595
1.0083 = 1.2995
1.0083 = 0.9307
Sum of indices
= 3.9998
Quality
21
Seasonal adjustment illustrated: 3 years
of quarterly sales of Wolfpack Red Soda
Step 6. Deseasonalize data
t
1
2
3
4
Qtr.
1
2
3
4
5
6
7
8
1
2
3
4
9
10
11
12
1
2
3
4
Xt
53
83
95
72
/ Index
/ 0.7101
/ 1.0595
/ 1.2995
/ 0.9307
=
=
=
=
=
Des. Xt
74.6
78.3
73.1
77.4
50 / 0.7101
75 / 1.0595
102 / 1.2995
66 / 0.9307
=
=
=
=
70.4
70.8
78.5
70.9
=
=
=
=
77.5
76.5
71.6
81.7
55
81
93
76
/
/
/
/
0.7101
1.0595
1.2995
0.9307
Quality
22
How to start up a control chart system
1.
Identify quality characteristics.
2.
Choose a quality indicator.
3.
Choose the type of chart.
4.
Decide when to sample.
5.
Choose a sample size.
6.
Collect representative data.
7.
If data are seasonal, perform seasonal adjustment.
8.
Graph the data and adjust for outliers.
Quality
23
How to start up a control chart system
(cont.)
9. Compute control limits
10. Investigate and adjust special-cause variation.
11. Divide data into two samples and test stability of limits.
12. If data are variables, perform a process capability study:
a. Estimate the population standard deviation.
b. Estimate natural tolerance.
c. Compute process capability indices.
d. Check individual observations against specifications.
13. Return to step 1.
Quality
24
Quick reference to quality formulas
• Control factors
n
A
A2
2
2.121 1.880
3
1.732 1.023
4
1.500 0.729
5
1.342 0.577
D3
0
0
0
0
D4
3.267
2.574
2.282
2.114
• Process capability analysis
σx = R / d2
Cp = (USL – LSL) / 6σx
CpL = ( X – LSL) / 3σx
d2
1.128
1.693
2.059
2.316
d3
0.853
0.888
0.880
0.864
CpU = (USL – X ) / 3σx
Cpk = Minimum (CpU, CpL)
Quality
25
Quick reference to quality formulas
(cont.)
• Means and ranges
UCL = X + A2R
LCL = X – A2R
UCL = D4R
LCL = D3R
• Percentage defective in a sample
SP = [( P (1-P ))/n]1/2 UCL = P + 3SP
LCL = P – 3SP
• Individual quality observations
σx = R / d2
UCL = X + 3σX
LCL = X – 3σX
• Number of defects per unit
SC = ( C / n)1/2
UCL = C + 3SC
LCL = C – 3SC
Quality
26