Quality Control and Realiability_all in One

7/31/2019 Quality Control and Realiability_all in One

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MAN- 302

Quality and ReliabilityManagement

Text Book: Introduction to Statistical Quality Control

by

Douglas C. Montgomery.

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4/19/2012 5:28 PMB. Shahul Hamid Khan, IIITDM2

Definition of Quality Dimensions of quality Quality control

- Seven statistical tools of quality, Control charts for variablesand attributes, New seven management tools, Process

capability concepts, Concept of six sigma, Concept of

Product Life cycle, Basic concept of ISO 9000 and other

quality systems

Reliability Introduction Definitions Reliability evaluation

- Failure data analysis Mean Time to Failure, Maintainability

& Availability concepts Reliability improvement techniques Design for reliability


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Steps for Producing Products

1. Planning and forecasting2. Conceptualization

3. Feasibility Assessment

4. Establishing the Design Requirements

5. Preliminary Design (Embodiment Design)

6. Detailed Design

7. Process planning

8. Production Planning and Tool Design

9. Prototyping

10. Production

11. Testing and Inspection



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The production processes are not perfect!

Which means that the output of these processes will not be perfect.

(non identical and non-deterministic)

Successive runs of the same production process will produce non-

identical parts.



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Alternately, similar runs of the production process will vary, by some

degree, and impart the variation into the some product characteristics.

Because of these variations in the products, we need probabilistic models

and robust statistical techniques to analyze quality of such products.

As quality measurements will vary from item to item, and there will

be a probability distribution associated with the population of such

measurements.



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Objective of Quality control

Objective ofquality control is

To develop a scheme for sampling a process,

Making a quality measurement of interest on sample items

and

Making a decision whether the process is in control or not.

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QUALITY CONTROL

OFF-LINE QUALITY

CONTROL

STATISTICAL PROCESS

CONTROLACCEPTANCE

SAMPLING PLANS



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This is a traditional definition

This is a modern definition of quality



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Two Different Approaches



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An equivalent definition is that quality improvement is the

elimination of waste. This is useful in service or transactional

businesses.



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Statistics is a way to get information from data

Data Statistics

Information

Data: Facts, especially

numerical facts, collected

together for reference or

information.

Information: Knowledge

communicated concerning

some particular fact.

Statistics is a tool for creating new understanding from a set

of numbers.



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Statistical Concepts

Population

A population is the group of all items that possess a certain

characteristic of interest.

Size: very large; sometimes infinite.

Sample

A sample is a set of data drawn from the population.

Size: Small (sometimes large but less than the population)



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Parameter

A descriptive measure of a population.

A parameter is a characteristic of a population, something that

describes it.

Statistic

A descriptive measure of a sample.



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Populations have Parameters

Samples have Statistics

Parameter

Population Sample

Statistic

Subset



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Statistical

Methods

Descriptive

Statistics

Inferential

Statistics

EstimationHypothesis

Testing



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Descriptive Statistics describe the data set that is being analyzed, but doesnt

allow us to draw any conclusions or make any interferences about the data.

Hence we need another branch of statistics: inferential statistics.

Inferential statistics is also a set of methods, but it is used to draw conclusions or

inferences about characteristics ofpopulations, based on data from a sample.



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Measures of central tendency Mean, median, mode, etc.

Quartile

Measure of variation

Range, interquartile range, variance and standard deviation,coefficient of variation

Shape

Symmetric, skewed, using box-and-whisker plots

Coefficient of correlation



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Statistical Inference

Statistical inference is the process of making an estimate, prediction, or decision

about a population based on a sample.

Parameter

Population

Sample

Statistic

Inference



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Population

Sample

Sample

Statistics

Estimates



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Mean, , isunknown

Population Random SampleI am 95%confidentthat is

between40 & 60.

Mean

X= 50



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Central Tendency

MeanMedian

Mode

Quartile

Summary Measures

Variation

Variance

Standard Deviation

Coefficient of

Variation

Range



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Mean (arithmetic mean) of data values

Sample mean

Population mean

1 1 2

n

i

i n

X

X X XX

n n

= + + += =

1 1 2

N

i

i N

XX X X

N N

=+ + +

= =

Sample Size

Population Size



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The most common measure of central tendency

Acts as Balance Point

Affected by extreme values (outliers)

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Mean = 5 Mean = 6



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Robust measure of central tendency

Not affected by extreme values

The value of Middle when the observations are ranked.

