NATL INST. OF STAND & TECH

AlllDb 33MTbD

NBS Special Publication 673

NBS

pBL/CAT/ONS

Economic Model of

Calibration Improvements

for Automatic Test Equipment

U.S. Department of Commerce

National Bureau of Standards

100

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1934
NATIONAL BUREAU OF STANDARDS

The Nations! Bureau of Standards' was established by an act of Congress on March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their effective application for public benefit. To this end, the Bureau conducts

research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau's technical work is per- formed by the National Measurement Laboratory, the National Engineering Laboratory, and

the Institute for Computer Sciences and Technology.

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! Some divisions within the center are located at Boulder, CO 80303.
I NATIONAL BUREAU

OFnv STAHDABBfl

LI2RAT-Y

Economic Model of

Calibration Improvements

for Automatic Test Equipment K*4

Stephen F. Weber

Anne P. Hillstrom

Center for Applied Mathematics

National Engineering Laboratory

National Bureau of Standards

Washington, DC 20234

Sponsored by

:

Center for Electronics and Electrical Engineering

Center for Applied Mathematics

National Bureau of Standards

Washington, DC 20234

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued April 1984
Library of Congress Catalog Card Number: 84-601027

National Bureau of Standards Special Publication 673

Natl. Bur. Stand. (U.S.), Spec. Publ. 673, 94 pages (Apr. 1984)

CODEN: XNBSAV

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1984

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402
Abstract

This paper presents a model for measuring the benefits of improved

accuracy in test equipment. The model permits calculation of the probability

of accepting a unit under test (UUT) that is out of specifications (i.e.,

Consumer's Loss) and the probability of rejecting a UUT that is within

specifications (i.e., Producer's Loss) for alternative levels of test

equipment accuracy. Accuracy is defined in terms of both the systematic and

the random measurement error of the equipment. Other parameters that are taken into account by the model are the mean and variance of the UUT attribute

of interest, the performance specifications of the UUT, and the test

specifications which define acceptance and rejection in terms of test

measurement results. A discussion of the economic consequences of Consumer's

Loss and Producer's Loss is included. The model may be used to optimize both

procurement policy for new test equipment as well as maintenance and

calibration policy for existing test equipment.

Key words: accuracy; automatic test equipment; calibration; Consumer's Loss;

economics; error analysis; measurement; Producer's Loss; quality

assurance; random error; systematic error; testing.

iii

Acknowledgements

This research is jointly funded by the Center for Applied Mathematics

(CAM) and the Center for Electronics and Electrical Engineering (CEEE), both

of the National Bureau of Standards. The authors are grateful for the useful

suggestions and guidance offered by a number of colleagues and reviewers.

Those deserving special appreciation are Judson French and Ronald Powell of

CEEE headquarters; Barry Bell, Paul Lederer, Thomas Leedy, and Oskars

Petersons of the Electrosystems Division of CEEE; Janet Russell of CAM headquarters; James Filliben, James Lechner, Mary Natrella, and Charles Reeve of

the Statistical Engineering Division of CAM; Javier Bernal, Robert Chapman,

Jacqueline Kee, Laurene Linsenmayer, Barbara Lippiatt, Harold Marshall, and

Rosemary Massengill of the Operations Research Division of CAM; George Dines

of the Computer Services Division of CAM; James Barnett of the Building

Physics Division of the Center for Building Technology; Donald Owen of

Southern Methodist University; and Ingram Olkin of Stanford University.

