Download - Control charts for attributes 1
-
7/28/2019 Control charts for attributes 1
1/75
Control Charts
For Attributes
To Accompany Russell and Taylor, Operations Management, 4th Edition, 2003 Prentice-Hall, Inc. All rights reserved.
Mohammed Mokbil
July 2008
TOSHIBA EL-ARABY
-
7/28/2019 Control charts for attributes 1
2/75
Course Outline
Session 1.1
Session 1.2
Session 2.1
Session 2.2 Control charts for Attributes with variable sample size
The Control Chart for Nonconformity
Basic Principles of Control Charts
The Control Chart for Fraction Nonconforming
Day 1
Day 2
-
7/28/2019 Control charts for attributes 1
3/75
Session 1.1 :
Basic Principles of Control Charts
-
7/28/2019 Control charts for attributes 1
4/75
Session Objectives :
When You complete this session you should be able to :
Identify or Define :
Describe or Explain :
Quality
Process
Statistical Process Control
Quality Improvement
Variation
Causes of Variation
the Basic Concept of a Control Chart
How To Choose the Control Chart Type
-
7/28/2019 Control charts for attributes 1
5/75
Definitions of Quality
Qualitymeans fitness for use
- quality of design- quality of conformance
Qualityis inversely proportional to variability.
-
7/28/2019 Control charts for attributes 1
6/75
Quality ImprovementQuality improvementis the reduction of
variability in processes and products.
Alternatively, quali ty improvementis also
seen as waste reduction.
-
7/28/2019 Control charts for attributes 1
7/75
Process :
-
7/28/2019 Control charts for attributes 1
8/75
Statistical Quali ty controlis Activities
undertaken to regulate quality of a product .
Statistical process controlis a collection oftools that when used together can result in
process stability and variance reduction.
Considers a subset of SQC
Product Quali ty controlis the Activities to
evaluate and regulate quality followingproduction inspect and reject inspect and reject
-
7/28/2019 Control charts for attributes 1
9/75
The seven major tools of SPC are :
1) Histogram
2) Pareto Chart
3) Cause and Effect Diagram
4) Defect Concentration Diagram
5) Control Chart
6) Scatter Diagram
7) Check Sheet
The Magnif icent Seven :
-
7/28/2019 Control charts for attributes 1
10/75
what are Types Of Data ?
In God we trust .... all others must bring data.
-- The Statisticians Creed
We may have lots of data, but .
Does it represent the process outputs we are interested in ?
Is it representative of our current process ?
Can we split it into subsets to aid problem solving ?Can it be paired with process inputs ?
Is there operational definitions for how measurements are
taken and data recorded ?
-
7/28/2019 Control charts for attributes 1
11/75
what are Types Of Data ?
1-Attribute (discrete) data : is that which can be countedExamples:
On orOff?
2- Variable (continuous) data : is that which can be physically
be measured on a continuous scale.
Examples:
Temperature
Weight
Broken orunbroken?
-
7/28/2019 Control charts for attributes 1
12/75
Attr ibute Vs. Variable data
Which type of data ?Length in millimeters
SMC (standard manufacturing cost)
Number of breakdowns per dayAverage daily temperature
Proportion of defective items
Number of spars with concession
Lead time (days)
Mean time between failure
Variable Attribute
-
7/28/2019 Control charts for attributes 1
13/75
Which is best ?
Variable data should be the preferred type as ittells us more about what is happening to a
process.
Attribute - tells us little about the process
Variable - gives plenty of insight into the
process
-
7/28/2019 Control charts for attributes 1
14/75
Variation I ts everywhere.
No 2 things are alike.
Variation exists - even if variation small and
appears same, precision instruments showdifferences.
Ability to measure variation necessary before cancontrol.
-
7/28/2019 Control charts for attributes 1
15/75
Basically there are 3 categories of variation in
piece part production :
1.Within piece - e.g. surface roughness
2.Piece to piece - eg. dimensions
3.Time to time - different outcomes e.g.morning & afternoon, tool wear, workers tired
Variation I ts everywhere.