Property:

50% of the values are Less than or equal to it.

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Median = 5 Median = 5



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A measure of central tendency Value that occurs most often


Used for either numerical or categorical data

There may be no mode or several modes

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mode = 9

0 1 2 3 4 5 6

No Mode



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Engineering example for Mode

A hardware store wants to determine what size of circular saws it should

stock. From the past sales data, a random sample of 30 pieces are below (Inmm)

80 120 100 100 150 120 80 150 120

80 120 100 120 120 150 80 120 100120 80 100 120 120 150 120 100 120

120 100 100

Mode 120



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Difference between the largest and the smallest observations:

Ignore the way in which data are distributedLargest SmallestRange X X=

7 8 9 10 11 12

Range = 12 - 7 = 5

7 8 9 10 11 12

Range = 12 - 7 = 5



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Q1, the first quartile, is the value such that 25% of the

observations are smaller, corresponding to (n+1)/4 ordered

observation

Q2, the second quartile, is the median, 50% of the observations

are smaller, corresponding to 2(n+1)/4 = (n+1)/2 orderedobservation

Q3, the third quartile, is the value such that 75% of the

observations are smaller, corresponding to 3(n+1)/4 ordered

observation



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Youre a financial analyst for Prudential-Bache Securities. You havecollected the following closing stock prices of new stock issues:

17, 16, 21, 18, 13, 16, 12, 11, 17.

Measure central tendencyandvariationusing quartiles.



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Interquartile Range (IQR)


Also known as mid-spread

Spread in the middle 50%

Difference between the first and third quartiles


3 1Interquartile Range 17.5 12.5 5Q Q= = =

Data in Ordered Array: 11 12 13 16 16 17 17 18 21



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Box-and-Whisker Plot

Graphical Display of Data Using 5-Number Summary.

Median

4 6 8 10 12

Q3Q1 XlargestXsmallest



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Variation

Variance Standard Deviation Coefficientof Variation

Population

Variance

Sample

Variance

Population

Standard

Deviation

Sample

Standard

Deviation

Range

Interquartile Range



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Important measure of variation

Shows variation about the mean

Sample variance:

Population variance:

( )2

2 1

N

i

i

X

N

=

=

( )22 1

1

n

i

i

X X

Sn

=

=



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Most important measure of variation Shows variation about the mean

Has the same units as the original data

Sample standard deviation:

Population standard deviation:

( )2

1

1

n

i

i

X X

Sn

=

=

( )2

1

N

i

i

X

N

=

=



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Data Collection

To control or improve a process we need information or Data.

Data on quality characteristics is described by a random variable

Random variable1. Discrete variable

2. Continuous variable

Discrete variableNo ofdefective rivets in assembly

No ofsatisfied customers in a retail shop



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Continuous variable

Thickness of a plate

Viscosity of an oil

Diameter of a shaft



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36.0 36.2 34.8 36.0 34.6 38.4 35.4 36.834.7 33.4 37.4 38.2 31.5 37.7 36.9 34.034.4 35.7 37.9 39.3 34.0 36.9 35.1 37.0

33.2 36.1 35.2 35.6 33.0 36.8 33.5 35.035.1 35.2 34.4 36.7 36.0 36.0 35.7 35.738.3 33.6 39.8 37.0 37.2 34.8 35.7 38.937.2 39.3

The following data represent the heights (ininches) of a random sample of 50 two-year oldmales.

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Mean = 15.5

s = 3.33811 12 13 14 15 16 17 18 19 20 21

11 12 13 14 15 16 17 18 19 20 21

Data B

Data A

Mean = 15.5

s = .9258

11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5

s = 4.57

Data C



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Central Limit Theorem

The central limit theorem is: Sampling distributions can be

assumed to be normally distributed even though the

population (lot) distributions are not normal.

The theorem allows use of the normal distribution to easily set

limits for control charts and acceptance plans for both attributes

and variables.