v
Table of Contents

Page

ABSTRACT iii

ACKNOWLEDGEMENTS v

LIST OF TABLES viii

LIST OF FIGURES ix

1 . INTRODUCTION 1

1.1. Background and Objective 1

1.2. Scope 2

1.3. Organization 5

2. MATHEMATICAL FORMULATION OF THE MODEL 7

2.1. The Basic Model 7

2.2. Extensions of the Model 15

3. RESULTS 19

3.1. Effects of Systematic Measurement Error 19

3.1.1. Direction of Effects 19

3.1.2. Pattern and Magnitude of Effects 30

3.2. Effects of Random Measurement Error 37

3.2.1. Direction of Effects 37

3.2.2. Pattern and Magnitude of Effects 40

4. SUMMARY AND APPLICATION 46

BIBLIOGRAPHY 48

APPENDIX A: BINORMAL . CL-PL , AN INTERACTIVE COMPUTER PROGRAM FOR

CALCULATING CONSUMER'S LOSS AND PRODUCER'S LOSS IN

TESTING 51

A.l. Purpose 51

A. 2. Scope 52

A. 3. Using the Program 53

A. 4. Program Characteristics 56

vi
Page

APPENDIX B: PLOTS OF CONSUMER'S LOSS AND PRODUCER'S LOSS AS

FUNCTIONS OF THE MEAN AND STANDARD DEVIATION OF

MEASUREMENT ERROR 63

B.l. Introduction 63

B.2. Scope 63

B.3. Interpretation of the Plots 65

B.4. Illustrative Applications of the Figures 79

vii
List of Tables

Page

2 1 Notation Used in the Model and in Appendix B 8

3.1 Parameter Values and Systematic Error Values Used to

Analyze the Direction of the Effect of Systematic

Error on CL and PL 21

3.2 Critical Values of TA/oe Below Which CL Decreases With

Increasing Systematic Error, For Various Symmetric

Performance Limits and Standard Deviations of Measurement Error... 22

3.3 Parameter Values and Systematic Error Values Used to Analyze

the Pattern and Magnitude of the Effect of Systematic Error on on CL and PL 31

3.4 Parameter Values and Standard Deviation of Measurement Error

Values Used to Analyze the Direction of the Effect of the

Standard Deviation of Measurement Error on CL and PL 38

3.5 Parameter Values and Standard Deviation of Measurement Error

Values Used to Analyze the Pattern and Magnitude of the Effect

of the Standard Deviation of Measurement Error on CL and PL 41

A. l Input Variables for BINORMAL . CL-PL 54

B. l Parameter Values and Resulting Values of CL and PL for Case

Examples 81

vm
List of Figures

Page

3.1. Comparison of the Relative Cost Ratio, CCL/CpL, With the

Absolute Value of the Relative Change Ratio, A PL/ACL,

for a Performance Specification of One Standard Deviation 27

3.2. Comparison of the Relative Cost Ratio, CCL/CPL , With the

Absolute Value of the Relative Change Ratio, A PL/ACL, for

a Performance Specification of Two Standard Deviations 28

3.3. Comparison of the Relative Cost Ratio, CCL/CpL, With the

Absolute Value of the Relative Change Ratio, A PL/ACL, for

a Performance Specification of Three Standard Deviations 29

3.4. Consumer's Loss as a Function of Systematic Measurement Error

(Bias), for Characteristic Values of Test Tolerance 33

3.5. Producer's Loss as a Function of Systematic Measurement Error

(Bias), for Characteristic Values of Test Tolerance 34

3.6. Consumer's Loss as a Function of Standard Deviation of

Measurement Error, for Characteristic Values of Test Tolerance.... 43

3.7. Producer's Loss as a Function of Standard Deviation of

Measurement Error, for Characteristic Values of Test Tolerance.... 45

A.l. Illustrative Computer Run of BINORMAL . CL-PL 55

A. 2. Computer Code Listing of BINORMAL. CL-PL 58

B. l. Effect of Systematic Measurement Error on Consumer's Loss

(o e =0.5, y u=0, 0 U=1, Su=2, SL=-2).... 67

B.2. Effect of Systematic Measurement Error on Consumer's Loss

(a e =0.25, u u=0, a u=l, Su=2, SL

=-2) 68

B.3. Effect of Systematic Measurement Error on Consumer's Loss

(ae=0.1, m u=0, o u=l, sy=2, SL

=-2) 69

B.4. Effect of Systematic Measurement Error on Producer's Loss

(o e =0.5, p u=0, o u=l, Su=2, SL

=-2) 70

B.5. Effect of Systematic Measurement Error on Producer's Loss

(a e =0.25, m u=0, a u=l, SD

=2 , SL

=-2) 71

B.6. Effect of Systematic Measurement Error on Producer's Loss

