Statistical Process Control Operations Management Dr. Ron Tibben-Lembke

Statistical Process Control

Operations Management

Dr. Ron Tibben-Lembke

Designed Size

10 11 12 13 14 15 16 17 18 19 20

Natural Variation

14.5 14.6 14.7 14.8 14.9 15.0 15.1 15.2 15.3 15.4 15.5

Theoretical Basis of Control Charts

95.5% of allX fall within ± 2

Properties of normal distribution

X

Theoretical Basis of Control ChartsProperties of normal distribution

99.7% of allX fall within ± 3

X

Skewness Lack of symmetry Pearson’s coefficient of

skewness: 0246810121416

0246810121416

0246810121416

Skewness = 0 Negative Skew < 0

Positive Skew > 0

s

Medianx )(3

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

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.

Process Capability and 6

A “capable” process has UTL and LTL 3 or more standard deviations away from the mean, or 3σ.

99.7% (or more) of product is acceptable to customers

LTL UTL

3 6

LTL UTL

Process Capability

LTL UTL LTL UTL

Capable Not Capable

LTL UTL LTL UTL

Process Capability Specs: 1.5 +/- 0.01 Mean: 1.505 Std. Dev. = 0.002 Are we in trouble?

Process Capability Specs: 1.5 +/- 0.01

LTL = 1.5 – 0.01 = 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

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

Process Capability, Cpk

Tells how well parts produced fit into specs

33min

XUTLor

LTLXC pk

ProcessSpecs

3 3LTL UTLX

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

33min

XUTLor

LTLXC pk

006.0

505.151.1

006.0

49.1505.1min orC pk

Process Capability: Re-centered If process were properly centered Specs: 1.5 +/- 0.01

LTL = 1.5 – 0.01 = 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

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

Packaged Goods What are the Tolerance Levels? What we have to do to measure capability? What are the sources of variability?

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.

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?

Your Job Write down package #

Weigh package and candies, all together, in grams and ounces

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 for weighing

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

Ro

un

de

d W

eig

ht

- g

ram

s

0.50

0.60

0.70

0.80

Ro

un

de

d W

eig

ht

- O

un

ce

s

g - rounded

oz - rounded 0.7 Ounces

20 grams

0.6 Ounces

19 grams

18 grams

21 grams

Peanut Color Mix website

Brown 17.7% 20% Yellow 8.2% 20% Red 9.5% 20% Blue 15.4% 20% Orange 26.4% 10% Green 22.7% 10%

Classwebsite

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

So who cares? Dept. of Commerce National Institutes of Standards & Technology NIST Handbook 133 Fair Packaging and Labeling Act

Acceptable?

Package Weight “Not Labeled for Individual Retail Sale” If individual is 18g MAV is 10% = 1.8g Nothing can be below 18g – 1.8g = 16.2g

Goal of Control Charts collect and present data visually allow us to see when trend appears see when “out of control” point occurs

0102030405060

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

0102030405060

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

Natural Variation

UCL

LCL

Time

X

Definitions of Out of Control1. No points outside control limits

2. Same number above & below center line

3. Points seem to fall randomly above and below center line

4. Most are near the center line, only a few are close to control limits

1. 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

Attributes vs. VariablesAttributes: Good / bad, works / doesn’t count % bad (P chart) count # defects / item (C chart)

Variables: measure length, weight, temperature (x-bar

chart) measure variability in length (R chart)

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)

p Chart Control Limits

# Defective Items in Sample i

Sample iSize

UCLp p zp 1 p

n

p X i

i1

k

ni

i1

k



Sample iSize

z = 2 for 95.5% limits; z = 3 for 99.7% limits

# Samples

n

ppzpUCLp

1

p X i

i1

k

ni

i1

k

n ni

i1

k

k



# Samples

Sample iSize

z = 2 for 95.5% limits; z = 3 for 99.7% limits

n

ppzpUCLp

1

n

ppzpLCLp

1

n ni

i1

k

k

p X i

i1

k

ni

i1

k

p Chart ExampleYou’re manager of a 500-room hotel. You want to achieve the highest level of service. For 7 days, you collect data on the readiness of 200 rooms. Is the process in control (use z = 3)?

