Statistical Process Control Operations Management Dr. Ron Lembke.
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Transcript of 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
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 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
p Chart Control Limits
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 =
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
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
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
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