(0 e =O.l, y u=0, o u=l, Su=2, SL=-2) 72

B.7. Effect of Standard Deviation of Measurement Error on Consumer's

Loss (y e =0.5, y u=0, o u=l, Su=2, SL=-2) 73

ix
Page

B.8. Effect of Standard Deviation of Measurement Error on Consumer's

Loss (m £ =0.25, y u=0, a u

=l , Su=2, STj=-2) 74

B.9. Effect of Standard Deviation of Measurement Error on Consumer's

Loss (ue =0, y u=0, a u

=l, Su=2, SL=-2) 75

B.10. Effect of Standard Deviation of Measurement Error on Producer's

Loss (ye =0.5, y u=0, a u

=l , Su=2, SL

=-2) 76

B.ll. Effect of Standard Deviation of Measurement Error on Producer's

Loss (y e =0.25, m u=0, a u

=l , Su=2, SL

=-2) 77

B.12. Effect of Standard Deviation of Measurement Error on Producer's

Loss (pe =0, u u=0, a u=l, Su=2, SL=-2) 78

x
1. Introduction

1.1 Background and Objective

This report presents a model for measuring the benefits of improved

accuracy in test equipment. Although the model could be applied to many types

of test equipment, the focus of this report is on that broad category of

electronic test equipment commonly referred to as Automatic Test Equipment

(ATE). For purposes of this discussion, ATE is defined as electronic test

equipment that is controlled by microprocessors or programmable computers and

is capable of high speed testing of individual electronic components, printed

circuit boards, wiring harnesses, or entire electronic system assemblies.!

The origin of expected improvements in the accuracy of ATE lies in a

proposal by the National Bureau of Standards (NBS) Center for Electronics and

Electrical Engineering (CEEE) to develop prototypes for and support the

implementation of a new calibration system that satisfies the special

requirements of ATE. This proposed system, called Dynamic Transport Standards

(DTS), would address two shortcomings of the current approach: "...(1) the

lack of adequate ATE calibration in its operational environment under

operational conditions at the UUT [unit under test] terminals and (2) the

extensive range extension and bootstrapping activities in the building up of

multi-parameter multi-level standards in numerous standards and calibration

This definition is adapted from an unsigned article, "Electronic Component

Testing with ATE" (Part I), Quality , Vol. 20, No. 10 (October 1981),

pp. 16-20.

1
laboratories in order to achieve traceability to national standards at NBS."1

Two types of improvements related to the performance of ATE could result

from the use of the DTS system of calibration, once it is fully implemented.

These potential improvements would both be in the form of reductions in the

overall measurement error of ATE. The first improvement is a decrease in the

systematic error or bias present in the test equipment measurements. The

second type of improvement possible in the accuracy of ATE involves a decrease

in the random measurement error of the test results. This second type of

improvement is equivalent to an increase in measurement precision. The

benefits of both these types of improved accuracy (reductions in measurement

error) are determined by the model presented in this report.

1.2 Scope

The model is stochastic in the sense that it is based on two probability

distributions and the measures of the benefits from improved accuracy are

stated in terms of probabilities. The probability distributions are those of

the UUT performance^ being tested and of the measurement error of the test

equipment. The model can be used to analyze accuracy improvements for

practically any application of test equipment as long as these two probability

Barry A. Bell and Oskars Petersons, "ATE Calibration by Means of Dynamic

Transport Standards," Proceedings of Autotestcon 81: International Automatic

Testing Conference (Orlando, FL, October 19-21, 1981), Institute of

Electrical and Electronic Engineers (IEEE Catalogue No. 81 CH 1716-0), 1981,

pp. 280-287.

