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BITSPilaniPilani Campus

Chapter 9 (Lecture No: 13)

Process Capability Analysis

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BITS Pilani, Pilani Campus

Required to analyze whether a product or service meets the specifications

given by the customer by indicating the measure of process performance

Process capability analysis has to be conducted only when the process is in

a state of statistical control

Specification limit

Generally used interchangeably with tolerance limit

Limitsthat define the conformance boundaries for an individual unit of a

manufacturing or service operation(ANSI, A1, 1987)

Are used for categorizing materials, product or services in terms of their

stated requirementsAre determined by the needs of the customer

Are placed on the product characteristic by engineers or designers to

ensure proper functioning of product

Represents the desired bound on the variability for individual items

Introduction

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Tolerance limitAre subset of specifications

They pertain to physical requirements (length, diameter, thickness etc),

but specification limits includes all

Are preferred in evaluating the manufacturing or service requirements

Can be two sided or one sided

Lower tolerance limit (LTL) defines the lower conformance boundary for

an individual unit of a manufacturing or service and UTL refers to upper

conformance boundary

Example: Specification of a building crane is a hoist load of 5000 +/- 300 kg.

To satisfy this, the diameter of the steel cable should be 4 +/- 0.2 cm

Control limits

Identify the variation that exists between samples or reflect the variability

of the process

No relationship with the specification limits

IntroductionContd..

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Determination of process capability begins only after the process has been brought to a state of statisticalcontrol

Process capability

Represents the performance of a process in a state of statistical control

Determined by the total variability that exists due to all common causes present in the system

It can be viewed as the variation in the product quality characteristic that remain after all special causes

has been removedIt can also be a measure of uniformity of a quality characteristic of interest

A common measure of process capability is given by 6also called as process spread

Distance 6encompasses virtually all values of the output quality characteristic

Process capability Analysis

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Involves estimating the process mean and standard deviation

Also the form of the relative frequency distribution of the characteristic of

interest is estimated

If specification limits are known, proportion of non conforming products

can also be estimated To be precise, a process capability study involves observing a quality

characteristic of the product and hence should be called product analysis

study

A true process capability study should involve collecting and analyzing

data related to process parameters like feed, depth of cut etc. Objective of this study is to find the relationship between process

parameter and the product characteristics

Process capability analysis

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Process capability analysisContd..

Benefits:

Continuous estimation and monitoring of these parameters will ensure the best performance

that the process is capable of achieving.

Used to take decision regarding buying of new machine or about quality of raw materials etc,

based on the process average and process spread

Uniformity of output: helps to control the variability and hence more uniform output is

obtained

Maintained or improved quality

Product and process design facilitated: Information from process capability provides feedbackto design thereby making them aware of inherent variation and hence reduces lead time of

product by designing for given tolerance.

Helpful in vendor selection and control

Reduction in total cost: helps in monitoring the production of non conforming items

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Natural Tolerance Limits

Called as process capability limits, which are established orinfluenced by the process itself

Product Specification

LowerSpecification

Limit

Nominal

Specification

UpperSpecification

Limit

Target

(Ideal level for use in product)

Tolerance

x

(Product

characteristic)

(Maximum range of variation of the product

characteristic that will still work in the product.)

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Statistical tolerance limits

Defined by ANSI / ASQC as the limits of an interval that (witha given level of confidence ) contains at least a specified

proportion (1-)of the population

Limits are found from sampling information

Example Level of confidence of 0.98, while sample size is

10 and 95% of the part length will be between 30 and 35mm

Limits are influenced by sample size

As sample size increase, statistical tolerance limit approach

the values found from population parameters

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Technically, there might not be a mathematical relationshipbetween the process capability limits (NTL) and the specification

limits

NTL is determined by the condition of the process and its inherent

variability Specification limits are influenced by customer

A desired relationship is that specification limits are preferably

outside the natural tolerance limits, in which case, most of the units

produced will be acceptable

Case i: Process spread less than specification spread

Process is quite capable and a preferred situation

Spread between specification (USL-LSL) > spread between

natural limits (UNTLLNTL)

Specifications and process capability

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Specifications and process capabilityContd..

