tm 620: quality management session eight – 23 november 2010 control charts, part ii –attributes...
TRANSCRIPT
TM 620: Quality Management
Session Eight – 23 November 2010
• Control Charts, Part II– Attributes– Special Cases
Shewhart’s Assumptions
• The data generated by the process when it is in control:– Are normally distributed– Are independent– Have a mean and standard deviation that are
fixed and unknown
Data on Quality Characteristics
• Attribute data– Discrete– Often a count of some type
• Variable data– Continuous– Often a measurement, such as length,
voltage, or viscosity
Control Chart Concept MapQuality Characteristic
Q-Sum Chart
n>10
n >1
Sm. shfts
X, Moving R
type of attribute
ni = n
p, np
ni = n
c
pvar u
X, S
X, R
no
yes
no
yes
yes
no
variable attribute
defective defects
yes yes
no no
Names for Abnormalities
• Variable data– Defective
• Attribute data– Nonconforming– Does the item meet the requirements on one
or more quality characteristics
Charts for Attributes
• Fraction nonconforming (p-chart)– Fixed sample size– Variable sample size
• np-chart for number nonconforming
• Charts for defects– c-chart– u-chart
Fraction Nonconforming
• The ratio of the number of nonconforming items in a population to the total number of items in the population– These items may have several quality
characteristics simultaneously inspected
pp
The P-Chart
ppUCL 3
ppLCL 3where
pd
nm
ii
m
1p
n
( )1
Example; P-Chart
• Operators of a sorting machine must read the zip code on a letter and diver the letter to the proper carrier route. Over one month’s time, 25 samples of 100 letters were chosen, and the number of errors was recorded. Error counts for each of the 25 days follows.
Example
Day Error %Defect Day Error %Defect1 3 0.03 14 2 0.022 1 0.01 15 3 0.033 0 0.00 16 3 0.034 0 0.00 17 2 0.025 2 0.02 18 0 0.006 3 0.03 19 1 0.017 5 0.05 20 5 0.058 2 0.02 21 4 0.049 4 0.04 22 3 0.0310 3 0.03 23 2 0.0211 0 0.00 24 4 0.0412 1 0.01 25 3 0.0313 4 0.04
Example
2503....00.01.03. p = 0.024
Example
2503....00.01.03. p
pp p
070.0100
)024.1(024.3024.
)1(3
nUCLp
= 0.024
Example
p pp0.0022.
)1( n
LCLp 3
2503....00.01.03. p
pp p
070.0100
)024.1(024.3024.
)1(3
nUCLp
= 0.024
Example; P-Chart
P-Chart; Mail Sort
-0.02
0.00
0.02
0.04
0.06
0.08
0 10 20 30
Day
% E
rro
rs
Variable P-Charts Idea
• Recall,
• For different sample size, compute CL for each sample
pppCL
n
3
1( )
pppCL
ni
i
31( )
Sample ni Di pi
p LCL UCL
1 100 12 0.120 0.0299 0.0095 0.18872 80 8 0.100 0.0334 -0.0011 0.19933 80 7 0.088 0.0334 -0.0011 0.19934 100 9 0.090 0.0299 0.0095 0.18875 110 10 0.091 0.0285 0.0136 0.18456 110 12 0.109 0.0285 0.0136 0.18457 100 9 0.090 0.0299 0.0095 0.18878 100 10 0.100 0.0299 0.0095 0.18879 90 10 0.111 0.0315 0.0046 0.193610 90 8 0.089 0.0315 0.0046 0.193611 110 12 0.109 0.0285 0.0136 0.184512 120 11 0.092 0.0273 0.0173 0.180913 100 10 0.100 0.0299 0.0095 0.188714 90 8 0.089 0.0315 0.0046 0.193615 110 12 0.109 0.0285 0.0136 0.1845
Sum = 1490 148p bar = 0.099
Variable Size P Chart
Variable Size P Chart
-0.050
0.000
0.050
0.100
0.150
0.200
0.250
0 2 4 6 8 10 12 14 16
Time
% D
efe
ctiv
e
Number Nonconforming
• np-chart
• Many non-statistically trained people find the np chart easier to interpret that the p-chart
The np Control Chart
• UCL =
• CL = np
• LCL =
)1(3 pnpnp
)1(3 pnpnp
C - Chart
• p chart shows % of parts that are defective in a lot
• np chart shows # parts defective in a lot• Suppose more than one defect can occur
in a particular part• C-chart shows # defects in a given lot
– based on the Poisson– subgroup size constant > 25
Poisson
• Distribution of rare events
• Idea: let c = # defectsc
c x
E X[ ]
x
p xe
x
x
( )!
