Control charts for attributes 2

7/28/2019 Control charts for attributes 2

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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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Session 2.1 :

Control Charts for

Nonconformity


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A nonconformity is a quality characteristic that

does not meet specifications A nonconforming item has one or more

nonconformities

If there are 3 scratches on an item then number ofnonconformities is counted as 3

Size of the sample is called as area of opportunity

A unit may not be nonconforming, even though it hasseveral nonconformities.

So, nonconformingdefects or nonconformities


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Area of opportunityis the area that you arelooking for nonconformities

Examples are 100 m2 of fabric, 10 TV sets,

1 roll of paper This area must be chosen wide enough that

there exist a number of nonconformities

If the average number of nonconformities for aTV set is 0.08, then sample sizes of 50 would

make sense (rather than 10)


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Control Chart For

Nonconformities (Defects)

C Chart u Chart

Control Chart forThe total number of

nonconformities in a unit

Control Chart forThe Average number of

nonconformities per unit


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Chart parameters :

With standards given:

Without standards given:

Control chart for nonconformities

{ c Chart }


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Example :In a process of manufacturing the circuit boards, the number of

nonconformities was observed in 26 samples, each sample for

reason of convenience was 100 boards. Construct a control

chart to control the process.

No. of

nonconformities

Sample No.No. of

nonconformities

Sample No.

1914211

1015242

1716163

1317124

2218155

181956

3920287

3021208

2422319

16232510

19242011

1725241215261613


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Since the 26 samples contain 516 total nonconformities, we

estimate c by

c = 516/26 = 19.85

Therefore, the trial control limits are given by :

= 19.85 + 3 19.85 = 33.22

= 19.85

= 19.85 - 3 19.85 = 33.22

Plotting the points on the control chart results the following :


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We noticed that two points plot outside the control limits. 6 and 20

Investigation of sample 6 revealed that new inspector hadexamined the boards in this sample and he didnt recognize severalof the types of nonconformities that could have been present.

Furthermore, the unusually large number of nonconformities in

sample 20 resulted from a temperature control problem in thewave soldering machine.

Therefore, it seems reasonable to exclude these two samples and

revise the trial control limits.


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c = 472/24 = 19.67

And the revised control limits are :

= 19.67 + 3 19.67 = 32.97

= 19.67

= 19.67 - 3 19.67 = 6.37

the estimate ofc now computed as

These becomes the standard values against which production in

the next period can be compared.


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20 new samples, each of 100 boards are subsequently collected.

No. of

nonconformities

Sample No.No. of

nonconformities

Sample No.

18371627

21381828

16391229

22401530

19412431

12422132

14432833

9442034

16452535

21461936

These points are plotted on the control chart .


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No lack of control is indicated, the process is in

control on this level. However, The number of

nonconformities per board is still unacceptably

high.

Management action is necessary

to improve the process.


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The number of nonconformities found on final inspection of a cassette deck

is shown here. Can you conclude that the process is in statistical control?What center line and control limits would you recommend for controlling

future production? What are the center line and control limits for a control

chart for monitoring future production based on the total number of defects

in a sample of 4 cassette decks?

Deck No no of Nonconformities Deck No no of Nonconformities

1 0 10 1

2 1 11 0

3 1 12 3

4 0 13 2

5 2 14 5

6 1 15 1

7 1 16 2

8 3 17 1

9 2 18 1

Workshop :


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Control chart for Average number of

nonconformities per unit { u Chart }

There is no reason why the sample size must be restricted toone inspection unit.

We would often prefer to use several inspection units in thesample, thereby increasing the area of opportunity foroccurrence of nonconformities.

Sample size should be chosen according to :

- it should be large enough to ensure positive LCL.

- to obtain a particular probability of detecting a process shift.

- economic factors could inter into sample size determination.


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to illustrate this suppose that we were to specify a subgroupsize of n = 2.5 inspection units.

then, the sample size becomes 2.5 * 100 = 250 boards.

to construct a chart once a new sample size has been selected.

You can use a control chart based on the average no of

nonconformities per inspection unit.

m

u

u;n

cu

m

1i

i

i

ii

ui: average number of nonconformities per unit in a sample.

ci: number of nonconformities in sample i (n is not necessary be integer

ni: size of sample i


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Example :A personal computer manufacturer wishes to establish a control chart for

nonconformities per unit on the final assembly line. The sample size is selected

to be 5 computers. Data was collected in the following table for 20 samples.

Average no. ofnonconformities per unit uiTotal no. ofnonconformitiesSample size nSample No. I

2.01051

2.41252

1.6853

2.81454

2.01055

3.21656

2.21157

1.4758

2.01059

3.015510

1.89511

1.05512

1.47513

2.211514

2.412515

1.26516

1.68517

2.010518

1.47519

1.05520


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m

u

u

m

1i

i

We would estimate the average no. of nonconformities to be:

= 38.60/20 = 1.93

n

u3uUCL

uCL

n

u3uLCL

= 1.93 + 31.93/5 = 3.79

= 1.93

= 1.93 - 31.93/5 = 0.07

The Control chart is shown in the following fig.