Case iContd.. If the process mean at the target value (midway between the

specification limits), all items produced are well with in specification

Even if process go out of control, process mean shift to 1 or

process standard deviation increase to 1 still, items produced will

still meet specificationsBut use of control chart can help in bringing back the out of control

situation to controlled situation

Case iiProcess spread equal to specification spread

(USL LSL) = (UNTL LNTL) - Acceptable or adequate situation

and there is no room for errorFor process in control, virtually all items (99.74%) produced will be

with in specification

For out of control process, say shift in mean or standard deviation

increases, will result in a proportion of the product non conforming

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Case iii: Process spread greater than specification spread(USLLSL) > (UNTLLNTL)

Undesirable situation

Inherent variability of the process exceeds the

specification spread even though the process is incontrol

Always some proportion of items produced will not

meet specification

Shift in mean or increase in standard deviation result in

increasing proportion of the product not meeting

specification

Such process is not capable

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Corrective measurescontd..Leave the process unchanged, perform 100%

inspection to eliminate the non conforming parts

Not an ideal solution, as 100% inspection may not get rid of allnon conforming items

100% inspection is merely a sorting activity and it does not get

to the root causes

It cannot be used to analyse the causes and to determine

corrective measures

Best approach is to do things right the first time and

reduce reliance on inspection

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NTLs, SLs and CLs

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Process capability index

The ability of a process to meet specification limit should beestimated only, when it is in state of statistical control

The process has no special causes and hence variability is

the reflection of what the process can achieve

First step is to analyse whether the process is in control

before we go for process capability analysis

Assumptions

Process output (distribution of quality characteristic)

is normal, which helps to estimate the proportion of

non conforming Assumption of normality can be validated by

empirical plots of histogram, normal probability parts,

statistical test for goodness of fit like chi-square test,

Klomogrov test etc.

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Process capability indexContd..

Ability to meet specification is the criteria for measuring theattractiveness of the process.

Capability indices are non-dimensional, makes them more

versatile and appealing because they do not depend on specific

process parameter units.

It takes care of location and the variation of the process

Cp index Common measure to describe the potential of a

process to meet specification

Relates process spread (difference between natural

tolerance limits) to specification spread Cp=USLLSL / 6

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Cp indexcontd.. When is unknown, it is replaced by its estimate say sample

standard deviation (Rbar/d2 or sbar /c4)

A process centred between specification limits will produce a

minimum proportions of items that fall outside these limits

It is desirable to have Cp >= 1, If Cp = 1, the process spread equalsspecification spread and process is said to be barely capable

If process is not centred, it is possible that even for process with Cp

> 1, some proportion of the product will be non-conforming

If Cp < 1, inherent variability in the process as measured by process

spread 6,is greater than the specification limit Cp is the ratio of allowable process spread and actual process

spread

It does not take to account the location of the process and is a

measure of process potential and not process performance

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Upper and Lower capability indices

Used when only one specification limit

Indices can be derived that measure shifts in the process

mean relative to the process spread

For a given upper specification limit, upper capability index isgiven by

It is desirable to have CPU greater than 1


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Upper and Lower capability indicescontd..

Similarly if the lower specification limit is given,

the lower capability index (CPL) is given by

It is desirable to have CPL greater than or equal

to 1

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CpkIndex

It accounts for the location of the process mean and used when

it is not at the target mean

Cpkis given by

Desirable values of Cpkis greater than or equal to 1

It represents the actual capability of the process with theexisting parameter values. It measures process performance

For the given process Cpk < 1, the value of Cp > 1 indicating

process can potentially meet specifications (Refer figure)

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CpkIndexcontd..