-
7/28/2019 Control charts for attributes 1
16/75
Equipment : tool wear, Vibrations etcMaterial : tensile strength, moisture
content etc
Environment : temperature, light, humidity etc.
Operator : method, motivation level,
training etcInspection : inspector, inspection
equipment, environment etc
Sources of Variation :
-
7/28/2019 Control charts for attributes 1
17/75
Causes Of Variation : Chance & Assignable
Chance or random causes are unavoidable
As long as fluctuate in natural/expected/stable patternof chance causes of variation which are small .
This is in state of statistical control
When causes of variation large in magnitude; can beidentified, classified as assignable causes of variation.If present, process variation is excessive (beyond
expected natural variation)
state of out of control assignable cause
Example : Body temperature - 36.5oC ~ 37.5oC
-
7/28/2019 Control charts for attributes 1
18/75
Common Causes vs. Special Causes
Process in control vs. Process out of control
A process in control.
What management likes.
Boring predictability.
The same today, tomorrowand every day.
A process out of control.
its interesting & exciting.
unpredictable and great for firefighting .
Not so good for planningthrough.
-
7/28/2019 Control charts for attributes 1
19/75
Data Distr ibution :
DATA CAN BE GROUPED TO PROVIDE EASIER ANALYSIS
Average
Dispersion
Grouped Frequency Dispersion
-
7/28/2019 Control charts for attributes 1
20/75
Distr ibutions can vary in :
1- Location.
2- Shape.
3- Spread.
Location Spread Shape
Size Size Size
-
7/28/2019 Control charts for attributes 1
21/75
MEASURES OF CENTRAL TENDENCY
Mode =Median =MeanMode
Median
Mean
Normal Distribution Skewed Distribution
-
7/28/2019 Control charts for attributes 1
22/75
MEASURES OF DI SPERSION
Range: The difference between thelargest and smallest values.
Variance: Equal to the sum of thesquared deviations from the mean,
divided by the sample size.
Standard Deviation: The square root
of the variance
-
7/28/2019 Control charts for attributes 1
23/75
Rational Subgroups
Subgroups or samples should be selected
so that if assignable causes are present, thechance for differencesbetween subgroups
will be maximized, while the chance for
differences due to these assignable causes
within a subgroup will be minimized.
-
7/28/2019 Control charts for attributes 1
24/75
As the percentage of lots in samples is increased:
the sampling and sampling costs increase.
the quality of products going to customersincreases.
Typically, very large samples are too costly.
Extremely small samples might suffer from
statistical imprecision.
Larger samples are ordinarily used whensampling for attributes than for variables.
-
7/28/2019 Control charts for attributes 1
25/75
Constructing Rational Subgroups
Select consecutive units of production. Provides a snapshot of the process.
Good at detecting process shifts.
Select a random sample over the entire samplinginterval.
Good at detecting if a mean has shifted
out-of-control and then back in-control.
-
7/28/2019 Control charts for attributes 1
26/75
Consecutive
Samples
RandomSamples
-
7/28/2019 Control charts for attributes 1
27/75
What is a Control Chart ? A control chart is a statistical tool used to distinguish between
variation in a process resulting from common causes and variationresulting from special causes.
It presents a graphic displayof process stability or
instability over time.
1 2 3 4 5 6 7 8 9 10Sample number
Uppercontrol
limit
Process
average
Lowercontrol
limit
Out of control
Upper Specification Limit
Upper Control Limit
Centerline or
Average
Lower Control Limit
Lower Specification Limit
USL
UCL
LCL
LSL
X
-
7/28/2019 Control charts for attributes 1
28/75
Histograms do not
take into account
changes over time.