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Sample

SampleCoun

tPerUnit

191715131197531

4

3

2

1

0

_U=1.93

UCL=3.794

LCL=0.066

a control chart for nonconformities per unit


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The preliminary data dont exhibit lack of

statistical control ,

The trial control limits above would be adopted

for current control purposes. the process is in

control on this level.

Although the process is in control, the average

number of nonconformities per unit is still

unacceptably high.

Management action is necessary

to improve the process.


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Demerit Systems for Defects


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Session 2.2 :

Control charts for Attributes

with variable sample size


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In some applications of the control chart for

the fraction nonconforming, the sample is a100% inspection of the process output over

some period of time.

Since different numbers of units could be

produced in each period, the control chart

would then have a variable sample size.


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Three Approaches for Control Charts with

Variable Sample Size

1. Variable Width Control Limits2. Control Limits Based on Average Sample Size

3. Standardized Control Chart


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Variable Width Control L imits

Determine control limits for each individual

sample that are based on the specific sample

size. The upper and lower control limits are

in)p1(p3p


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Control L imits Based on an Average Sample Size

Control charts based on the average sample size

results in an approximate set of control limits.

The average sample size is given by

The upper and lower control limits are

m

nn

m

1ii

n

)p1(p3p


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The Standardized Control Chart

The points plotted are in terms of standard deviation

units. The standardized control chart has the follow

properties:

Centerline at 0

UCL = 3 LCL = -3

The points plotted are given by:

E l


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ExampleNo. of nonconforming units DiSample size niSample No. i

121001

8802

6803

91004

101105

121106

111007

161008

10909

69010

2011011

1512012

912013

812014

611015

88016

108017

78018

59019

810020

510021

810022

1010023

69024

99025

==

S l ti


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SolutionLCLUCLziDi / niDiniSample No. i

0.11830.0090.0290.120121001

0.19500.0330.1008802

0.19500.0330.0756803

0.1830.0090.0290.09091004

0.1800.0120.0280.091101105

0.1800.0120.0280.109121106

0.1830.0090.0290.110111007

0.1830.0090.0290.160161008

0.1890.0030.0310.11010909

0.1890.0030.0310.06769010

0.1800.0120.0280.1822011011

0.1770.0150.0270.1251512012

0.1770.0150.0270.075912013

0.1770.0150.0270.067812014

0.1800.0120.0280.055611015

0.19500.0330.10088016

0.19500.0330.125108017

0.19500.0330.08878018

0.1890.0030.0310.05659019

0.1830.0090.0290.080810020

0.1830.0090.0290.050510021

0.1830.0090.0290.080810022

0.1830.0090.0290.1001010023

0.1890.0030.0310.06769024

0.1890.0030.0310.10099025

Control chart using the var iable width control l imits :


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Sample

Proportion

252321191715131197531

0.20

0.15

0.10

0.05

0.00

_P=0.0955

UCL=0.1885

LCL=0.0026

1

Control chart using the var iable width control l imits :

Control char t using the average sample size :


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Sample

Proportion

252321191715131197531

0.20

0.15

0.10

0.05

0.00

_P=0.0955

UCL=0.1846

LCL=0.0064

1

Control char t using the average sample size :


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Notes on control char ts with var iable sample size :

We must be careful in analyzing runs or abnormalpatterns on control charts with variable sample sizes.

The problem is that a change in the sample fraction

nonconforming must be interpreted relative to the

sample size.

Example: p = 0.2

p1 = 0.28 p2 = 0.24

n1 = 50 n2 = 250

1 = 1.41 2 = 1.58

It is clear that looking for runs or other random patterns

is virtually meaningless here.


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Tests for runs and pattern could safely be applied to the

standardized control charts.

The difficulty in the standardized control chart is large for

operating personal to understand and interpret. As the actual

fraction nonconforming has been lost.

The standardized control charts is also recommended whenthe length of production runs is short.

In Control charts for nonconformities with variable sample

size, it will be very difficult to use these procedures with cchart because both the center line and control limits will

vary with the sample size.

The correct procedure is to use u chart.


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Process Capabil i ty

Tolerancesdesign specifications reflecting product

requirements

Process Stabil ity and Capabil i ty

Once a process is stable, the next emphasis

is to ensure that the process is capable.

Process capability refers to the ability of a

process to produce a product that meets

specifications.


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The Difference Between Capabil i ty

and Stabil i ty?

Once again, a process is capable if

individual products consistently meetspecifications.

A process is stable if only common

variation is present in the process.


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(b) Design specificationsand natural variation thesame; process is capableof meeting specificationsmost of the time.

DesignSpecifications

Process

(a) Natural variationexceeds designspecifications; processis not capable ofmeeting specificationsall the time.


Process


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(c) Design specificationsgreater than naturalvariation; process iscapable of alwaysconforming tospecifications.


Process

(d) Specifications greaterthan natural variation, butprocess off center; capablebut some output will notmeet upper specification.


Process


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Control

Capability

Capable

Not Capable

In Control Out of Control

IDEAL

Process Capabil i ty and control


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Control charts for attributes 2

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