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Sampling Distributions

The sampling distribution can be assumed to be normally distributedunless sample size (n) is extremely small.

The mean of the sampling distribution ( x ) is equal to the population

mean (

).

The standard error of the sampling distribution (x ) is smaller than

the population standard deviation (x ) by a factor of

1/

-

n

=


x

x

=

n


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Mean of Sampling Distribution X

Sample Mean1 3.55

2 3.59

3 3.48

4 3.51

5 3.496 3.46

7 3.48

8 3.52

9 3.51

10 3.49

=

The sampling distribution of sample mean

is approximately normal



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The difference between specification limits and control limits

Specification limits ---- the voice of the customer

Control limits ----- the voice of the process.



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Confidence interval for the Average

What is the 90% confidence interval for the average, where sample size n

= 15, S= 1.2 and Sample average X = 25.

(Note: When sample size is less than 30; use t- distribution. If greater

than or equal to 30; use normal distribution)

__

(1 )

- Risk4/19/2012 5:28 PM50

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51

1 23

2 28

3 30

4 30

5 20

6 267 29

8 21

9 26

10 24

11 24

12 24

13 22

14 30

15 22

Average value = 25


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A random sample of 20 observations on welding time is given (in mins). Find

IQR for these data

2.2 2.5 1.8 2.0 2.1 1.7 1.9 2.6 1.8 2.3

2.0 2.1 2.6 1.9 2.0 1.8 1.7 2.2 2.4 2.2

Location of Q1 = 0.25 (n+1) = 0.25 (21) = 5.25

Location of Q3 = 0.75 (n+1) = 0.75 (21) = 15.75

Q1 = 1.825

Q2 = 2.275

IQR =Q3-Q1= 0.45

Quartiles


f d Sk


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Measurement of Kurtosis and Skewness

Skewness coefficient



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Kurtosis coefficient

Kurtosis is a measure of Peakedness of data set



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Graphical Methods of Data Representation and

Quality Improvement tools

Check sheet and Histogram

Pareto diagram

Run chart Box plot

Scatter diagram

Control charts Cause and effect diagram



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N l Di t ib ti d t


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Normal Distribution data

29 39 56 50 48

54 47 75 39 29

35 42 56 44 42

68 29 60 41 41

55 51 41 72 34

49 61 54 44 55

49 59 41 40 50

40 40 55 51 52

55 61 53 36 49

36 35 52 55 59



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Box-Whisker plot


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Box Whisker plot

Steps

1. Determine first quartile Q1. This value determines the lower edge of the box

2. Determine Third quartile Q3. This value determines the upper edge of the box

3. Find IQR

4. Find median of the set Q2. Draw a line at median to divide the box

5. Two lines known as whiskers, are drawn outward from the box.

one end extended from Q3 -- to either a Maximum data value (or) Q3+1.5 (IQR)

(whichever is lower)



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Box andWhisker Plot Example


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Box and Whisker Plot Example

Suppose you wanted to compare the performance of threelathes responsible for the rough turning of a shaft.

The design specification is 18.85 +/- 0.1 mm.

Diameter measurements from a sample of shafts taken from

each roughing lathe are displayed in a box and whisker plot.


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Applications


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Lathe 1 appears to be making good parts, and is centered in the

tolerance.

Lathe 2 appears to have excess variation, and is making shafts

below the minimum diameter.

Lathe 3 appears to be performing comparably to Lathe 1.

However, it is targeted low in the tolerance, and is making shafts

below specification.


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18.6

18.65

18.7

18.75

18.8

18.85

18.9

18.95

Data Set # 1 Data Set # 2

Lower Quartile

Minimum

Median

Maximum

Upper Quartile


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Scatter Plots

The Scatter plot is another problem analysis tool. Scatter plots are also calledcorrelation charts.

A Scatter plot is used to uncover possible cause-and-effect relationships.

It is constructed by plotting two variables against one another on a pair of

axes.

A Scatter plot cannot prove that one variable causes another, but it does show

how a pair of variables is related and the strength of that relationship.



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Scatter Diagram


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Scatter Diagram



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Pareto Analysis is used to assist in prioritizing or focusing activities.