UUT performance refers to the value of the particular UUT attribute that is

being measured by the test equipment.

2
distributions can be assumed to be normal. That is, the attribute of the UUT

that is being tested as well as the test equipment measurement error must both

be approximately normally distributed.

The measures of benefits determined by the model are based on reductions

in the probabilities of the two types of errors possible with every test

situation: Consumer's Loss (CL) and Producer's Loss (PL).* CL is the

probability of accepting a UUT that does not conform to the performance

specifications for the attribute being tested. PL is the probability of

rejecting a UUT that does in fact conform to the performance specifications.

The terms "consumer" and "producer" are used to indicate which interest group

would typically bear the cost of each error type in a production testing

context. During product warrantee periods, however, most of the cost arising

from CL would typically be borne by producers. Moreover, when testing is

being conducted as part of a maintenance function, the costs of both CL and PL

usually remain within the organization.

The model expresses both probabilities, CL and PL, as explicit functions

of measurement bias (systematic error) , represented by the mean of measurement

error, and of measurement imprecision, represented by the standard deviation

of measurement error. These functions can be used to calculate directly the

benefits, as measured by lower probabilities of errors, from reduced measurement bias and/or imprecision. When applied to a case study, an economic value,

in terms of dollars saved per time period, can be established for these

1 These probabilities are also referred to as Consumer's Risk and Producer's

Risk, respectively.

3
reduced probabilities. The economic value depends on the number of tests

conducted in a time period and on the particular test situation being

analyzed. Multiplication of each probability reduction by the number of items

to be tested gives the expected reduction in the number of occurrences of each

type of test error. The economic value of each reduced error occurrence can

then be determined by identifying and categorizing the cost consequences of a

single CL and PL occurrence. Separate estimates of the value of each cost

consequence have to be derived. Examples of such cost consequences for a CL

occurrence are field repair costs, accidents due to equipment failure, and for

military applications the possible effects on readiness and mission success

rates. Examples of cost implications for a PL occurrence are unnecessary

diagnostic and repair costs, the costs of unnecessary retesting (including

packing and shipping to higher echelon laboratories) , excessive inventory of

parts and UUTs because of downtime, and reduced production yields.

Because testing is conducted for a variety of purposes, the applicability

as well as the dollar value of each of these cost implications (among others

that may be identified) can vary greatly depending on the particular test

situation being analyzed. Examples of the many test applications possible are

production testing both at intermediate stages and at the final product stage

(i.e., outgoing tests), incoming testing of components to be used in

manufacturing, and maintenance testing. The economic model presented in this

report can be used to analyze all of these test applications, provided the

basic nature of the measurement process is analog rather than digital.

4
That is, the principal characteristic of the UUT that is being measured must

be a continuous variable rather than a binomial or other type of discrete

variable. This requirement is not too restrictive, however, because even when

semiconductor devices that are primarily digital in design are being tested,

some critical analog measurements must be made.

This economic model may be employed to guide managers in making

cost-effective decisions regarding research planning, maintenance and

procurement policy, and product quality assurance functions. For example, by

determining the effects of alternative calibration support strategies on CL

and PL, the model can assist program managers in selecting areas of

calibration research that have the highest potential payoff. In the area of

maintenance and procurement policy, the model can be used to optimize the

frequency and quality of calibration procedures with existing test equipment,

as well as to select the best new test equipment through a comparison of

additional costs with the economic benefits of expected accuracy improvements.

Moreover, in the area of quality assurance, the model can be applied to

determine, for a particular product and test environment, which test

specifications optimize the economic trade-off between CL and PL.

1.3 Organization

The next section describes the mathematical formulation of the model for

calculating CL and PL as functions of the mean and the standard deviation of

the measurement error. Section 3 discusses the results of applying the model

to an analysis of how changes in the mean and standard deviation of the

measurement error can be expected to affect CL and PL. Section 4

summarizes the results and describes several useful applications of the model.

