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### Quality Control

Dr. Everette S. Gardner, Jr.

x

Strong positive

Positive

x

x

x

Negative

x

x

Strong negative

*

Competitive evaluation

Engineering characteristics

Source: Based on John R. Hauser and Don Clausing, “The House of Quality,” Harvard Business Review, May-June 1988.

Acoustic trans., window

Energy needed to open door

Check force on level ground

Energy needed to close door

x

Water resistance

= Us

Door seal resistance

Importance to customer

= Comp. A

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

AB

x

x

Doesn’t leak in rain

3

B

A

x

No road noise

2

B

A

Importance weighting

3

2

10

9

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

x

Technical evaluation (5 is best)

A

x

3

A

x

2

A

x

1

Quality

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

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

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

Quality

value

0.13%

Upper control limit

Normal

tolerance

of

process

99.74%

Process mean

Lower control limit

0.13%

7

6

8

1

3

4

5

2

0

Sample number

Quality

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 (σx) is:

σx = R / d2

Quality

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

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

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

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

Quality

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

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

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

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

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

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 (seasonally-adjusted) data.

Quality

Seasonal adjustment illustrated: 3 years of quarterly sales of Wolfpack Red Soda

Step 1. Moving averages

t Qtr. Xt4-Qtr. moving average

1 1 53 NA

2 2 83 NA

3 3 95 (53 + 83 + 95 + 72) / 4 = 75.75

4 4 72 (83 + 95 + 72 + 50) / 4 = 75.00

5 1 50 (95 + 72 + 50 + 75) / 4 = 73.00

6 2 75 (72 + 50 + 75 + 102) / 4 = 74.75

7 3 102 (50 + 75 + 102 + 66) / 4 = 73.25

8 4 66 (75 + 102 + 66 + 55) / 4 = 74.50

9 1 55 (102 + 66 + 55 + 81) / 4 = 76.00

10 2 81 (66 + 55 + 81 + 93) / 4 = 73.75

11 3 93 (55 + 81 + 93 + 76) / 4 = 76.25

12 4 76 NA

Quality

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 = 0.6849

75 / 74.75 = 1.0033

102 / 73.25 = 1.3925

66 / 74.50 = 0.8859

55 / 76.00 = 0.7237

81 / 73.75 = 1.0983

93 / 76.25 = 1.2197

NA

Quality

Seasonal adjustment illustrated: 3 years of quarterly sales of Wolfpack Red Soda

Step 3. Mean ratios

Qtr.Sum of ratios for each qtr. / Nbr.

1 (0.6849 + 0.7237) / 2 = 0.7043

2 (1.0033 + 1.0983) / 2 = 1.0508

3 (1.2542 + 1.3925 + 1.2197) / 3 = 1.2888

4 (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

Seasonal adjustment illustrated: 3 years of quarterly sales of Wolfpack Red Soda

Step 5. Final seasonal indices

Qtr.Mean ratio x Factor = Index

1 0.7043 x 1.0083 = 0.7101

2 1.0508 x 1.0083 = 1.0595

3 1.2888 x 1.0083 = 1.2995

4 0.9230 x 1.0083 = 0.9307

Sum of indices = 3.9998

Quality

Seasonal adjustment illustrated: 3 years of quarterly sales of Wolfpack Red Soda

Step 6. Deseasonalize data

t Qtr. Xt / Index = Des. Xt

1 1 53 / 0.7101 = 74.6

2 2 83 / 1.0595 = 78.3

3 3 95 / 1.2995 = 73.1

4 4 72 / 0.9307 = 77.4

5 1 50 / 0.7101 = 70.4

6 2 75 / 1.0595 = 70.8

7 3 102 / 1.2995 = 78.5

8 4 66 / 0.9307 = 70.9

9 1 55 / 0.7101 = 77.5

10 2 81 / 1.0595 = 76.5

11 3 93 / 1.2995 = 71.6

12 4 76 / 0.9307 = 81.7

Quality

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

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

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

σx = R / d2

Cp = (USL – LSL) / 6σx CpU = (USL – X ) / 3σx

CpL = ( X – LSL) / 3σx Cpk = Minimum (CpU, CpL)

Quality

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 ))/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

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