Solution is to move the process mean towards midpoint m

A measure of this deviation of the process mean from target m

is given by scaled distance k

Where m is sum of USL and LSL divided by 2

Relationship between Cpand Cpkis given by

If process mean is at the target m, k = 0 and hence Cp= Cpk

and if process mean is at the USL or LSL then Cpk= 0

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Capability Ratio A measure of the ability of the process to produce item within

specification limit is based on the specification range (USL LSL),

that is used by the process

As process spreads increase, it tend to use more of this

specification bandCapability ratio CR is given by

It is expressed as a % to indicate the % of specification range used

A CR

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Comparison of Capability indicescontd..Calculate the Cp, Cpk, Cpm and % yield

Cpk is 1 for each process, while value of Cp changes and hence

process C has more potential

When process is centre, it is known that Cp = Cpk (process A)

Process mean changes for B and C, still Cpk is same, due to the fact t hat

its affected also by std. deviation

For a given value of Cpk, we have a upper bound and lower bound on

the process yield, given by

Lower bound is achieved when the process mean is centred between the

specification limits, while the upper bound is obtained, when process mean

approaches USL or LSL and std. deviation approaches zero

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Comparison of Capability indicescontd..

For a process in mid of spec.limit and at target value, then we get Cp =

Cpk = Cpm (Process A)

As the process mean moves away from the target value, value of Cpm

decreases

Cpk is not a suitable measure for process centering

Similarities between Cpk and Cpm

For a fixed value of std. deviation, both indices coincides with Cp when the

process mean is at target T

Both decrease, as the process mean moves away from target

For mean > USL or mean < LSL, Cpk

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Determining the tolerance or specification limit that should be set on assemblies orsub assemblies, given the tolerance of individual components and vice versa

Assume that the processes making the assemblies or components are barely

capable

Tolerance on Assemblies and Sub assemblies

Assemblies and sub assemblies are formed by combining 2 or more

components

Dimension of interest in an assembly may be the sum or the difference of

individual components

Eg. 4 rods welded together and the quality characteristic of assembly I s

interest given by Y= sum of all rod length

Generally the dimension is expressed as a linear combination of

some individual component dimensions given by

Setting tolerances on assemblies

and components

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Setting tolerances on assemblies andcomponentsContd..

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Setting tolerances on assemblies and

componentsContd..

Tolerance limit on individual componentsTolerance or specification limits on assembly are often determined

from product function or customer needs

Designer determines individual component tolerance that will give

specified assembled tolerances

Tolerances on Mating parts

Mating parts shaft and bearing, piston and a cylinder, represents a

special form of assembly and it is based on the type of fit

Clearance fit

Size of the hole prior to assembly is always larger than the size of the shaft and

hence there is some room for play.

NTL of hole are outside the NTL of shaft

Normality is assumed for both distribution of shaft and hole diameters

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Setting tolerances on assemblies andcomponentsContd..

Tolerances on Mating parts contd..Interference fit

Size of the hole prior to assembly is always smaller than the size of the shaft and

hence shaft has to be forced into hole

Range of NTL of hole is below that of NTL of shaft

Eg. Assembly of a pin to a sleeve

Transition fit

There is either a clearance or an interference in an assembly

NTL range of the hole and shaft can overlap

Depends on the relative location of the means and variability of the hole and shaft

diameter, an assembly will be either clearance or interference fit

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Estimating Natural Tolerance Limitof a Process

NTL define the inherent capability of a process and it is estimated once a processis in statistical control

If control charts are used to bring a process to control, information from these

charts can be used to determine the NTL of a process

Assumption is that reliable estimates based on elaborate historical data are

available for the process mean and process std. deviation

If small sample are used to generate estimates of the process mean and std.

deviation, then statistical tolerance limits can be found

When sample size increases, STL will approach NTL

Even the value of Cp can be used to estimate NTL and Spec. limit.