Poisson Distribution
• Idea: let c = # defects
Control Limits c c 3
c
c x
ExampleSample # Defects LCL UCL
1 2 0 9.772 4 0 9.773 3 0 9.774 5 0 9.775 2 0 9.776 3 0 9.777 5 0 9.778 4 0 9.779 3 0 9.7710 6 0 9.7711 3 0 9.7712 3 0 9.7713 6 0 9.7714 5 0 9.7715 4 0 9.77
c bar = 3.867
Example
C - Chart
0
2
4
6
8
10
12
0 2 4 6 8 10 12 14 16
Time
De
fect
s
U - Chart
• Like the c - chart, except it removes the restriction of equal sample sizes
uc c c
n n nk
k
1 2
1 2
...
...
uS nui i /
u uControl Limits ni i 3 /
ExampleSample # Defects Sample SizeDefects/Unit LCL UCL
1 2 100 0.020 0 0.102 4 80 0.050 0 0.113 3 90 0.033 0 0.104 5 105 0.048 0 0.105 2 95 0.021 0 0.106 3 85 0.035 0 0.117 5 100 0.050 0 0.108 4 105 0.038 0 0.109 3 90 0.033 0 0.1010 6 105 0.057 0 0.1011 3 110 0.027 0 0.1012 3 85 0.035 0 0.1113 6 95 0.063 0 0.1014 5 100 0.050 0 0.1015 4 90 0.044 0 0.10
Sum = 58 1435u bar = 0.040
Example
U - Chart
0.000
0.020
0.040
0.060
0.080
0.100
0.120
0 2 4 6 8 10 12 14 16
Time
De
fect
s p
er
Un
it
CUSUM Chart
• X-bar, R charts work well to detect significant shifts in the mean (1.5 - 2.0)
• However, suppose we wish to detect smaller shifts in the process
• Consider a piston ring process with target value at 10.0 cm.– First 10 sampled at normal with = 10– Second 10 sampled at normal with = 11– Control limits computed at LCL = 7, UCL = 13
ExampleTarget = 10.0Sample x LCL UCL
1 9.45 7.00 13.002 7.99 7.00 13.003 9.29 7.00 13.004 11.66 7.00 13.005 12.16 7.00 13.006 10.18 7.00 13.007 8.04 7.00 13.008 11.46 7.00 13.009 9.20 7.00 13.0010 10.34 7.00 13.0011 10.90 7.00 13.0012 9.33 7.00 13.0013 12.29 7.00 13.0014 11.50 7.00 13.0015 10.60 7.00 13.0016 11.08 7.00 13.0017 10.38 7.00 13.0018 11.62 7.00 13.0019 11.31 7.00 13.0020 10.52 7.00 13.00
Example
X bar Chart
0.002.004.006.008.00
10.0012.0014.00
0 5 10 15 20 25
Time
Me
an
CUSUM Idea
• Show the cumulative effects of relatively small changes
i
xCi jj
o ( )
1
xxi o jj
i
o
( ) ( )1
1
x Ci o i ( ) 1
Example, CUSUM TableTarget = 10.0Sample x x - Ci = (x-)+Ci-1
1 9.45 -0.55 -0.552 7.99 -2.01 -2.563 9.29 -0.71 -3.274 11.66 1.66 -1.615 12.16 2.16 0.556 10.18 0.18 0.737 8.04 -1.96 -1.238 11.46 1.46 0.239 9.20 -0.80 -0.5710 10.34 0.34 -0.2311 10.90 0.90 0.6712 9.33 -0.67 0.0013 12.29 2.29 2.2914 11.50 1.50 3.7915 10.60 0.60 4.3916 11.08 1.08 5.4717 10.38 0.38 5.8518 11.62 1.62 7.4719 11.31 1.31 8.7820 10.52 0.52 9.30
Example; CUSUM Chart
Q - Sum Chart
-4.00
-2.00
0.00
2.00
4.00
6.00
8.00
10.00
0 5 10 15 20 25
Time
Cu
mu
lati
ve
Questions to Ask When Implementing Control Charts
• Which process characteristics to control• Where the charts should be implemented in the
process• What is the proper type of control chart for your
process• What mechanisms are there to take action
based on the analysis of the control charts• What data collection systems and computer
software should be used
Control Chart Design Issues
• Basis for sampling
• Sample size
• Frequency of sampling
• Location of control limits
SPC Implementation Requirements
• Top management commitment
• Project champion
• Initial workable project
• Employee education and training
• Accurate measurement system
Example: Catalog Company
• A catalog distributer ships a variety of orders each day. The packing slips often contain errors such as wrong purchase order numbers, wrong quantities, or incorrect sizes. The data on the next slide was collected during the month of August.
