capability indicies for processes subject to unilateral and positional tolerances

8/10/2019 Capability Indicies for Processes Subject to Unilateral and Positional Tolerances

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Quality Engineering 2 4) , 46 -471 1990)

C P BILITY INDICES FOR PROCESSES

SUBJECT TO UNIL TER L ND

POSITION L TOLER NCES

K

S KRISHNAMOORTHI

Industrial Engineering

Bradley University

Peoria, Illinois 61625

ey Words

Capability indices; Unilateral tolerance; True position tolerance .

Introduction

A process

is

said to be capable when it

is in

control and meets a given set

of

specifications; and indices such as Cp and Cpk are used to measure the degree

of

such capability

in

processes. Cp

is

the ratio

of

the variability allowed

by

the

specification i.e., upper specification limit minus lower specification limit) to the

natural variability present

in

the process. The natural variability is represented by

6a

where

a

is the estimated standard deviation

of

the process. Cpk is the ratio

of

the distance between the process center and the nearest specification limit to one

1

half of the process variability

3a).

Cp gives an assessment

of

how good a pro- _

ht::JS

cess is with respect to the allowed variability and Cpk gives,

in

addition, an 3

_

assessment

of

how centered the process

is

with respect to the given

t a r g e ~

and Cpk are used to identify problem processes, prioritize processes for resource

allocation, and monitor processes in a department, plant, or company.

In some situations such as when unilateral tolerances or positional tolerances

are specified, these indices seem to be inadequate for measuring process capabili-

46

Copyright 1990

by

Marcel Dekker, Inc .


2/10

462

KRISHNAMOORTHI

ties. This article examines this inadequacy and proposes some alternative meas

ures for process capability in such cases.

Measuring apability of Processes with Unilateral Tolerance

In

a machine shop, several product characteristics have print tolerances that

allow variations only on one side

of

the target such as 3.5 0.010/-0.000. Fig

ure 1 shows two conditions of a process with one-sided tolerance and with meas

urements assumed

to

be normally distributed. In case (a), the process center

is

located near the target and the variability is small, and in case (b), the process

center is farther away and the process variability

is

large. However, the Cpk cal

culated for the two process conditions are the same. The Cpk index, considered

superior to the Cp index because

of

its ability to judge process centering in addi

tion to process variability, is not able to discriminate between a good process

condition and a poor one in the example where the process is subject to unilateral

tolerance.

The inability of the Cpk index to discriminate between the two process condi

tions can be explained as follows. The Cpk index evaluates a process location

with respect to an ideal center which, in the normal bilateral case, is the midpoint

of

the specification limits. In the case

of

processes with unilateral tolerance, as

explained later, the ideal location for the process is not a constant, but changes

with process variability. The Cpk index does not recognize this changing ideal

center, and so is unable to recognize a good location from a poor one. It is neces

sary to define a capability measure that will differentiate between good and poor

process conditions when the process is subject to unilateral tolerance.

Cpk .002/.002 1 0

Cpk =005/.005 =

O

3 5

3.502

3 51

3 5

3 505

3 51

Target Target

(aj b)

igure 1 Examples of process conditions subject to unilateral tolerance.


3/10

CAPABILITY INDICES

AND

POSITIONAL TOLERANCES

46

We first define the ideal condition for a process

subject

to unilateral

tolerance

and

then search for a measure that would distinguish process conditions that are

close

to

and those that

are

farther

away from

this best

condition.

When used as

guidance

for

process impr.ovement,

the

improving

values

of the

index

should indi-

cate

process

movement toward

the

best

condition.

The ideal

condition for

a process

with

unilateral

tolerance changes

with

varia-

bility.

When the

process variability

is

zero,

the ideal

condition

is the target. f

there is any variability at all, the process cannot be

centered

on target, a situation

different

from the bilateral

case.

Then the ideal

center

is at u

distance

from tar-

get so that there is no production

below

target. Besides the

ideal

location, there

are many acceptable locations where all production

is

within specification.

The

acceptable

locations range from the ideal location to u

distance inside

the upper

limit.

Figure 2 a)

shows

some examples

of processes

with

different variabilities

and

their acceptable locations. Figure 2 b) shows the case

with

extreme accept-

able

variability,

with only one acceptable position, which

is

also the ideal loca-

tion.

