Quality Control

1
Quality Control

• Dr. Everette S. Gardner, Jr.

2
Correlation
x
Strong positive
Positive
x
x
x
Negative
x
x
Strong negative

Competitive evaluation
Engineering characteristics
Acoustic trans., window
Check force on level ground
Energy needed to open door
Energy needed to close door
Water resistance
x
Us
Door seal resistance
Importance to customer
A
Comp. A
Comp. B
B
Customer requirements
(5 is best)
1 2 3 4 5
x
Easy to close
7
AB
Stays open on a hill
x
AB
5
Easy to open
3
x
AB
Doesnt leak in rain
3
x
B
A
x
No road noise
2
A
B
Importance weighting
10
9
2
3
6
6
Relationships
Strong 9
Medium 3
Target values
Reduce energy level to 7.5 ft/lb
Reduce energy to 7.5 ft/lb
Small 1
Maintain current level
Maintain current level
Maintain current level
Reduce force to 9 lb.
5
BA
B
BA
x
x
A
4
B
B
B
Technical evaluation (5 is best)
x
A
x
3
A
2
x
A
x
1
3
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
• EL(x) k(s2 D2)
• where
• s2 Variance of quality characteristic
• D2 ( x T)2
• Note x is the mean quality characteristic. D2
  is zero if the mean equals the target.

4
Taguchi analysis (cont.)

• Average or expected loss, variance unkown
• EL(x) kS ( 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)

5
Introduction to quality control charts

• Definitions
• Variables Measurements on a continuous scale,
  such as length or weight
• Attributes Integer counts of quality
  characteristics, such as nbr. good or bad
• Defect A single non-conforming quality
  characteristic, such as a blemish
• Defective A physical unit that contains one or
  more defects
• Types of control charts
• Data monitored Chart name Sample
  size
• Mean, range of sample variables MR-CHART
  2 to 5 units
• Individual variables I-CHART 1 unit
• of defective units in a sample P-CHART
  at least 100 units
• Number of defects per unit C/U-CHART 1
  or more units

6

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

• n A A2 D3 D4 d2 d3
• 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
• 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
  (sx) is
• sx R / d2

8
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

9
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
  (sx)
• sx R / d2
• 4. Estimate the natural tolerance of the
  process
• Natural tolerance 6sx
• 5. Determine the specification limits
• USL Upper specification limit
• LSL Lower specification limit

10
Process capability analysis (cont.)

• 6. Compute capability indices
• Process capability potential
• Cp (USL LSL) / 6sx
• Upper capability index
• CpU (USL X ) / 3sx
• Lower capability index
• CpL ( X LSL) / 3sx
• Process capability index
• Cpk Minimum (CpU, CpL)

11
Mean-Range control chartMR-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

12
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 ))/n1/2
• 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

13
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

14
Control chart for individual observations
I-CHART (cont.)

• 4. Estimate the population standard deviation
  (sX)
• sX R / d2
• Note Sample size is always 2, so d2 1.128.
• 5. Set 3-std.-dev. control limits
• UCL X 3sX
• LCL X 3sX

15
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.

16
Quick reference to quality formulas

• Control factors
• n A A2 D3 D4 d2 d3
• 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
• sx R / d2
• Cp (USL LSL) / 6sx CpU (USL X ) /
  3sx
• CpL ( X LSL) / 3sx Cpk Minimum (CpU,
  CpL)

17
Quick reference to quality formulas (cont.)

• Means and ranges
• UCL X A2R UCL D4R
• LCL X A2R LCL D3R
• Percentage defective in a sample
• SP ( P (1-P ))/n1/2 UCL P 3SP
• LCL P 3SP
• Individual quality observations
• sx R / d2 UCL X 3sX
• LCL X 3sX
• Number of defects per unit
• SC ( C / n)1/2 UCL C 3SC
• LCL C 3SC

18
Multiplicative seasonality

• 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

19
Multiplicative seasonal adjustment

• 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.

20
Additive seasonality

• 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

21
Additive seasonal adjustment

• 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.

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.

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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.