statistical process control operations management dr. ron tibben-lembke
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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
p Chart Control Limits
# Defective Items in Sample i
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
p Chart Control Limits
# Defective Items in Sample i
# 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
p Chart Control Limits
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 =
R &X Chart Hotel Data
SampleDay Delivery TimeMean Range
1 7.30 4.20 6.10 3.45 5.555.32 3.85
7.30 - 3.45Sample Range =
Largest Smallest
R &X Chart Hotel Data
SampleDay Delivery TimeMean Range
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
Sample Range at Time i
# Samples
Sample Mean at Time i
X Chart Control Limits
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
X Chart Control Limits
UCL X A R
LCL X A R
X
X
kR
R
k
X
X
ii
k
ii
k
2
2
1 1
Sample Range at Time i
# 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
R &X Chart Hotel Data
SampleDay Delivery TimeMean Range
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
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
. . ..
. . ..
X Chart Control Limits
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
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