Control charts
can tell us when a
process changes
-
7/28/2019 Control charts for attributes 1
29/75
A Process I s I n Control I f :
No sample points are outside control limits
Most points are near the process average
About an equal points are above & below the
centerline
Points appear randomly distributed
-
7/28/2019 Control charts for attributes 1
30/75
Typical Out-of-Control Patterns
Point outside control limits
Sudden shift in process average
Cycles
Trends
Hugging the center line
Hugging the control limits
Instability
-
7/28/2019 Control charts for attributes 1
31/75
Zones For Pattern Tests
UCL
LCL
Zone A
Zone BZone C
Zone C
Zone B
Zone A
x + 2 sigmax + 1 sigma
x + 3 sigma
x - 1 sigma
x - 2 sigma
x - 3 sigma
C.L
-
7/28/2019 Control charts for attributes 1
32/75
1. 8 consecutive points on one side of thecenter line
2. 8 consecutive points up or downacross zones
3. 14 points alternating up or down
4. 2 out of 3 consecutive points in Zone A
but still inside the control limits
5. 4 out of 5 consecutive points in Zone A or B
Identifying Potential Shifts
-
7/28/2019 Control charts for attributes 1
33/75
Identifying Potential Shifts
-
7/28/2019 Control charts for attributes 1
34/75
Shift in Process Average
-
7/28/2019 Control charts for attributes 1
35/75
Cycles
-
7/28/2019 Control charts for attributes 1
36/75
Trend
-
7/28/2019 Control charts for attributes 1
37/75
UCL
LCL
1/3
1/3
1/3
Process
Average
Hugging the Centerline or Control Limit
-
7/28/2019 Control charts for attributes 1
38/75
Control Charts and the normal Distr ibution :
-
7/28/2019 Control charts for attributes 1
39/75
Why Use a Control Chart?To monitor, control, and improve process performance over
time by studying variation and its source.
What Does a Control Chart Do?Focuses attention on detecting and monitoring process
variation over time;Distinguishesspecialfrom common causes of variation, as a
guide to local or management action;
Serves as a tool for ongoing control of a process;
Helps improve a process to perform consistently andpredictably for higher quality, lower cost, and higher effectivecapacity;
Provides a common language for discussing process
performance.
-
7/28/2019 Control charts for attributes 1
40/75
Developing Control Charts
1. Prepare
Choose measurement
Determine how to collect data, sample size,and frequency of sampling
Set up an initial control chart
2. Collect Data
Record data
Calculate appropriate statistics Plot statistics on chart
-
7/28/2019 Control charts for attributes 1
41/75
Next Steps
3. Determine trial control limits
Center line (process average)
Compute UCL, LCL
4. Analyze and interpret results
Determine if in control Eliminate out-of-control points
Recompute control limits as necessary
-
7/28/2019 Control charts for attributes 1
42/75
F inal Steps
5. Use as a problem-solving tool
Continue to collect and plot data
Take corrective action when necessary
6. Compute process capability
-
7/28/2019 Control charts for attributes 1
43/75
68.3%
+/- 1 Std Dev = 68.3%
-4 -3 -2 -1 0 1 2 3 4
2s
68.3% of data should be within 1 standard deviations of the mean if no special
cause variation is present
Choice of Control L imits
-
7/28/2019 Control charts for attributes 1
44/75
95.5%
+/- 2 Std Dev = 95.5%
-4 -3 -2 -1 0 1 2 3 4
4s
95.5% of data should be within 2 standard deviations of the mean if no special
cause variation is present
Choice of Control L imits
-
7/28/2019 Control charts for attributes 1
45/75
99.74%
+/- 3 Std Dev =99.74%
-4 -3 -2 -1 0 1 2 3 4
6s
99.74% of data should be within 3 standard deviations of the mean if no specialcause variation is present.
Control limits are an estimation of 3 standard deviations either side of the mean.
Choice of Control L imits
-
7/28/2019 Control charts for attributes 1
46/75
99.7% of the Data :
The use of 3-sigma limits generally gives good resultsin practice.
If approximately 99.7% of the data lies within 3 of the
mean { i.e., 99.7% of the data should lie within thecontrol limits}, then 1 - 0.997 = 0.003 or 0.3% of the
data can fall outside 3 {or 0.3% of the data can fall
outside the control limits}. Actually, we should use the more exact value 0.0027
The limits are often referred to as action limits.
