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Statistical Process Control

Operations Management

Dr. Ron Tibben-Lembke

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Designed Size

10 11 12 13 14 15 16 17 18 19 20

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Natural Variation

14.5 14.6 14.7 14.8 14.9 15.0 15.1 15.2 15.3 15.4 15.5

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Theoretical Basis of Control Charts

95.5% of allXfall within 2

Properties of normal distribution

X

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Theoretical Basis of Control ChartsProperties of normal distribution

99.7% of allXfall within 3

X

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Skewness Lack of symmetry

Pearsons coefficient of

skewness:

0

2

4

6

8

10

12

14

16

0

2

4

6

8

10

12

14

16

0

2

4

6

8

10

12

14

16

Skewness = 0 Negative Skew < 0

Positive Skew > 0

sMedianx )(3

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Kurtosis Amount of peakedness

or flatness

Kurtosis < 0 Kurtosis > 0

Kurtosis = 04

4)(

ns

xx

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-6 -4 -2 0 2 4 6

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Design Tolerances Design tolerance:

Determined by users needs

UTL -- Upper Tolerance Limit

LTL -- Lower Tolerance Limit

Eg: specified size +/- 0.005 inches

No connection between tolerance and

completely unrelated to natural variation.

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Process Capability and 6

A capable process has UTL and LTL 3 or more

standard deviations away from the mean, or 3.

99.7% (or more) of product is acceptable to

customers

LTL UTL

3 6

LTL UTL

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Process Capability

LTL UTLLTL UTL

Capable Not Capable

LTL UTL LTL UTL

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Process Capability

Specs: 1.5 +/- 0.01

Mean: 1.505 Std. Dev. = 0.002

Are we in trouble?

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Process Capability

Specs: 1.5 +/- 0.01

LTL = 1.50.01 = 1.49

UTL = 1.5 + 0.01 = 1.51

Mean: 1.505 Std. Dev. = 0.002

LCL = 1.505 - 3*0.002 = 1.499

UCL = 1.505 + 0.006 = 1.511

1.499 1.511.49 1.511

ProcessSpecs

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Capability Index

Capability Index (Cpk) will tell the position of

the control limits relative to the design

specifications. Cpk>= 1.0, process is capable

Cpk< 1.0, process is not capable

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Process Capability, Cpk

Tells how well parts

produced fit into specs

33min

XUTLor

LTLXCpk

ProcessSpecs

3 3

LTL UTLX

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Process Capability

Tells how well parts produced fit into specs

For our example:

Cpk= min[ 0.015/.006, 0.005/0.006]

Cpk= min[2.5,0.833] = 0.833 < 1 Process not capable

33minXUTL

or

LTLX

Cpk

006.0505.151.1

006.0

49.1505.1

min orCpk

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Process Capability: Re-centered

If process were properly centered

Specs: 1.5 +/- 0.01

LTL = 1.50.01 = 1.49

UTL = 1.5 + 0.01 = 1.51

Mean: 1.5 Std. Dev. = 0.002

LCL = 1.5 - 3*0.002 = 1.494

UCL = 1.5 + 0.006 = 1.506

1.494 1.511.49 1.506

ProcessSpecs

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If re-centered, it would be Capable

1.494 1.511.49 1.506

ProcessSpecs

67.1006.0

01.0,

006.0

01.0min

006.0

5.151.1,

006.0

49.15.1min

pk

pk

C

C

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Packaged Goods

What are the Tolerance Levels?

What we have to do to measure capability?

What are the sources of variability?

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Production Process

Make Candy

Package Put in big bagsMake Candy

Make Candy

Make Candy

Make Candy

Make Candy

Mix

Mix %

Candy irregularity

Wrong wt. Wrong wt.

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Processes Involved Candy Manufacturing:

Are M&Ms uniform size & weight?

Should be easier with plain than peanut

Percentage of broken items (probably from printing)

Mixing:

Is proper color mix in each bag?

Individual packages:

Are same # put in each package?

Is same weight put in each package?

Large bags:

Are same number of packages put in each bag?

Is same weight put in each bag?

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Your Job Write down package #

Weigh package and candies, all together, in grams andounces

Write down weights on form

Optional: Open package, count total # candies

Count # of each color

Write down

Eat candies

Turn in form and empty complete wrappers forweighing

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The effects of rounding

17.00

18.00

19.00

20.00

21.00

22.00

23.00

24.00

25.00

14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0 22.5

Original Weight in grams

RoundedWeight-grams

0.50

0.60

0.70

0.80

RoundedWeight-Ounces

g - rounded

oz - rounded 0.7 Ounces

20 grams

0.6 Ounces

19 grams

18 grams

21 grams

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Peanut Color Mixwebsite

Brown 17.7% 20%

Yellow 8.2% 20% Red 9.5% 20%

Blue 15.4% 20%

Orange 26.4% 10%

Green 22.7% 10%

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Class website

Brown 12.1% 30%

Yellow 14.7% 20%

Red 11.4% 20%

Blue 19.5% 10%

Orange 21.2% 10%

Green 21.2% 10%

Plain Color Mix

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So who cares?