A capability

index is used in practice, first to

differentiate

an

acceptable condi-

tion from

an

unacceptable condition of a process, and then, within

the

acceptable

set,

to

distinguish

the

one closer

to the ideal

condition

from the

one farther

from

it.

The requirements for process acceptability

shown in

Figure 2 can be translated

into

notations

using properties of the normal distribution. If X and u

represent

process average and standard deviation, and T

and

U represent

the

target

and

upper

limit respectively;

for acceptability, the

requirement

on variability

translates to:

Target Target

US

a)

b)

Figure 2 Acceptable

process conditions with unilateral

tolerance.

a

- -


4/10

464

6u

U T

== >

6u/U - T) J 0

== >

Cp 1 0

KRISHNAMOORTHI

This requirement

on variability is the same as in

the

bilateral

case.

The requirement

for

acceptability on process centering translates to:

T

+

3u X U - 3u

== > [ X

-

T /3u] 1.0 AND [ U - X /3u] 1 0

== >

CpT 1 0

AND

CpU

1.0

Or, min

[CpT, CpU] 1.0

This again is the same requirement used for the cases with

bilateral tolerances

except that the target is used as the

lower

specification limit. Satisfying this

requirement will

force the

process

to acceptability. Yet, this will not

force

the

process to

move

to the

ideal

condition for its current variability). In order to

push the process to the ideal condition, the following

requirement

needs to be

added:

1.0 CpT 1.0

The

larger

the value

of

CpT

is

than

1.0,

the

smaller

the

variability. The larger

the

value of

the ratio CpU/CpT), the

closer the process

center is to the

target.

It is important that the value for CpT be larger than 1.0, but too large a value

will

indicate

that the

process

center is too far from

the target

for the

current

vari-

ability.

When

CpT= 1

and

CpU/CpT)= 1 the process is just at the limit of

acceptability.

For a

given

value of CpT, increases

in the value

of the

ratio CpU/

CpT) would indicate that the process is

moving

toward the ideal condition. Figure

3 shows

some

example process conditions and

the

corresponding values for these

indices.

In

practical terms, for

processes

subject to unilateral tolerance, first

the

process

standard deviation

must

be estimated. If

6u

is larger

than the

spread in the toler-

ance,

steps must be

taken to

reduce the process variability.

Once the

process vari-

ability

6u

is

smaller

than the specification width, the process must be

centered at

T +

3u

or slightly above it. Further reduction

in variability

must be attempted

so

that CpT will assume values larger than 1.0. Too large a

value

for CpT would

indicate that the process center

needs to

be

moved

closer to target but not closer

than T + 3a When this procedure of reducing

process

variability and then mov-

ing

the process center closer to

the

target

is

repeated,

the

process

will move


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CAPABILITY INDICES AND POSITIONAL TOLERANCES

CpT

1.0

CpU/CpT

=

.0

Target:

T

Target:

T

CpT>l.0

CpU/CpT

1.0

CpU/CpT > 1.0

u

u

465

Figure 3

Examples

of

process conditions with corresponding values for the ratio CpU/CpT).

closer and closer to the best condition. The measures suggested here would help

in

monitoring progress

in

this effort.

For audit purposes, the requirement can be imposed that:

1.0 CpT 1.33

and

CpU/CpT) 1.0

The above analysis assumes the process

is

normally distributed, and hence, a

verification for normality

of

the process

is

necessary. Although the analysis was

made with reference to a tolerance with

an

upper limit, the analysis would apply

to a tolerance with a lower limit as well. The condition for a lower limit L would

be:

1.0 CpT 1.33

and

CpL/CpT) 1 0


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66

KRISHN

MOORTHI

Measuring Capability of Processes Subject to Positional Tolerance

When a hole has to be drilled or a stud has to be located in a given position,

designers generally provide a tolerance region, usually a circle around the target

position, to allow for the variability in locating the hole or stud in actual

manufacture. When the performance

of

the process

is

to be compared against

such a tolerance, a measure is needed to compare the variability and centering in

actual production with that specified

by

the designer. n other words, a capability

index has to be defined for processes subject to such positional tolerances.

Figure 4 shows an example where the position for a hole

is

specified along

with a tolerance region. In this example the target position TP) is specified by

the coordinates a,b) for the center of a hole, and the tolerance

by

the circle with

diameter

D.