-
7/28/2019 Control charts for attributes 1
47/75
Control Chart For
Attr ibutes Selection
c chart u chart p or np chart p chart
Defect or
Nonconformity Data
Defective or
Nonconforming Data
Constant Variable Constant Variable
sample size sample size n > 50 n > 50
-
7/28/2019 Control charts for attributes 1
48/75
Commonly used control charts :
For Variables datax-bar(mean) and R- (range) charts
x-bar and s- (standard deviation) charts
Charts for individuals (x-charts)
(MR-charts)Moving range charts
For Attributes data
For defectives (p-chart, np-chart)For defects (c-chart, u-chart)
-
7/28/2019 Control charts for attributes 1
49/75
Control Charts for attr ibutes
Fordefectives p-chart : Control chart for fraction nonconforming.
np-chart : Control Chart for Number of
nonconforming.For defects
c-chart : Control Chart for Nonconformities.
u-chart : Control Chart for Average Number ofNonconformities per Unit.
-
7/28/2019 Control charts for attributes 1
50/75
-
7/28/2019 Control charts for attributes 1
51/75
Session 1.2 :
The Control Chartfor
Fraction Nonconforming
-
7/28/2019 Control charts for attributes 1
52/75
Fraction of Nonconformingis the Ratio of the
number of nonconforming items in a population
to the total number of items in that population
Control Chart for F raction Nonconforming
p Chart
The Sample Fraction of Nonconformingis the
Ratio of the number of nonconforming items in
the sample {D} to the sample size {n}
-
7/28/2019 Control charts for attributes 1
53/75
Mean & Variances
Mean Variance
With specified standard p value :
-
7/28/2019 Control charts for attributes 1
54/75
Whenp is not known, itmust be estimated from
collected data
Average of theseindividual sample
fractions nonconforming
Fraction Nonconforming
control chart: NoStandard Given
Trial Control Limit
-
7/28/2019 Control charts for attributes 1
55/75
The np Control Chart :
Alternative top Control Chart
Based on the number nonconforming rather
than the fraction nonconforming
If standard valuep is not known, use the
estimator
-
7/28/2019 Control charts for attributes 1
56/75
Development and operation of the control Chart:
Example:
Frozen Orange juice is packed in cans formed on a machine by spinning them
from a cardboard stock and attaching a metal bottom panel. By inspection of cans
it could possibly leak and thus it is nonconforming. We wish to setup a control
chart to improve the fraction of nonconforming cans produced by this machine.
Answer:
We will first collect data for trial control limits,
With sample size n=50 the following 30 samples data were
collected.
-
7/28/2019 Control charts for attributes 1
57/75
No. of
nonconforming
cans, Di
SampleNo.
No. of
nonconforming cans,
Di
SampleNo.
816121
1017152
51883
1319104112045
202176
1822167
242398
1524149
92510101226511
727612
13281713
9291214
6302215
data for trial control limits
-
7/28/2019 Control charts for attributes 1
58/75
P=0.2313
I ni tial fraction nonconforming control chart
-
7/28/2019 Control charts for attributes 1
59/75
We note that two points from samples 15 and 23 plot abovethe UCL, so the process in out of control.
These points must be investigated to see whether assignablecause can be determined.
Analysis of the data from sample 15 indicates that a new
material was put into production during that half-our sample,it caused irregular production performance.
Furthermore, during the half-hour period in which sample 23
was obtained, a relatively inexperienced operator had beentemporarily assigned to the machine.
Consequently, samples 15 and 23 are eliminated and the newcenterline and revised control limits are calculated as :
-
7/28/2019 Control charts for attributes 1
60/75
= Points not included in control
limit calculations.
control chart with revised control l imits
-
7/28/2019 Control charts for attributes 1
61/75
Now the sample 21 exceeds the UCL .
But analysis didnt produce any assignable causes.Therefore, we decided to retain the point. And to use the newcontrol limits for future samples.
Sometimes examination of data reveals information thataffects other point.
for example : the new operator assigned again to the machineat point 24.
