Statistical Quality Control presented by Dr. Eng. Abed Schokry

Islamic University, Gaza - Palestine

Department of Industrial Engineering

Statistical Quality Control

presented by

Dr. Eng. Abed Schokry


Department of Industrial Engineering

Statistical Quality Control

C and U Chart

presented by

Dr. Eng. Abed Schokry


Control Charts for Nonconformities (Defects)

• There are many instances where an item will contain nonconformities but the item itself is not classified as nonconforming.

• There is a single situation in which any number of incidents may occur, each with a small probability. Typical of such may occur, each with a small probability. Typical of such incidents are scratches in tables, fire alarms in a city, typesetting errors in a newspaper.

• It is often important to construct control charts for the total number of nonconformities or the average number of nonconformities for a given “area of opportunity”. The inspection unit must be the same for each unit.


• Why need it?– Control the number of nonconformities

• A nonconforming product does not satisfy one or more of the specifications

• nonconformity/defect: Each specific point at which a specification is not satisfied, e.g.,

– stones on a bottle

Control Charts for Nonconformities (Defects)- C and U Charts

– stones on a bottle– defective fuse on a car body– typos on a paper

• A unit may not be “nonconforming”, even though it has several nonconformities. So, nonconforming defects or nonconformities

• Assumption: The occurrence of nonconformities in samples of constant size is well modeled by the Poisson distribution. – The number of potential location for nonconformities is infinitely

large, and the probability of occurrence of a nonconformity at any location is small and constant


Control Charts for Attribute Data: C Chart

Number Nonconforming per Unit: c Chart Reasons for Using a c Chart

1. One or more types of nonconformities2. Poisson distribution


Control Charts for Nonconformities (Defects)

Poisson Distribution

• The number of nonconformities in a given area can be modeled by the Poisson distribution. Let c be the parameter for a Poisson distribution, then the mean and variance of the Poisson distribution distribution, then the mean and variance of the Poisson distribution are equal to the value c.

• The probability of obtaining x nonconformities on a single inspection unit, when the average number of nonconformities is some constant, c, is found using:

!x

ce)x(p

xc


The c Chart Construction

1. Collect m samples of data, each of size n

2. Determine the number of nonconformities for the ith unit. Call this value ci

3. Find the average number of nonconformities per unit, c

4. Compute the 3-sigma control limits

5. Construct the chart


c Chart Equations

cc ==∑∑ccii

mm

UCLUCL = = cc + + 3 3 ccUCLUCL = = cc + + 3 3 cc

Center LineCenter Line = = cc

LCLLCL = = cc -- 3 3 cc


Door Panels Example

99 –

88 –

77 –

66 –

c Chart for Door panelsc Chart for Door panels

UCLUCL66 –

55 –

44 –

33 –

22 –

11 –

00 –

UCLUCL

LCLLCL

I I II I I I I I I I I I I II I I I I11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919 2020

Sample NumberSample Number

I I I I I2121 2222 2323 2424 2525


c Chart Equations

cc ==∑∑ccii

mm

UCLUCL = = cc + + 3 3 ccUCLUCL = = cc + + 3 3 cc

Center LineCenter Line = = cc

LCLLCL = = cc -- 3 3 cc


Procedures with Constant Sample Size

c-chart• Standard Given:

cCL

c3cUCL

• No Standard Given:c3cLCL

cCL

c3cLCL

cCL

c3cUCL


Control Chart for Temperature



A paper mill uses a control chart to monitor the imperfection in finished rolls of paper. Production output is inspected for 20 days, and the resulting data are shown below. Use these data to set up a control chart for nonconformities per roll of paper. Does the process appear to be in statistical control? What center line and control limits would you recommend for controlling current production?

Day RollsProduced

Number ofImperfections

Day RollsProduced

Number ofImperfections

1 18 12 11 18 182 18 14 12 18 143 24 20 13 18 94 22 18 14 20 105 22 15 15 20 146 22 12 16 20 137 20 11 17 24 168 20 15 18 24 189 20 12 19 22 20

10 20 10 20 21 17


Example (Cont’d)

18 [0.1088, 1.2926]

20 [0.1392, 1.2622]

ni [LCLi, UCLi]

1.5

Control Chart for Paper Imperfections

21 [0.1527, 1.2487]

22 [0.1653, 1.2361]

24[0.1881, 1.2133]

0 4 8 12 16 20

Day

0

0.3

0.6

0.9

1.2

1.5

U0.70073


Procedures with Constant Sample Size

Choice of Sample Size: The u Chart

• If we find c total nonconformities in a sample of n inspection units, then the average number of nonconformities per inspection unit is u = c/n.