Procedure

Decide the objectives of Pareto analysis

Develop list of the responses to be classified

Collect data

Rank the categories

Compute cumulative frequency

Plot the diagram


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Dr. Shahul Hamid Khan

Introduction to Control Charts

Introduction to Control chart


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t o uct o to Co t o c a t


Symbols

parameter

- estimator

- probability of type I error

- probability of type II error

- process standard deviation

x - standard deviation of sample mean


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Rule 1

A process is assumed to be out of control if a single point

plots outside the control limits.


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Construct two lines at two sigma deviations above and

below center line


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Rule 2

A process is assumed to be out of control if two out of three

consecutive points falls outside the two sigma warning limits on

the same side of the center line.


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Rule 5

A process is assumed to be out of control if there is a run of six or

more consecutive points steadily increasing or decreasing .


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Control Charts for Mean and Range


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g No of Samples

Control Charts for Mean and Range

Average

Range


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RD=LCL

RD=UCL

LimitsControlChartR

3

4

Trial Control Limits


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Revised Control Limits


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RA-x=LCL

RA+x=UCL

LimitsControlChartx

2

2

X- bar chart


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Proportion of Nonconforming items (P- Chart)

No of Nonconforming items (np- Chart)

These above two charts are based on Binomial

Distribution

Charts for Attributes

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3

Dr. SHAHUL HAMID KHAN


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4/19/2012 5:30 PMDr. SHAHUL HAMID KHAN

4

Total Number of nonconformities (c- chart)

Nonconformities per unit (u - chart)

Based on poisson distribution


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Terms used


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n Sample size

g - No of samples

p - Population proportion nonconformingp - Sample proportion nonconforming^

p

^- Standard deviation of p

Var (p) =^ E (p) = p^

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6

Dr. SHAHUL HAMID KHAN

Attribute


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Defect is appropriate for use when evaluation is

in terms of usage.

Nonconformity is appropriate for conformance to

specifications.

The term Nonconforming Unitis used to describe

a unit of product or service containing at least

one nonconformity.


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The P Chart


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The fraction nonconforming, p, is usually small, say,

0.10 or less.

Because the fraction nonconforming is very small,

the subgroup sizes must be quite large to produce

a meaningful chart.


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Revised Control limits


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17

P- chart for the Standard Specified


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18

The standard or Target value


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23

The Actual average proportion of Nonconforming

items is

P Chart (Variable Sample Size)


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24

1) Control Limits for Individual Samples2) Control Limits based on Average sample size

Control Limits for Individual Samples


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Example - Problem


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4/19/2012 5:30 PMDr. SHAHUL HAMID KHAN26


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For Instance, the calculations for sample 1 are as follows


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Control Limits based on Average sample size


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= 4860/ 20 = 243


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The Standardized value of proportion nonconforming for


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The Standardized value of proportion nonconforming for

the ith sample may be expressed as

Special considerations for p-chart

32


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32

Necessary assumptions

1. Each items are assumed to be independent of

each other with respect to meet the specifications

This assumption may not be valid if products are

manufactured in batches

Observations below LCL for p-chart


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Comparison with specified Standard (Po)

Impact of Design Specifications

Average P value may be High, even if the process is stable and in control.

Only some change in Design Specifications may reduce p value.

Tolerances can be loosen without changing Specification limits.

It gives Information about overall Quality Level

Chart for Number of Nonconforming (np Chart)

34


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34

The control limits for np chart are


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40

The c-Chart


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40

Nonconformity is defined as a quality characteristic that does

not meet some specifications

It is possible for a product to have one or more non-

conformities and still acceptable

C chart is used to track the Total No of non-conformities in a

sample of constant size.

