quality lecture notes

8/4/2019 Quality Lecture Notes

1/23

Quality Control

Dr. Everette S. Gardner, Jr.


2/23

Quality 2

Doorseal

resistance

Checkforceon

levelground

Energyneeded

toopendoor

Acoustictrans.,

window

Maintain

current

level

Reduce

force

to9lb.

Reduce

energy

to7.5

ft/lb

Engineeringcharacteristics

Customerrequirements

54321

Easy to close

Stays open on a hillEasy to open

Doesnt leak in rain

No road noise

Importance weighting

7

53

3

2

10 6 6 9

Source: Based on John R. Hauser

and Don Clausing, The House ofQuality,Harvard Business Review,May-June 1988.

2 3

x

x

x x

xx

*Competitiveevaluationx

AB(5 is best)1 2 3 4 5

= Us= Comp. A= Comp. B

Target values

Technical evaluation(5 is best)

Correlation:

Strong positivePositiveNegative

Strong negative

xxx

x

x

x

x

x

xx

x

AB

AB

AB

BABA

AA

A

A

A

A

BBBB

B

B

Relationships:Strong = 9

Medium = 3Small = 1


3/23

Quality 3

Taguchi analysis

Loss function

L(x) = k(x-T)2where

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(2 + D2)

where

2

= Variance of quality characteristicD2 = ( x T)2

Note: x is the mean quality characteristic. D2 is zero if the meanequals the target.


4/23

Quality 4

Taguchi analysis (cont.)

Average or expected loss, variance unkownE[L(x)] = k[ ( 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)


5/23

Quality 5

Introduction to quality control charts

Definitions

Variables Measurements on a continuous scale, such as length orweight

Attributes Integer counts of quality characteristics, such as nbr. good orbad

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


6/23

Quality 6

Sample mean

value

Sample number

99.74%

0.13%

0.13%

Upper control limit

Lower control limit

Process mean

Normal

tolerance

ofprocess

0 1 2 3 4 5 6 7 8


7/23

Quality 7

Reference guide to control factorsn A A2 D3 D4 d2 d3

2 2.121 1.880 0 3.267 1.128 0.8533 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 deviationof individual observations (x) is:

x = R / d2


8/23

Quality 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 + 3d3D1 = d2 3d3, unless the result is negative, then D1 = 0


9/23

Quality 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 (x):

x = R / d2

4. Estimate the natural tolerance of the process:

Natural tolerance = 6x

5. Determine the specification limits:

USL = Upper specification limitLSL = Lower specification limit


10/23

Quality 10

Process capability analysis (cont.)

6. Compute capability indices:

Process capability potential

Cp = (USL LSL) / 6x

Upper capability index

CpU = (USL X ) / 3x

Lower capability index

CpL = ( X LSL) / 3x

Process capability index

Cpk= Minimum (CpU, CpL)


11/23

Quality 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


12/23

Quality 12

Control chart for percentage defectivein 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


13/23

Quality 13

Control chart for individualobservations 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


14/23

Quality 14

Control chart for individualobservations I-CHART (cont.)

4. Estimate the population standard deviation (X):

X = R / d2

Note: Sample size is always 2, so d2 = 1.128.


UCL = X + 3XLCL = X 3X


15/23

Quality 15

Control chart for number of defectsper 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.


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.


16/23

Quality 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

x = R / d2

Cp = (USL LSL) / 6x CpU = (USL X ) / 3xCpL = ( X LSL) / 3x Cpk= Minimum (CpU, CpL)


17/23

Quality 17

Quick reference to quality formulas(cont.)

Means and rangesUCL = X + A2R UCL = D4RLCL = X A2R LCL = D3R

Percentage defective in a sample

SP = [( P (1-P ))/n]1/2

UCL = P + 3SPLCL = P 3SP

Individual quality observationsx = R / d2 UCL = X + 3X

LCL = X 3X

Number of defects per unitSC = ( C / n)

1/2 UCL = C + 3SCLCL = C 3SC


18/23

Quality 18

Multiplicative seasonality

The seasonal index is the expected ratio of actual datato the average for the year.

Actual data / Index = Seasonally adjusted data

Seasonally adjusted data x Index = Actual data


19/23

Quality 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 theysum to 4 (quarterly data) or 12 (monthly data).

5. Multiply mean ratios by normalization factor to get finalseasonal indexes.

6. Deseasonalize data by dividing by the seasonal index.

7. Forecast deseasonalized data.

8. Seasonalize forecasts from step 7 to get final forecasts.


20/23

Quality 20

Additive seasonality

The seasonal index is the expected differencebetween actual data and the average for the year.

Actual data - Index = Seasonally adjusted data

Seasonally adjusted data + Index = Actual data


21/23

Quality 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 sothey sum to zero.

5. Compute final indexes: Mean difference normalizationfactor.

6. Deseasonalize data: Actual data seasonal index.7. Forecast deseasonalized data.

8. Seasonalize forecasts from step 7 to get final forecasts.


22/23

Quality 22

How to start up a control chart system1. 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.


23/23

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.

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