© 1995 Corel Corp.

p Chart Hotel DataNo. No. Not

Day Rooms Ready Proportion

1 200 16 16/200 = .0802 200 7 .0353 200 21 .1054 200 17 .0855 200 25 .1256 200 19 .0957 200 16 .080


n ni

i1

k

k

1400

7200

p Chart Control Limits16 + 7 +...+ 16

p X i

i1

k

ni

i1

k

121

14000.0864

n ni

i1

k

k

1400

7200

p Chart Solution16 + 7 +...+ 16

p X i

i1

k

ni

i1

k

121

14000.0864

n ni

i1

k

k

1400

7200

p zp 1 p

n 0.0864 3

0.0864 1 0.0864 200

p Chart Solution16 + 7 +...+ 16

p zp 1 p

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 X i

i1

k

ni

i1

k

121

14000.0864

n ni

i1

k

k

1400

7200

0.00

0.05

0.10

0.15

1 2 3 4 5 6 7

P

Day

p Chart

UCL

LCL

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

You’re manager of a 500-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 the process in control?

Hotel Example

Hotel Data

Day Delivery Time

1 7.30 4.20 6.10 3.455.552 4.60 8.70 7.60 4.437.623 5.98 2.92 6.20 4.205.104 7.20 5.10 5.19 6.804.215 4.00 4.50 5.50 1.894.466 10.10 8.10 6.50 5.066.947 6.77 5.08 5.90 6.909.30

R &X Chart Hotel Data

SampleDay Delivery TimeMean Range

1 7.30 4.20 6.10 3.45 5.555.32 7.30 + 4.20 + 6.10 + 3.45 + 5.55

5Sample Mean =



1 7.30 4.20 6.10 3.45 5.555.32 3.85

7.30 - 3.45Sample Range =

Largest Smallest



1 7.30 4.20 6.10 3.45 5.555.32 3.85

2 4.60 8.70 7.60 4.43 7.626.59 4.27

3 5.98 2.92 6.20 4.20 5.104.88 3.28

4 7.20 5.10 5.19 6.80 4.215.70 2.99

5 4.00 4.50 5.50 1.89 4.464.07 3.61

6 10.10 8.10 6.50 5.06 6.947.34 5.04

7 6.77 5.08 5.90 6.90 9.306.79 4.22

R Chart Control Limits

UCL D R

LCL D R

R

R

k

R

R

ii

k

4

3

1

Sample Range at Time i

# Samples

From Exhibit 6.13

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

R

R Chart Control Limits

R

k

ii

k

1 3 85 4 27 4 227

3 894. . .

.

R Chart Solution

From 6.13 (n = 5)

R

R

k

UCL D R

LCL D R

ii

k

R

R

1

4

3

3 85 4 27 4 227

3 894

(2.11) (3.894) 8 232

(0)(3.894) 0

. . ..

.

02468

1 2 3 4 5 6 7

R, Minutes

Day

R Chart Solution

UCL

X Chart Control Limits

k

RR

k

XX

RAXUCL

k

ii

k

ii

X

11

2


# Samples

Sample Mean at Time i


UCL X A R

LCL X A R

X

X

kR

R

k

X

X

ii

k

ii

k

2

2

1 1

From Table 6-13


UCL X A R

LCL X A R

X

X

kR

R

k

X

X

ii

k

ii

k

2

2

1 1


# Samples

Sample Mean at Time i

From 6.13

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



1 7.30 4.20 6.10 3.45 5.555.32 3.85

2 4.60 8.70 7.60 4.43 7.626.59 4.27

3 5.98 2.92 6.20 4.20 5.104.88 3.28

4 7.20 5.10 5.19 6.80 4.215.70 2.99

5 4.00 4.50 5.50 1.89 4.464.07 3.61

6 10.10 8.10 6.50 5.06 6.947.34 5.04

7 6.77 5.08 5.90 6.90 9.306.79 4.22


X

X

k

R

R

k

ii

k

ii

k

1

1

5 32 6 59 6 797

5 813

3 85 4 27 4 227

3 894

. . ..

. . ..


From 6.13 (n = 5)

X

X

k

R

R

k

UCL X A R

ii

k

ii

k

X

1

1

2

5 32 6 59 6 797

5 813

3 85 4 27 4 227

3 894

5 813 0 58 * 3 894 8 060

. . ..

. . ..

. . . .

X Chart Solution

From 6.13 (n = 5)

X

X

k

R

R

k

UCL X A R

LCL X A R

ii

k

ii

k

X

X

1

1

2

2

5 32 6 59 6 797

5 813

3 85 4 27 4 227

3 894

5 813 (0 58)

5 813 (0 58)(3.894) = 3.566

. . ..

. . ..

. .

. .

(3.894) = 8.060

X Chart Solution*

02468

1 2 3 4 5 6 7

`X, Minutes

Day

UCL

LCL

Thinking ChallengeYou’re manager of a 500-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

Redesign the luggage delivery process Use TQM tools

Cause & effect diagrams Process flow charts Pareto charts

Solution

Method People

Material Equipment

Too Long

Documents

Statistical Process Control Operations Management Dr. Ron Tibben-Lembke