41

Poisson Distribution


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41

The mean and variance of poisson distribution are given below

Probability of observing x nonconformities are

42

The c-Chart


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( ) ( )

.3

:becomethenlimitscontrolestimatedThe

.1

:isofestimatorthen theinspected,areitemsIf

.3

:arelimitscontrolthe,andSince

1

CC

Cn

C

n

CVarCE

n

i

i

ii

=

==

=

Poisson distribution Example

43


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Example


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1-/2 = 1 0.01/2 = 1 0.005 = 0.995


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Chart for No of Nonconformities per Unit (u - Chart)

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(1)

(2)


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S- Chart (With no Given Standard)


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By Eqn (1)


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Unstable and Stable Processes


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Illustration of processes that are (a) unstable or out of control and (b) stable or in control.Note in sketch (b) that all distributions have lower standard deviations and have meanscloser to the desired value.


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Construct OC curve for increase in process mean from120 kg (take the same problem)


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Z = 130.74 123.578

3.58

UCL Mean Z Z130.733 123.578 1.998603352 2UCL = 130 74


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130.733 127.156 0.999162011 1130.733 130.733 0 0

130.733 134.311 -0.999441341 -1

130.733 137.888 -1.998603352 -2

130.733 141.466 -2.998044693 -3

109.26 123.578 -3.999441341 -4

109.26 127.156 -4.998882682 -5

109.26 130.733 -5.998044693 -6

109.26 134.311 -6.997486034 -7

109.26 137.888 -7.996648045 -8109.26 141.466 -8.996089385 -9

UCL 130.74

LCL = 109.26


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Average Run Length


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The Average Run Length (ARL) is the average number

of points that most be plotted before a point indicates

an out-of-control condition.

For chart this can be calculated as

ARL=1/pwhere p is the probability that any one point exceedsthe control limit.

OC Curve for p- Chart


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Where,

n sample sizep - sample proportion nonconforming

X No of nonconforming items

^

Ability to detect a shift in the process fraction nonconforming from anominal value p to some other value p

Example Plastic Injection Molding process


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Given:Sample size (n) = 50

No of samples (g) = 25

Total no of nonconforming items = 90


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P np

0.08 4

0.09 4.5

0.1 5

0.15 7.5

0.2 10

0.28 14

0.4 20


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OC curve for C chart


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Control Charts for Individual Units


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Moving Range found from two successive observations

For No Given Standard:

= MR/ d2

For g individual observations, there will be g-1 moving ranges.

n=2 here


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For n=2 ,

For Given Standard


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If the standard values specified are


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PROCESS CAPABILITY

ANALYSIS

Dr. B. SHAHUL HAMID KHAN


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Specifications and Process capability

Case 1: process spread less than specification spread

Case 2: process spread equal to specification spreadCase 3: process spread greater than specification

spread


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Case 2: Process spread equal to specification spread


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Case 3: Process spread greater than specification spread


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PROCESS CAPABILITY INDEX

Process capability index is an easily understood aggregate

measure of the Goodness of the process performance


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For practical applications is unknown, so use estimator of

Use S or R/ d2


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When Cp = 1, the process spread equals the specificationspread and the process is said to be barely capable


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This value of Cp, which is less than 1, indicates that the process is

t bl f ti th ifi ti


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not capable of meeting the specifications

Capability ratio

Percentage of specification band

CR = 1/ Cp

A process in control has an estimated standard deviation of 3

mm. The specification limits for the corresponding product are

100 7 mm. Estimate the capability ratio of the process and

comment on the process potential


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comment on the process potential.

The percent of the specification range used by the process is

128.6%, which is 28.6% more than what is permissible. Even if

the process were centered at the target value of 100, which isthe most favorable situation, it would still not meet the

specifications


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where denotes the standard normal cumulative distribution function

Recall that Cpk measures actual rather than potential process capability.

Let's consider Figure 1.0, where USL =62 mm, LSL = 38 mm,and T =50 three processes, A, B, and C, with different meansand standard deviations


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Effect of Measurement Error on Capability Indices

Whenever measurements are involved, the variability of the

observations is depends on the variability of the product


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observations is depends on the variability of the product

characteristic.

AIM

To study the effect of measurement error on the process

potential as it impacts the Cp index and the capability ratio CR.