Dept. of Commerce

National Institutes of Standards & Technology

NIST Handbook 133

Fair Packaging and Labeling Act

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Acceptable?

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Package Weight

Not Labeled for Individual Retail Sale

If individual is 18g

MAV is 10% = 1.8g

Nothing can be below 18g1.8g = 16.2g

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Goal of Control Charts

collect and present data visually

allow us to see when trend appears

see when out of control point occurs

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010

20304050

60

1 2 3 4 5 6 7 8 9 10 11 12

Process Control Charts

Graph of sample data plotted over time

UCL

LCL

Process

Average

3

Time

X

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010

20304050

60

1 2 3 4 5 6 7 8 9 10 11 12

Process Control Charts

Graph of sample data plotted over time

Assignable

Cause

Variation

NaturalVariation

UCL

LCL

Time

X

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Definitions of Out of Control

1. No points outside control limits

2. Same number above & below center line

3. Points seem to fall randomly above andbelow center line

4. Most are near the center line, only a few are

close to control limits1. 8 Consecutive pts on one side of centerline

2. 2 of 3 points in outer third

3. 4 of 5 in outer two-thirds region

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Attributes vs. VariablesAttributes:

Good / bad, works / doesnt

count % bad (P chart) count # defects / item (C chart)

Variables:

measure length, weight, temperature (x-barchart)

measure variability in length (R chart)

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Attribute Control Charts

Tell us whether points in tolerance or not

p chart: percentage with given characteristic

(usually whether defective or not) np chart: number of units with characteristic

c chart: count # of occurrences in a fixed area of

opportunity (defects per car)

u chart: # of events in a changeable area of

opportunity (sq. yards of paper drawn from a

machine)

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p Chart Control Limits

# Defective

Items in

Sample i

Sample i

Size

UCLp pzp 1p

n

p Xi

i1

k

ni

i1

k

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# Defective

Items in

Sample i

Sample i

Size

z = 2 for

95.5% limits;

z = 3 for

99.7% limits

# Samples

n

ppzpUCLp

1

p Xi

i1

k

ni

i1

k

n

nii1

k

k

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# Defective

Items in

Sample i

# Samples

Sample i

Size

z = 2 for

95.5% limits;

z = 3 for

99.7% limits

n

ppzpUCLp

1

n

ppzpLCLp

1

n ni

i1

k

k

p Xi

i1

k

ni

i1

k

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p Chart Example

Youre manager of a 500-

room hotel. You want to

achieve the highest level

of service. For 7 days,

you collect data on the

readiness of200 rooms. Is

theprocess in control (usez= 3)?

1995 Corel Corp.

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p Chart Hotel DataNo. No. Not

Day Rooms Ready Proportion

1 200 16 16/200 = .080

2 200 7 .035

3 200 21 .105

4 200 17 .085

5 200 25 .1256 200 19 .095

7 200 16 .080

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n ni

i1

k

k

1400

7 200

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16 + 7 +...+ 16

p Xi

i1

k

nii1

k

121

1400

0.0864n

nii1

k

k

1400

7 200

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p Chart Solution

16 + 7 +...+ 16

p Xi

i1

k

nii1

k

121

1400

0.0864n

nii1

k

k

1400

7 200

pz

p 1p n 0.0864 3

0.0864 1 0.0864 200

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p Chart Solution

16 + 7 +...+ 16

pz

p 1p n 0.0864 3

0.0864 1 0.0864 200

0.0864 3*0.01984 0.0864 0.01984

0.1460, and 0.0268

p Xi

i1

k

nii1

k

121

1400

0.0864n

nii1

k

k

1400

7 200

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0.00

0.05

0.10

0.15

1 2 3 4 5 6 7

P

Day

p Chart

UCL

LCL

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R Chart

Type of variables control chart

Interval or ratio scaled numerical data

Shows sample ranges over time Difference between smallest & largest values

in inspection sample

Monitors variability in process

Example: Weigh samples of coffee &

compute ranges of samples; Plot

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room hotel. You want to

analyze the time it takes to

deliver luggage to the room.

For 7 days, you collect data

on 5 deliveries per day. Is

theprocess in control?