When the holes are drilled, the centers

of

the holes will not all exactly coincide

with the specified target position, but will be distributed in an area around a

center, say, CC for center

of

centers). Let us assume first that this CC coincides

with the specified target position TP, and the actual location of hole centers are

distributed as bivariate normal around the CC . On the assumption that the devia

tions of the hole centers from CC on X and Y axes have equal variance and are

uncorrelated, the footprint

of

the bivariate normal distribution will be a circle

with center at CC. Figure

5

illustrates this case when the CC coincides with the

TP.

t might seem appropriate to compare the area of the tolerance circle and the

area of the circle of natural variability to get a measure of capability of the pro

cess . The assumption that the CC coincides with the target position can be

verified by taking a sample

of

holes, measuring the X and Y coordinates of their

centers and calculating

X,Y).

f

X,Y) _

a,b) then the assumption can be con-

D

TP: a,b)

a

Figure 4

Example of a positional specification.


7/10


TP=(a.b)

CC=( X,Y)

467

Tolerance region

Region of natural

variability

.

_

Figure S.

A

positioning process: natural variability vs. tolerance.

: ::: ~ s T u c r o a ~ ~ a l i d

The

area of natural variability can

be

approximated

by:

?r 3u )

2

= ra

2

where u is the equal) standard deviation of the X and Y coordi

nates of the sample holes. When the standard deviations of the X and Y coordi

nates are

not

equal, the larger of

crx

and u

is

recommended.

This

will

give a cir

cular

area

for natural variability

to be

compared

with the

circular tolerance area

while giving

conservative

larger) estimate for natural variability.

Thus,

the capability index

PCp

Positional

Cp) can be

defined as:

1r/40

2

1 0

2

V

PCp

911 u2 36

u2

When

PCp

= 1.0

the

natural variability is equal to the variability in tolerance.

When

it

is

larger

than

1.0,

the natural

variability

is

sqialler

than the

tolerance

variability. A

value

for PCp greater

than

1.33

can be required to make all

hole

locations

fall within

tolerance.

Increasing

values for

PCp

would indicate improv

ing

process

conditions.

Consider

next the

case when CC does not coincide with TP. Figure 6 shows an

example where the CC does not coincide with the target position specified. Tfle

figure

shows

the off-centered

location in

an exaggerated fashion for the

sake

of

clarity.

Again, i we make

the

assumption

that the

centers are distributed as bivariate

normal with

variability

in the

two

axes equal

and deviations

uncorrelated, the

PCpk

Positional Cpk)

can

be defined

as:

PCpk

=

7r/4

02

7r V X - a)

2

+

Y

- b)

2

+ 3u]

2

02

; : : : : : : : ; : : : = = : : : ; : : = = : : ; : : : = = ~ ~

4(V(X - a)

2

Y - b)

2

of

I

I

I


8/10

468

KRISHNAMOORTHI

Tolerance region

TP a b)

CC X, Y

Figure

6

A positioning process where the center

of

actual centers misses the target position.

t should be noted that when the process center CC coincides with target posi

tion TP, PCp

= PC

pk.

The requirement that PCpk be larger than 1.33 can be imposed for audit pur

poses. The process center will have to be closer to the target position and the pro-

cess variability

has to

be smaller than the variability in tolerance in order

to

meet

this requirement.

The following example illustrates the method of computing the capability

indices and interpreting their meaning for a positioning process.

Example

The following coded) data represent X and Y coordinates. of 30 holes drilled

against a true position 0,0) and tolerance diameter D

=

2.

X: -.514 - .372

-.412

-.412

-.356

-.638

-.274

- .346 -

.845

-

.462

- .400 -

.532

- .530 -.347

- .440 -

.372

- .795 -

.829

- .500 -

.506

-.436

-

.632

- .

729

- .934 - .546 - .

313

- .510

Y:

-.622

- .663

-.685

-.462

-.169 -.502 -.576

-.448 - .618

-.786 -.671 -.528

-

.333

-.386

-.524

- .290

-.480 -.384

-.683

- .

588 -.885

- .376 - .

574

-.594 - .618

- .

574

- .514

x -

.5034

y = - .5237

S X) = 1755 S Y) =

1542

Estimate

of J =

.1755 larger

of

the two sample standard deviatiom.

PCp

=

l/36) 4/.175s2)

=

3.6

-.494

- .