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Measurement errors are usually normally distributed

An estimate of measurement error is obtained through an index known

as the Precision- to Tolerance ratio (r)


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as the Precision to Tolerance ratio (r)

Observed Cp


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where CR* represents the capability ratio as calculated from the measured

observations

Assumptions in Process capability Ratio

1) The Quality Characteristic has a normal distribution


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) The Quality Characteristic has a normal distribution

2) Process is in statistical control

3) In case of two sided specifications, the process mean is

centered between the lower and upper specification limits

Use confidence Intervals for Process capability Ratio

Confidence Intervals for Process capability Ratio (Cp)


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Use S instead of R/d2

NOTE


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Confidence Intervals for Process capability Index (Cpk)


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n sample size

Example Problem

Sample size n = 20

Cpk = 1.33


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p

Determine 95 % confidence interval on Cpk value.


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Accuracy

The predicted difference on average between the measurement and the true value.

Accuracy is also known as bias

Repeatability


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p y

The variation that occurs among measurements made by the same operator.

Repeatability is a form of random variation. Repeatability is also known as

Equipment variation (EV).

Reproducibility

The difference in the average of groups of repeated measurements made by

different operators. Reproducibility is also known as appraiser variation (AV). Variation between operators.

Example of Repeatability

Operator 1 measures the diameter of steel shaft


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(5 different measurements on same part)

10. 01510. 009

10. 012

10. 021

10. 011

Example of Reproducibility


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Example of Reproducibility

Operator 1

Operator 2

Operator 3

Operator 4

Calculate

Standard deviation for operator 1

Standard deviation for all operators


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5 samples are chosen for measurement. Two operators are chosen. Each of 5 parts ismeasured two times.


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Traditionally, the Repeatability (EV) as well as reproducibility (AV) is reported as 5.15

timesvalue to reflect a 99 % level of confidence

EV = 5.15 * ev


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Here we are taking one sample; two operators

m - No of Operators


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N Op

g - No of samples

m = 2

g = 1

1 0.889

2 0.855

3 0.8684 0.888

5 0.867

6 0.886

7 0.859

8 0.87

9 0.8910 0.87

Average 0.8742

Operator A


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Confidence Interval for Repeatability, Reproducibilityand Combined R&R


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ACCEPTANCE SAMPLING

B. Shahul Hamid Khan

QUALITY CONTROL


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QUALITY CONTROL

OFF-LINE QUALITY

CONTROL

STATISTICAL PROCESS

CONTROLACCEPTANCE

SAMPLING PLANS


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ADVANTAGES AND DISADVANTAGESOF SAMPLING

ADVANTAGES

1. If inspection is destructive, 100% inspection is not feasible.


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2. Sampling is more economical and causes less damage due to handling

3. Sampling reduces inspection error

DISADVANTAGES

1. There is a risk of rejecting "good" lots or accepting "poor" lots, identified as the

producer's risk and consumer's risk, respectively.

2. There is less information about the product, compared to that obtained from 100%

inspection

3. The selection and adoption of a sampling plan require more time and effort in

planning and documentation.

Acceptance Sampling

Acceptance sampling is a method used to accept or reject

product based on a random sample of the product.


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The purpose of acceptance sampling is to make decision on

lots (accept or reject) rather than to estimate the quality of alot.

Acceptance sampling plans do not improve quality

P Proportion nonconforming, or Lot Quality

AOQ Average outgoing quality


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ATI Average total inspection

AOQL Average outgoing quality limit

N Lot sizec Acceptance number for a single samplingplan

ASN Average sample number

AQL Acceptable quality levelLTPD Lot Tolerance Percent Defective

Producers and Consumers Risk

AQL or acceptable quality level

proportion defect the customer will accept a given lot


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LTPD or lot tolerance percent defective

limit on the number of defectives the customer will accept

or producers risk

probability of rejecting a good lot

or consumers risk

probability of accepting a bad lot


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Operating Characteristic Curve

n = 990.70.8

0.9

1

ce

= 0.05 (producers risk)


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c = 4

AQL LTPD

0

0.1

0.20.3

0.4

0.5

0.6

1 2 3 4 5 6 7 8 9 10 11 12

Percent defective

Probability

ofacceptan

=0.10(consumers risk)

Acceptable quality

levellot tolerance percent

defective

Single sampling plan for Attributes

Example


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Lot size N = 10000

Sample size n = 89

Acceptance No c = 2

d - No of defective or nonconforming items

OC Curve


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The Ideal OC Curve


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Effect of n and c on OC Curves


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Type A and Type B OC Curve


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Lot

Accept Reject

T I E


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GoodL

BadLot

No ErrorType I ErrorProducer Risk

Type II ErrorConsumers Risk

No Error

AQL (Acceptable Quality Level)

Poorest level of quality for vendors process that the customer

ld id b bl


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would consider to be acceptable .