Then we should discard both the two points even if the otherpoint is between control limits.
-
7/28/2019 Control charts for attributes 1
62/75
Before we conclude the process is in control, we must examin
the remaining 28 samples for runs.
We find that : the largest run is one of length 5 above the centerline. Its Ok.
The process is in control at level P=0.2150 and with the revised
control limits.
Note : The process is in control , Where the Fraction of
nonconforming is too high, but in a stable manner.
That is the Top Management and the Engineering Staff to analyze
the process and try to improve the Yield.
After the Machine adjustments, the data from the next 3 shifts
was colleted as shown in the following table.
{ 24 samples with n=50 }
-
7/28/2019 Control charts for attributes 1
63/75
No. of
nonconforming
cans, Di
SampleNo.
No. of
nonconforming cans,
Di
SampleNo.
446931
847632
5481233
649534750635
551436
652637
353338
554739
640241
442
343
644
545
-
7/28/2019 Control charts for attributes 1
64/75
Continuation of f raction nonconforming control chart
-
7/28/2019 Control charts for attributes 1
65/75
From the last control chart, our immediate impression is that
the process may be out of control.
But with no reasonable causes, the only logical reason is the
machine adjustments made by the engineering staff, and
possibly the operators themselves.
It seems logical to revise the control limits again.
Calculations should be with the most recent samples ( No. 31 to
54 ) . This result in the following chart.
-
7/28/2019 Control charts for attributes 1
66/75
New control l imits on the fraction nonconforming control chart
-
7/28/2019 Control charts for attributes 1
67/75
No. of nonconforming
cans, Di
Sample
No.
No. of nonconforming
cans, Di
Sample
No.
575855
876756
1177557
978658
779459380560
581261
282362
183463
484764
585665
386566787567
688368
489769
490970
691671
8921072
593473694374
Data for the process during the next five shifts are shown in the
following table.
-
7/28/2019 Control charts for attributes 1
68/75
Completed fraction nonconforming control chart
-
7/28/2019 Control charts for attributes 1
69/75
The control chart should be continued, by marking the timescale of the control chart when a process change is made.
The control chart becomes a logbookin which the timing ofprocess interventions and their subsequent effect on process
performance are easily seen.
-
7/28/2019 Control charts for attributes 1
70/75
Control Chart for number of Nonconforming
np Chart
Alternative to p Control Chart
Based on the number nonconforming rather than the fraction
nonconforming
If standard value p is not known, use the estimator p
-
7/28/2019 Control charts for attributes 1
71/75
Revisit the first data table in the past example. You can find that:
p = 0.2313 n = 50
Therefore, the parameters of the np control chart would be :
UCL = np + 3 np(1-p)
= 50(0.2313) + 3(50)(0.2313)(0.7687)
= 20.510
C.L = np = (50)(0.2313) = 11.565
LCL = np - 3 np(1-p)= 50(0.2313) + 3(50)(0.2313)(0.7687)
= 2.620
-
7/28/2019 Control charts for attributes 1
72/75
3020100
25
20
15
10
5
0
Sample Number
1
1
NP=11.57
3.0SL=20.51
-3.0SL=2.621
I ni tial number of nonconforming (np) control chart
-
7/28/2019 Control charts for attributes 1
73/75
Some practitioners prefer to use integer values in control limitsinstead of decimal values.
In the last example use 2 and 21 as LCL and UCL.
The np chart requires that the sample size of each subgroup
be the same each time a sample is drawn.
When subgroup sizes are equal, either thep ornp chart can
be used. They are essentially the same chart.
Ch t ti
-
7/28/2019 Control charts for attributes 1
74/75
Advantages
np chart is a scaling of the vertical axis by the constant n,
provide the same information as p chart
np chart needs less calculation ( no need to calculate Di/ni)
often used when n is constant and p is small
Limitations
not easy for interpretation when n is varied (UCL LCL and CL
all vary) only plot of defects without considering sample size, hard to
take action
np Chart properties :
-
7/28/2019 Control charts for attributes 1
75/75