257

- .370

J l9

-.480

- .380

PCpk

=

4/4[Y - .5034)

2

.5237)

2

3x.1755]

2

=

0.63


9/10


469

PCp = 3

6

indicates that

the

variability

in

locating the holes

is

small compared

with

the

tolerance allowed. The value

o

PCpk

=

0.63 indicates, especially since

PCp is large, that the actual hole centers are off the specified true position. This

can be seen by comparing (X, Y) = - .5034, - .5237) with the specified target

position (0.0). The steps

to

be taken

to

improve the capability

o

the process

become obvious.

Meaning o PCpk

A point needs

to

be made about the meaning

o

PCpk.

t is

only

an

index

which indicates whether the actual centers are centered around and close to the

specified target position. When the value of PCpk is larger than 1.0, there is the

assurance that all holes will

fall

within the tolerance region. When the value

is

much larger than 1.0, say larger than 1.33, the degree

o

assurance

is

larger.

f

its value is smaller than 1.0, all holes are not within the tolerance region.

Although a process with PCpk

o

0.5 will be considered worse than a process

with PCpk o 0.75, it should not be interpreted that one half o the holes drilled

in

the fotmer case or one fourth

o

those drilled in the latter are outside

specification limits. Process improvements should aim at larger and larger values

for PCpk.

Validity

o

ssumptions Made

First, the assumption

o

bivariate normal for the actual location

o

the hole

centers is probably valid in many situations and can also be verified. The devia

tions on the two axes should

be

tested for independence and when independence

can be taken to be true, they can be tested for normality as two one-dimensional

variables .

Second, equality o variability on the Xand Y axes: This assumption requires

that

ax

and

y

be equal. When the deviations on the X and Y axes have equal

variance and are uncorrelated, the region

o

natural variability is a circle. Since

the tolerance region

is usually given as a circle, the derivation and computation o

the indices are simple. When these assumptions are not true, the region o natural

variability will be an

ellipse. Calculating

the

area

o

the ellipse and comparing it

with the tolerance area might give a more correct comparison, at least for the

case when

T

and CC coincide. When they

do

not coincide, derivation and com

putation o the PCpk becomes complicated. However, taking the larger o the two

standard deviations simplifies calculation, and gives a conservative estimate for

the indices (i.e., smaller values than would be obtained by calculating the area o

the elliptical region). This may be erring on the safer side.


10/10

47

KRISHNAMOORTHI

onclusions

Many machine shop processes are subject

to

the types

o

tolerances considered

above. Measures are necessary to compare the centering and variability generated

in

those processes in actual production with the variability and target specified by

the designer. Several informal rules have been used for this purpose. The ratio

o

the tolerance diameter to the major axis

o

the ellipse

o

natural variability has

been used as a capability measure. This oversimplifies the problem by not using

the information from the other axis . A comparison

o

the area

o

the tolerance

region with the ellipse

o

natural variability without reference to the centering

o

hole centers has been used. This has the obvious drawback

o

giving wrong indi

cations when the hole centers are off the target center. Further, simple compari

son

o

areas o tolerance circle with the natural ellipse suffers from the fact that,

when the deviation in one axis

is

very small , while that

o

another

is

very large,

the area

o

the natural ellipse may be small but there may be out-of-spec values

on the larger axis. The capability index will not be able to detect this. The meas

ures suggested in this article avoid these handicaps .

t

has been recently brought to the attention o the author that Gilliland 1989)

is suggesting a measure for the two-dimensional case that would, for example,

evaluate the proportion

o

hole centers located within the tolerance region using

bivariate normal distribution, and invert the proportion into a univariate capability

measure through the N 0,1) distribution . Such an index would give additional

meaning to capability measurement although the concept and the computational

details will be more difficult

to

sell

to

the engineers and technicians . However, i

simplified formulas are made available to minimize computational effort, the

above line

o

thinking should produce some good indices.

It

is

recognized that there are many other special process situations, besides the

two considered above , where the currently available capability indices are not

adequate. Processes that are not normally distributed and processes that have

unequal, bilateral tolerance are examples

o

processes that offer difficulty. A

major purpose

o

this article

is

to highlight the need for defining capability meas

ures for such special processes on a rational basis. The methods suggested here

may be the first steps in handling this class

o

problems.

cknowledgment

The author extends thanks to the Manufacturing and Quality General Offices o

Caterpillar Inc., Peoria, Illinois, for bringing the problem to his attention and

providing financial support for the research.

capability indicies for processes subject to unilateral and positional tolerances

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