AQL property of vendors Mfg process

Not a property of sampling plan

Based on AQL we can design the sampling plan

Design a sampling plan, such that OC curve gives a


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higher probability of acceptance at AQL

Pa = 95 %

Producers risk refers to the probability of rejecting a good lot. In order to

calculate this probability there must be a numerical definition as to what

constitutes good


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AQL (Acceptable Quality Level) - the numerical definition of a good lot.

The ANSI/ASQC standard describes AQL as the maximum percentage or

proportion of nonconforming items or number of nonconformities in a batch

that can be considered satisfactory as a process average

Consumers Risk refers to the probability of accepting a bad

lot where:

LTPD (Lot Tolerance Percent Defective) - the numerical


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definition of a bad lot described by the ANSI/ASQC

standard as the percentage or proportion of

nonconforming items or nonconformities in a batch for which

the customer wishes the probability of acceptance to be a

specified low value.

LTPD Poorest level of quality that the customer is


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willing to accept in an individual lot

Low probability of acceptance (10 %)

Evaluating sampling plans

OC curve is the measure of performance of a sampling plan.

We use other measures to evaluate the goodness of a


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sampling plan.

Rejected lots go through 100 % inspection where non

conforming items are replaced with conforming items.

Non conforming items found in the sample are also replaced

Rectifying Inspection


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QUALITY CONTROL


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OFF-LINE QUALITY

CONTROL

STATISTICAL PROCESS

CONTROLACCEPTANCE

SAMPLING PLANS

Acceptance Sampling3

Lot received for inspection

Sample selected and analyzed


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Sample selected and analyzed

Results compared with acceptance criteria

Accept the lot

Send to productionor to customer

Reject the lot

Decide on disposition

ADVANTAGES AND DISADVANTAGES

OF SAMPLING

ADVANTAGES

1. If inspection is destructive, 100% inspection is not feasible.

2. Sampling is more economical and causes less damage due to handling


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3. Sampling reduces inspection error

DISADVANTAGES

1. There is a risk of rejecting "good" lots or accepting "poor" lots, identified as the

producer's risk and consumer's risk, respectively.

2. There is less information about the product, compared to that obtained from 100%

inspection

3. The selection and adoption of a sampling plan require more time and effort in

planning and documentation.

Acceptance Sampling

Acceptance sampling is a method used to accept or reject



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The purpose of acceptance sampling is to make decision on

lots (accept or reject) rather than to estimate the quality of a

lot.

Acceptance sampling plans do not improve quality


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AQL or acceptable quality level

proportion defect the customer will accept a given lot


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LTPD or lot tolerance percent defective

limit on the number of defectives the customer will accept

or producers risk

probability of rejecting a good lot

or consumers risk

probability of accepting a bad lot

Types of sampling plan

Single sampling plan


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Double sampling plan

Multiple sampling plan

Lot formation

Lots should be homogeneous


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Larger lots are preferred over smaller ones

Lots should be conformable to the material handling

systems used in both vendor and customer facilities

Operating Characteristic Curve

n = 99

c = 40.6

0.7

0.8

0.9

1

tance

= 0.05 (producers risk)


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c 4

AQL LTPD

0

0.1

0.20.3

0.4

0.5

1 2 3 4 5 6 7 8 9 10 11 12

Percent defective

Probability

ofaccept

=0.10

(consumers risk)

Acceptable quality

levellot tolerance percent

defective

Single sampling plan for Attributes

Example

Lot size N = 10000


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Sample size n = 89

Acceptance No c = 2

d - No of defective or nonconforming items

OC Curve


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The Ideal OC Curve


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Effect of n and c on OC Curves


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Type A and Type B OC Curve


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dLot

Accept Reject

No ErrorType I Error


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Good

BadLot

No ErrorProducer Risk

Type II ErrorConsumers Risk

No Error

AQL (Acceptable Quality Level)

Poorest level of quality for vendors process that the customer

would consider to be acceptable .