• The control limits for the average number of nonconformities is

n

u3uLCL

uCLn

u3uUCL


Procedures with Variable Sample Size

Three Approaches for Control Charts with Variable Sample Size

1. Variable Width Control Limits

2. Control Limits Based on Average Sample Size

3. Standardized Control Chart


• If sample size varies, it is always to use a u chart rather than a c chart

• Approaches

– Control limits varies with each sample size, but the center line is constant

– Use a control limits based on an average sample size

uCL;n

u3uUCL;

n

u3uLCL

ii

– Use a control limits based on an average sample size

– Use a standardized control chart (this is preferred option), with UCL=3, LCL=-3, Center line=0.

• This chart can be used for pattern recognition

m

nn

m

1ii

i

ii

nu

uuZ



Variable Width Control Limits

• Determine control limits for each individual sample that are based on the specific sample size.

• The upper and lower control limits are

in

u3u



Control Limits 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

u3u



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– Centerline at 0

– UCL = 3 LCL = -3

– The points plotted are given by:

i

ii

nu

uuZ


Demerit Systems

• When several less severe or minor defects can occur, we may need some system for classifying nonconformities or defects according to severity; or to weigh various types of defects in some reasonable manner.


Demerit Systems

Demerit Schemes

1. Class A Defects - very serious2. Class B Defects - serious2. Class B Defects - serious3. Class C Defects - Moderately serious4. Class D Defects - Minor

• Let ciA, ciB, ciC, and ciD represent the number of units in each of the four classes.


Demerit Systems

Demerit Schemes

• The following weights are fairly popular in practice:– Class A-100, Class B - 50, Class C – 10, Class D - 1

di = 100ciA + 50ciB + 10ciC + ciD

di - the number of demerits in an inspection unit


Demerit Systems

Control Chart Development

• Number of demerits per unit:

Dwhere n = number of inspection units

D =n

Dui

n

1iid


ExampleThe 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 4cassette decks?

Deck # # of Nonconformities Deck # # of NonconformitiesDeck # # of Nonconformities Deck # # of Nonconformities

2412 0 2421 1

2413 1 2422 0

2414 1 2423 3

2415 0 2424 2

2416 2 2425 5

2417 1 2426 1

2418 1 2427 2

2419 3 2428 1

2420 2 2429 1


Demerit Systems

Control Chart Development

u

uCL

ˆ3uUCL

where

and uˆ3uLCL

DCBA uu10u50u100u

2/1

DC2

B2

A2

u n

uu10u50u100ˆ


The Operating-Characteristic Function

• The OC curve (and thus the P(Type II Error)) can be obtained for the c- and u-chart using the Poisson distribution.

• For the c-chart:• For the c-chart:

where x follows a Poisson distribution with parameter c (where c is the true mean number of defects).

P x UCL c P X LCL c( | ) ( | )


The Operating-Characteristic Function

• For the u-chart:

)u|LCLx(P)u|UCLx(P )u|nLCLc(P)u|nUCLc(P

)u|LCLx(P)u|UCLx(P


Dealing with Low-Defect Levels

• When defect levels or count rates in a process become very low, say under 1000 occurrences per million, then there are long periods of time between the occurrence of a nonconforming unit.

• Zero defects occur• Zero defects occur

• Control charts (u and c) with statistic consistently plotting at zero are uninformative.



Alternative

• Chart the time between successive occurrences of the counts –or time between events control charts.

• If defects or counts occur according to a Poisson distribution, then the time between counts occur according to an then the time between counts occur according to an exponential distribution.



Consideration

• Exponential distribution is skewed.• Corresponding control chart very asymmetric.• One possible solution is to transform the exponential random

variable to a Weibull random variable using x = y1/3.6 (where y is an variable to a Weibull random variable using x = y1/3.6 (where y is an exponential random variable) – this Weibull distribution is well-approximated by a normal.

• Construct a control chart on x assuming that x follows a normal distribution.


Guidelines for Implementing Control Charts

1. Determine which process characteristics to control.2. Determine where the charts should be implemented in the

process.3. Choose the proper type of control chart.4. Take action to improve processes as the result of SPC/control 4. Take action to improve processes as the result of SPC/control

chart analysis.5. Select data-collection systems and computer software.

Documents

Statistical Quality Control presented by Dr. Eng. Abed Schokry