• What type of chart should you use to analyze this data?
1 8 922 15 693 6 864 13 855 5 1236 5 877 3 748 8 839 4 103
10 6 6011 7 13612 4 8013 2 7014 11 7315 13 8916 6 12917 6 7818 3 8819 8 7620 9 10121 8 9222 2 7023 9 5424 5 8325 13 16526 5 13727 8 7928 6 7629 7 14730 4 8031 8 78
DayNumber of
DefectsSample
Size
Attribute (u) Chart
0.0000
0.0500
0.1000
0.1500
0.2000
0.2500
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Sample number
De
fec
ts p
er
un
itDefects per unit
Low er control limitUpper control limit
Center line
Example: Machine Breakdowns
• The number of machine breakdowns for a particular process are tracked per day over a 25 day period. The results are on the next slide.
• What type of chart should you use to analyze this data?
Day Breakdowns1 22 33 04 15 36 57 38 19 2
10 211 012 113 014 215 416 117 218 019 3220 221 122 423 024 025 3
Attribute (c) Chart
0
5
10
15
20
25
30
35
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Sample number
Nu
mb
er
of
de
fec
tsNumber of defectsLower control limitUpper control limitCenter line
Example: Silicon Wafer Production
• The thickness of silicon wafers used in the production of semiconductors must be carefully controlled. The tolerance of one such product is specified as ±0.0050 inches. In one production facility, three wafers were selected each hour and the thickness measured carefully to within one ten-thousandth of an inch.
• What type of chart should you use to analyze this data?
Sample Obs. 1 Obs. 2 Obs. 31 41 70 222 78 53 683 84 34 484 60 36 255 46 47 296 64 16 567 43 53 648 37 43 309 50 29 57
10 57 83 3211 24 42 3912 78 48 3913 51 57 5014 41 29 3515 56 64 3616 46 41 1617 99 86 9818 71 54 3919 41 2 5320 41 39 3621 22 40 4622 63 70 4623 64 52 5724 44 38 6025 41 63 62
X-bar Chart
0
20
40
60
80
100
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Sample number
Av
era
ge
s
AveragesLower control limitUpper control limitCenter line
R-Chart
0
10
20
30
40
50
60
70
80
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Sample number
Ra
ng
es
RangesLower control limitUpper control limitCenter line
Example: Orange Juice
• Frozen orange juice concentrate is packed in 6-oz. cardboard cans. These cans are formed on a machine by spinning them from cardboard stock and attaching a metal bottom panel. By inspection of a can, we may determine whether, when filled, it could possibly leak either on the side seam or around the bottom joint. Thirty samples of fifty cans each were selected at half-hour intervals over a three shift period in which the machine was in continuous operation.
• What type of chart should you use to analyze this data?
Sample Number of Errors1 122 153 84 105 46 77 168 99 14
10 1011 512 613 1714 1215 22
16 817 1018 519 1320 1121 2022 1823 2424 1525 926 1227 728 1329 930 6
Attribute (p) Chart
0.0000
0.1000
0.2000
0.3000
0.4000
0.5000
0.6000
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Sample number
Fra
cti
on
no
nc
on
form
ing
Fraction nonconformingLower control limitCenter lineUpper control limit
Number nonconforming (np) chart
0
5
10
15
20
25
30
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Sample number
Nu
mb
er
no
nc
on
form
ing
Number nonconformingLower control limitUpper control limitCenter line
Next Class
• Homework– Ch. 11 (12) Problems 13, 14, 21– Ch. 12 (13) Problems 16, 21, 22
• Topic– Six Sigma, FMEA, Reliability
• Preparation– Chapter 15 and Handout