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AQL property of vendors Mfg process

Not a property of sampling plan

Based on AQL we can design the sampling plan

Design a sampling plan, such that OC curve gives a

hi h b bili f AQL


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higher probability of acceptance at AQL

Pa = 95 %

Producers risk refers to the probability of rejecting a good lot. In order to

calculate this probability there must be a numerical definition as to what

constitutes good


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AQL (Acceptable Quality Level) - the numerical definition of a good lot.

The ANSI/ASQC standard describes AQL as the maximum percentage or

proportion of nonconforming items or number of nonconformities in a batch

that can be considered satisfactory as a process average


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LTPD Poorest level of quality that the customer is



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Low probability of acceptance (10 %)

Evaluating sampling plans

OC curve is the measure of performance of a sampling plan.

We use other measures to evaluate the goodness of a

sampling plan


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sampling plan.

Rejected lots go through 100 % inspection where non

conforming items are replaced with conforming items.

Non conforming items found in the sample are also replaced

Rectifying Inspection

Average Outgoing Quality (AOQ)

Average Outgoing Quality is the average quality level of a

series of batches that leaves the inspection station.


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4. p - Incoming lot quality


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Average Total Inspection

--- For single sampling plan

--- For Double sampling plan


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EXAMPLE


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AVERAGE SAMPLE NUMBER

For single sampling plan ASN is n


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P Chart36

The sample proportion nonconforming


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Example Plastic Injection Molding process

Given:

Sample size (n) = 50

No of samples (g) = 25


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Total no of nonconforming items = 90


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Construct OC Curve for this example problem


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de


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P np

0.08 4

0.09 4.5

0.1 5

0.15 7.5

0.2 10

0.28 14

0.4 20


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Find the Average Sample Number ASN for the following p values

0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.1, 0.12, 0.15, 0.17, 0.2


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Table value

Inequality


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Normal Inspection is used initially.

Tightened Inspection is instituted when the vendors recent quality

history has worsen


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Reduced Inspection is instituted when the vendors quality history

has been exceptionally good


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General Inspection Levels

Level I

Level II

Level III


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RELIABILITYDr. Shahul Hamid Khan

Generally defined as the ability of a product to

perform as expected over time.

Reliability


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Formally defined as the probability that a product,

piece of equipment, or system will perform its

intended function for a stated period of time under

specified operating conditions.


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LIFE CYCLE CURVE (Bath tub Curve)

F

A

I

L


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Time

U

R

E

R

A

T

E


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Mean time between failures (MTBF)

Mean time between failures (MTBF) is the predicted

elapsed time between inherent failures of a system during

operation.


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MTBF can be calculated as the arithmetic mean (average)

time between failures of a system. The MTBF is typically

part of a model that assumes the failed system is

immediately repaired (zero elapsed time), as a part of a

renewal process.

This is in contrast to the mean time to failure

(MTTF), which measures average time between


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failures with the modeling assumption that the

failed system is not repaired.

8

Mean Time to Failure: MTTF

1 n

iMTTF t

n=

0 0( ) ( )MTTF tf t dt R t dt

= =


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1in =

Time t

R(t)

1

0

1

22 is bet t er than 1?

9

Mean Time Between Failure: MTBF


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Exponential Distribution

Definition


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Mean and Variance


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EXPONENTIAL DISTRIBUTION


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APPLICATIONS

Exponential Distribution can be used to describe the time to

failure of the product of maturity phase, where the failure rate

is constant


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is constant

Probability density Function


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Weibull Distribution


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SYSTEM RELIABILITY

System with components in series

System with components in parallel

System with components in series and in parallel


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System with components in series


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Use of exponential model


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System with components in parallel

The probability of system failure


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Use of exponential model

If all components have the same failure rate, then


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If all components have the same failure rate, then

System with components in series and in parallel


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One standby component

For two standby components

Documents

Quality Control and Realiability_all in One