x
Engineering characteristics
Customer requirements
Easy to close
Stays open on a hill
Easy to open
Doesn’t leak in rain
No road noise
Importance weighting
7
5
3
3
2
10 x x
6
Target values
6 x x
9 2 3
Correlation: x
*
Strong positive
Positive
Negative
Strong negative x
A
Competitive evaluation
= Us
= Comp. A
B = Comp. B
(5 is best)
1 2 3 4 5 x AB x AB x
A x
A x AB
B
B
Relationships:
Strong = 9
Medium = 3
Small = 1
Technical evaluation
(5 is best)
3
2
5
4
1
B
A x
B
A x
Quality
B
A
B x A
Source: Based on John R. Hauser and Don Clausing, “The House of
Quality,” Harvard Business Review
May-June 1988.
,
2

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(σ 2 + D 2 ) where
σ 2 = Variance of quality characteristic
D 2 = ( x – T) 2
Note: x is the mean quality characteristic. D 2 is zero if the mean equals the target.
Quality 3

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) = kx 2
When larger is better (e.g., product life)
L(x) = k (1/x 2 )
Quality 4

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 Sample size
MR-CHART 2 to 5 units
I-CHART
P-CHART
C/U-CHART
1 unit at least 100 units
1 or more units
Quality 5

Sample mean value
0.13% Upper control limit
99.74% Process mean
Lower control limit
0.13%
0 1 2 3 4 5 6 7 8
Sample number
Quality
Normal tolerance of process
6

n
2
A A
2
D
3
D
4 d
2 d
3
2.121 1.880 0 3.267 1.128 0.853
3 1.732 1.023
0 2.574 1.693 0.888
4 1.500 0.729
0 2.282 2.059 0.880
5 1.342 0.577 0 2.114 2.316 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 (σ x
) is:
σ x
= R / d
2
Quality 7

• Note that control factors depend on the sample size n .
• Relationships amongst control factors:
A
2
D
4
D
3
= 3 / (d
2 x n 1/2
= 1 + 3 x d
3
/d
2
)
= 1 – 3 x d
3
/d
2
, unless the result is negative, then D
3
= 0
A = 3 / n 1/2
D
2
D
1
= d
2
= d
2
+ 3d
– 3d
3
3
, unless the result is negative, then D
1
= 0
Quality 8

1. Compute the mean of sample means ( X ).
2. Compute the mean of sample ranges ( R ).
3. Estimate the population standard deviation (σ x
):
σ x
= R / d
2
4. Estimate the natural tolerance of the process:
Natural tolerance = 6σ x
5. Determine the specification limits:
USL = Upper specification limit
LSL = Lower specification limit
Quality 9

6. Compute capability indices:
Process capability potential
C p
= (USL – LSL) / 6σ x
Upper capability index
C pU
= (USL – X ) / 3σ x
Lower capability index
C pL
= ( X – LSL) / 3σ x
Process capability index
C pk
= Minimum (C pU
, C pL
)
Quality 10

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 + A
2
R
LCL = X – A
2
R
4. Set 3-std.-dev. control limits for the sample ranges:
UCL = D
4
R
LCL = D
3
R
Quality 11

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 (S
P
S
P
= [( P (1-P ))/n] 1/2
) for each sample :
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 + 3S
P
LCL = P – 3S
P
Quality 12

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

4. Estimate the population standard deviation (σ
X
):
σ
X
= R / d
2
Note: Sample size is always 2, so d
2
= 1.128.
5. Set 3-std.-dev. control limits:
UCL = X + 3σ
X
LCL = X – 3σ
X
Quality 14

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 :
S
C
= ( 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 + 3S
C
LCL = C – 3S
C
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

• Control factors n A A
2
D
3
D
4 d
2 d
3
2 2.121 1.880 0 3.267 1.128 0.853
3 1.732 1.023
0 2.574 1.693 0.888
4 1.500 0.729
0 2.282 2.059 0.880
5 1.342 0.577 0 2.114 2.316 0.864
• Process capability analysis
σ x
C p
C pL
= R / d
2
= (USL – LSL) / 6σ x
= ( X – LSL) / 3σ x
C pU
C pk
= (USL – X ) / 3σ x
= Minimum (C pU
, C pL
)
Quality 16

•
Quick reference to quality formulas
(cont.)
Means and ranges
UCL = X + A
2
LCL = X – A
2
R
R UCL = D
4
LCL = D
3
R
R
• Percentage defective in a sample
S
P
= [( P (1-P ))/n] 1/2 UCL = P + 3S
P
LCL = P – 3S
P
• Individual quality observations
σ x
= R / d
2
UCL = X + 3σ
X
LCL = X – 3σ
X
• Number of defects per unit
S
C
= ( C / n) 1/2 UCL = C + 3S
C
LCL = C – 3S
C
Quality 17

The seasonal index is the expected ratio of actual data to the average for the year.
Actual data / Index = Seasonally adjusted data
Seasonally adjusted data x Index = Actual data
Quality 18

1.
Compute moving average based on length of seasonality (4 quarters or 12 months).
2.
Divide actual data by corresponding moving average.
3.
Average ratios to eliminate randomness.
4.
Compute normalization factor to adjust mean ratios so they sum to 4 (quarterly data) or 12 (monthly data).
5.
Multiply mean ratios by normalization factor to get final seasonal indexes.
6.
Deseasonalize data by dividing by the seasonal index.
7.
Forecast deseasonalized data.
8.
Seasonalize forecasts from step 7 to get final forecasts.
Quality 19

The seasonal index is the expected difference between actual data and the average for the year.
Actual data - Index = Seasonally adjusted data
Seasonally adjusted data + Index = Actual data
Quality 20

1.
Compute moving average based on length of seasonality
(4 quarters or 12 months).
2.
Compute differences: Actual data - moving average.
3.
Average differences to eliminate randomness.
4.
Compute normalization factor to adjust mean differences so they sum to zero.
5.
Compute final indexes: Mean difference – normalization factor.
6.
Deseasonalize data: Actual data – seasonal index.
7.
Forecast deseasonalized data.
8.
Seasonalize forecasts from step 7 to get final forecasts.
Quality 21

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 22

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 23