Say Cp = 1.33, which represents spec. limit is at 4 standard deviation

from mean while NTL is at 3 std. deviation from mean

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Estimating Natural Tolerance Limitof a Processcontd..

Statistical tolerance limit (STL) based on normal distributionLet Xbe normally distributed with mean and std. deviation

STL that encompass 100(1-)% of the product can be constructed as

+/- Z /2

But and are unknown, hence estimated from sample estimates

For a sample size n,sample mean is Xbarand std. deviation is s.

Though estimates, it is not necessarily true that Xbar +/- Z /2 s will

include 100(1- )% of the distribution

Also Xbar and s are random variables while and are constants

Different samples will lead to different values of X and s, which wil in t

urn lead to different estimates of limits

Some of these may be very different from +/- Z /2 and may not

include 100(1- )% of the product

Values of kare tabulated such that in a large proportion ( ) of these

intervals (Xbar +/- ks), atleast 100(1- )% of the distribution will be

included

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Estimating Natural Tolerance Limitof a Processcontd..

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Summary

S

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Summary

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Process Capability Index Example

A manufacturing process produces a certain part with a mean diameter of 2 inches

and a standard deviation of 0.03 inches. The lower and upper engineering

specification limits are 1.90 inches and 2.05 inches.

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Sol:

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A manufacturing process produces a certain part with a mean

diameter of 2 inches and a standard deviation of 0.03

inches. The lower and upper engineering specification limits

are 1.90 inches and 2.05 inches.

Example

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Sol:

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Effects of Reducing Variability on Process Capability

CAPABILITY ANALYSIS FOR

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Identification of Appropriate Distribution

In this approach, based on knowledge of the process by

which the quality characteristic is chosen, a distributional

fitting is attempted from a specified list of available

distributions, such as Weibull, gamma, lognormal, andexponential.

When an acceptable fit is identified through a statistical

goodness-of-fit test, that particular distribution, with

parameters estimated from the data, is used to obtainmeasures of capability and proportion nonconforming

CAPABILITY ANALYSIS FORNONNORMAL DISTRIBUTIONS

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The Box-Cox method estimates a value for thatminimizes the standard deviation of a standardized

transformed variable. It is defined as

Box-Cox Transformation

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Attribute charts are constructed for selected

specifications.

If the acceptable bounds for quality characteristics are

known, the chart for the proportion nonconforming (p-

chart) can be constructed to classify the product asconforming or not conforming.

The same holds true for the chart for the number

nonconforming (np-chart).

Similarly, the control chart for the number ofnonconformities (c-chart) can be developed if the

definition of a nonconformity is known.

Using Attribute Charts

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Using a Nonparametric

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Using a NonparametricApproach

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Contd.

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Refer to the four-component assembly shown in Figure

9-14. Suppose that the mean lengths of the four

components and their respective tolerances are as

shown in the following table:

Assuming a normal distribution for the individualcomponent dimensions, find the natural tolerance limits

for the assembly length. The design specifications for

assembly length are 20 0.3 cm. What proportion of the

assemblies will be nonconforming? Comment on theprocess capability to make assemblies that meet the

design specifications

Contd..

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Contd..

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Contd..

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ESTIMATING STATISTICAL

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The natural tolerance limits define the inherent capability

of a process.

Consequently, it is of interest to estimate these limits

once a process is in statistical control.

this approach we assume that reliable estimates based

on elaborate historical data are available for the process

mean and process standard deviation.

ESTIMATING STATISTICAL

TOLERANCE LIMITS OF A PROCESS

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Data from a small sample are used to generate

estimates of the process mean and standard deviation,

statistical tolerance limits can be found

CONTD..

Nonparametric Statistical

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Nonparametric statistical tolerance limits do not depend

on the distribution of the quality characteristic.

These limits are valid for any continuous probability

distribution.

They are based on die largest and smallest observations

in the sample.

Nonparametric StatisticalTolerance Limits

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CONTD

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CONTD..

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Thank You

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