Statistical Process Control Operations Management Dr. Ron Lembke.

Statistical Process Control

Operations Management

Dr. Ron 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

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

Heteroskedasticity

Sub-groups with different variances

Design Tolerances

Design tolerance: Determined by users’ needs USL -- Upper Specification Limit LSL -- Lower Specification 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 USL and LSL 3 or more standard deviations away from the mean, or 3σ.

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

LSL USL

3 6

LSL USL

Process Capability

LSL USL LSL USL

Capable Not Capable

LSL USL LSL USL

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

LSL = 1.5 – 0.01 = 1.49 USL = 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>= 1.33, process is capable

Cpk< 1.33, process is not capable

Process Capability, Cpk

Tells how well parts produced fit into specs

33min

XUSLor

LSLXC pk

ProcessSpecs

3 3LSL USLX

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 < 1.33 Process not capable

33min

XUSLor

LSLXC 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?

Weighing Package and all candies Before placing candy

on scale, press “ON/TARE” button

Wait for 0.00 to appear If it doesn’t say “g”,

press Cal/Mode button a few times

Write weight down on form

Candy colors1. Write Name on form

2. Write weight on form

3. Write Package # on form

4. Count # of each color and write on form

5. Count total # of candies and write on form

6. (Advanced only): Eat candies

7. Turn in forms and complete wrappers

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 Candy Weights Avg. 2.18, stdv 0.242, c.v. = 0.111

Peanut Individuals

0

1

2

3

4

5

6

7

8

9

1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3

Mass (g)

Co

un

t

Plain Candy Weights Avg 0.858, StDev 0.035, C.V. 0.0413

Individual Plain Candies

0

2

4

6

8

10

12

14

16

Mass (g)

Co

un

t

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 See if process is “in control”

Process should show random values No trends or unlikely patterns

Visual representation much easier to interpret Tables of data – any patterns? Spot trends, unlikely patterns easily

NFL Control Chart?

Control Charts

UCL

LCL

avg

Values

Sample Number

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

Control Charts

Normal Too Low Too high

5 above, or below Run of 5 Extreme variability

Control Charts

UCL

LCL

avg

1σ

2σ

2σ

1σ

Control Charts

2 out of 3 in the outer third

Out of Control Point? Is there an “assignable cause?”

Or day-to-day variability?

If not usual variability, GET IT OUT Remove data point from data set, and recalculate

control limits

If it is regular, day-to-day variability, LEAVE IT IN Include it when calculating control limits

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)

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)

p Chart Control Limits

# Defective Items in Sample i

Sample iSize

n

ppzpUCLp

1

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 1,700 room hotel. For 7 days, you collect data on the readiness of all of the rooms that someone checked out of. Is the process in control (use z = 3)?

© 1995 Corel Corp.

p Chart Hotel Data# Rooms No. Not Proportion

Day n Ready p

1 1,300 130 130/1,300 =.1002 800 90 .1133 400 21 .0534 350 25 .0715 300 18 .066 400 12 .037 600 30 .05


079.0150,4

326

150,4

30...90130

1

1

k

ii

k

ii

n

Xp

8.5927

150,4

7

600...80013001

k

nn

k

ii

068.7/)05.0...113.010.0( p

p Chart Solution

8.592,069.0 np

8.592

079.01079.03079.0

1CL

n

ppzp

0457.0LCL,1123.0UCL

0333.0079.00111.0*3079.0

n

pp

1

Hotel Room Readiness P-Bar

1 2 3 4 5 6 70

0.02

0.04

0.06

0.08

0.1

0.12

UCL

Actual

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 DataDay 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

Table 10.3, p.433

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 Chart Control Limits

894.37

22.4...27.485.31

k

RR

k

ii

0894.3*0*

232.8894.3*11.2*

3

4

RDLCL

RDUCL

R

R

10.3 Table from , 43 DD

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

Sample Range at Time i

# Samples

Sample Mean at Time i

X Chart Control LimitsA2 from Table 10-3

k

RR

k

XX

RAXLCL

RAXUCL

k

ii

k

ii

X

X

11

2

2

Table 10.3 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 Chart Control Limits

894.37

22.4...27.485.3

813.57

79.6...59.632.5

1

1

k

RR

k

XX

k

ii

k

ii

566.3894.3*58.0813.5*

060.8894.3*58.0813.5*

2

2

RAXLCL

RAXUCL

X

X

X Chart Solution*

02468

1 2 3 4 5 6 7

`X, Minutes

Day

UCL

LCL

Statistical Process Control Operations Management Dr. Ron Lembke.

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