control charts for attributes 1

7/28/2019 Control charts for attributes 1

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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


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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


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Session 1.1 :

Basic Principles of Control Charts


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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


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Definitions of Quality

Qualitymeans fitness for use

- quality of design- quality of conformance

Qualityis inversely proportional to variability.


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Quality ImprovementQuality improvementis the reduction of

variability in processes and products.

Alternatively, quali ty improvementis also

seen as waste reduction.


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Process :


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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


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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 :


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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 ?


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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?


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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


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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


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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.


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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.


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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 :


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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


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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.


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Data Distr ibution :

DATA CAN BE GROUPED TO PROVIDE EASIER ANALYSIS

Average

Dispersion

Grouped Frequency Dispersion


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Distr ibutions can vary in :

1- Location.

2- Shape.

3- Spread.

Location Spread Shape

Size Size Size


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MEASURES OF CENTRAL TENDENCY

Mode =Median =MeanMode

Median

Mean

Normal Distribution Skewed Distribution


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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


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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.


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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.


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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.


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Consecutive

Samples

RandomSamples


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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


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Histograms do not

take into account

changes over time.

Control charts

can tell us when a

process changes


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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


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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


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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


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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


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Identifying Potential Shifts


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Shift in Process Average


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Cycles


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Trend


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UCL

LCL

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Process

Average

Hugging the Centerline or Control Limit


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Control Charts and the normal Distr ibution :


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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.


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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


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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


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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


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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


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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



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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.



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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.


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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


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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)


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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.


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Session 1.2 :

The Control Chartfor

Fraction Nonconforming


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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}


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Mean & Variances

Mean Variance

With specified standard p value :


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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


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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


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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.


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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


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P=0.2313

I ni tial fraction nonconforming control chart


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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 :


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= Points not included in control

limit calculations.

control chart with revised control l imits


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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.


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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 }


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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


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Continuation of f raction nonconforming control chart


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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.


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New control l imits on the fraction nonconforming control chart


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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.


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Completed fraction nonconforming control chart


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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.


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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


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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


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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


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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


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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 :


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control charts for attributes 1

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