Hotel Example

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Hotel Data

Day Delivery Time

1 7.30 4.20 6.10 3.45 5.55

2 4.60 8.70 7.60 4.43 7.62

3 5.98 2.92 6.20 4.20 5.10

4 7.20 5.10 5.19 6.80 4.21

5 4.00 4.50 5.50 1.89 4.46

6 10.10 8.10 6.50 5.06 6.947 6.77 5.08 5.90 6.90 9.30

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R &X Chart Hotel DataSample

Day Delivery Time Mean Range

1 7.30 4.20 6.10 3.45 5.55 5.32

7.30 + 4.20 + 6.10 + 3.45 + 5.55

5

Sample Mean =

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1 7.30 4.20 6.10 3.45 5.55 5.32 3.85

7.30 - 3.45Sample Range =

Largest Smallest

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1 7.30 4.20 6.10 3.45 5.55 5.32 3.85

2 4.60 8.70 7.60 4.43 7.62 6.59 4.27

3 5.98 2.92 6.20 4.20 5.10 4.88 3.28

4 7.20 5.10 5.19 6.80 4.21 5.70 2.99

5 4.00 4.50 5.50 1.89 4.46 4.07 3.616 10.10 8.10 6.50 5.06 6.94 7.34 5.04

7 6.77 5.08 5.90 6.90 9.30 6.79 4.22

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R Chart Control Limits

UCL D R

LCL D R

RR

k

R

R

i

i

k

4

3

1

Sample Range

at Time i

# Samples

From Exhibit 6.13

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Control Chart Limits

n A2 D3 D4

2 1.88 0 3.278

3 1.02 0 2.57

4 0.73 0 2.28

5 0.58 0 2.11

6 0.48 0 2.00

7 0.42 0.08 1.92

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R

R Chart Control Limits

R

k

i

i

k

1 3 85 4 27 4 22

7 3 894

. . .

.

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R Chart Solution

From 6.13

(n = 5)

R

R

k

UCL D R

LCL D R

i

i

k

R

R

1

4

3

3 85 4 27 4 22

7 3 894

(2.11) (3.894) 8 232

(0) (3.894) 0

. . ..

.

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0

2

46

8

1 2 3 4 5 6 7

R, Minutes

Day

R Chart Solution

UCL

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X Chart Control Limits

k

R

Rk

X

X

RAXUCL

k

i

i

k

i

i

X

11

2

Sample

Rangeat Time i

# Samples

Sample

Mean atTime i

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UCL X A R

LCL X A R

XX

kR

R

k

X

X

i

i

k

i

i

k

2

2

1 1

From

Table 6-13

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UCL X A R

LCL X A R

XX

kR

R

k

X

X

i

i

k

i

i

k

2

2

1 1

Sample

Rangeat Time i

# Samples

Sample

Mean atTime i

From 6.13

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Exhibit 6.13 Limits

n A2 D3 D4

2 1.88 0 3.278

3 1.02 0 2.57

4 0.73 0 2.28

5 0.58 0 2.11

6 0.48 0 2.00

7 0.42 0.08 1.92

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1 7.30 4.20 6.10 3.45 5.55 5.32 3.85

2 4.60 8.70 7.60 4.43 7.62 6.59 4.27

3 5.98 2.92 6.20 4.20 5.10 4.88 3.28

4 7.20 5.10 5.19 6.80 4.21 5.70 2.99

5 4.00 4.50 5.50 1.89 4.46 4.07 3.616 10.10 8.10 6.50 5.06 6.94 7.34 5.04

7 6.77 5.08 5.90 6.90 9.30 6.79 4.22

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XChart Control Limits

X

X

k

R

R

k

i

i

k

i

i

k

1

1

5 32 6 59 6 79

75 813

3 85 4 27 4 22

73 894

. . ..

. . ..

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XChart Control Limits

From 6.13

(n = 5)

X

X

k

R

R

k

UCL X A R

i

i

k

i

i

k

X

1

1

2

5 32 6 59 6 79

75 813

3 85 4 27 4 22

73 894

5 813 0 58 * 3 894 8 060

. . ..

. . ..

. . . .

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XChart Solution

From 6.13

(n = 5)

X

X

k

R

R

k

UCL X A R

LCL X A R

i

i

k

i

i

k

X

X

1

1

2

2

5 32 6 59 6 79

75 813

3 85 4 27 4 22

73 894

5 813 (0 58)

5 813 (0 58)(3.894) = 3.566

. . ..

. . ..

. .

. .

(3.894) = 8.060

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XChart Solution*

0

2

46

8

1 2 3 4 5 6 7

X, Minutes

Day

UCL

LCL

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Thinking Challenge


room hotel. The hotel owner

tells you that it takes too

long to deliver luggage to the

room (even if the process

may be in control). What do

you do?

1995 Corel Corp.

N

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Redesign the luggage delivery process

Use TQM tools

Cause & effect diagrams Process flow charts

Pareto charts

Solution

Method People

Material Equipment

TooLong

06-Quality-B-09

Documents

Transcript of 06-Quality-B-09