5
Appendix A gives complete documentation for BINORMAL.CL-PL, the interactive

computer program developed to calculate CL and PL for a wide variety of test

situations. Appendix B provides plots of CL and PL as functions of the mean

and standard deviation of measurement error, and illustrates how the plots can

be used to derive estimates of CL and PL for certain cases.

6
2. Mathematical Formulation of the Model

This section begins by summarizing the work of statisticians who

developed the basic model for computing CL and PL in the context of 100

percent testing. Then this basic model is extended to incorporate the effect

of possible changes in the accuracy of the test equipment. These accuracy

changes are mathematically specified both in terms of uncorrected measurement

bias and random measurement error. The notation that will be used in the

model is summarized in table 2.1.

2.1 The Basic Model

Three key papers published in the 1950' s describe how to use CL and PL

when selecting appropriate test specification limits. Eagle defined the basic

concepts of CL and PL and published plots of their values as functions of the

test specification limits and the random measurement error for a fixed

performance specif ication. 1 Eagle's discussion was restricted to the case in

which both the upper and lower test specification limits and the performance

specification limits are symmetric with respect to the mean of the performance

distribution. In an article published in the same issue of Industrial Quality

Control , Grubbs and Coon maintained this symmetry restriction while applying

the model to optimize the decision of how to set the test specification in

relation to the performance specif ications. 2 They developed decision rules

Alan R. Eagle, "A Method for Handling Errors in Testing and Measuring,"

Industrial Quality Control , Vol. 10, No. 5 (March 1954), pp 10-15.

Frank E. Grubbs and Helen J. Coon, "On Setting Test Limits Relative to

Specification Limits," Industrial Quality Control , Vol. 10, No. 5 (March

1954), pp 15-20.

7
Table 2.1 Notation Used in the Model and in Appendix B.

Symbol Definition

UUT Unit Under Test: product or product component that is

being tested.

CL Consumer's Loss: the probability of accepting a UUT that

does not conform to the performance specifications for

the attribute being tested.

PL Producer's Loss: the probability of rejecting a UUT

that does in fact conform to the performance

specifications for the attribute being tested.

u Performance values of the particular UUT attribute that

is being tested.

e Error values that occur when u is measured with the test

equipment

.

in Observed measurement values of u as taken by the test

equipment , m = u + e

.

p Statistical correlation between u and m.

p u Mean of the UUT performance distribution.

a u Standard deviation of the UUT performance distribution.

ue Mean of the measurement error distribution, also known as undetected systematic measurement error or bias.

o e Standard deviation of the measurement error distribution,

also known as measurement imprecision.

S|j Upper specification limit for UUT performance.

Sl Lower specification limit for UUT performance.

TA Tolerance adjustment of the test equipment: the

difference between the performance specification limits

and the test specification limits. In this discussion,

TA is assumed to be equal for upper and lower

specifications. TA is an adjustment typically made to

allow for measurement error in the test equipment. A

positive TA value means that the test specifications

lie inside the performance specifications. Thus, a positive TA makes the test specifications more stringent (i.e, more likely to reject a UUT) than the

unadjusted performance specifications.

8
Table 2.1 Notation Used in the Model and in Appendix B (Cont'd)

Symbol Definition

Su-TA Upper test specification limit.

SL+TA Lower test specification limit.

M Standardized upper performance specification limit.

"•2 Negative It t» L> JL- V of\-/ J- the fcr 1 A V** standardized fc- t-t LAVA CA A. VA A u \-l lower -L. K> W V. A. Derf L/ t_ A. J. ormance V-/ A> A A V- ^

—

specification limit.

Standardized upper test limit.

<12 Negative of the standardized lower test limit.

9
for setting the test specification limits in order to optimize three

alternative objective functions: (1) to make CL and PL equal; (2) to minimize

the sum of CL and PL; and (3) to minimize the weighted sum of CL and PL, where

the weights could be interpreted as the relative costs of a single occurrence

of each type of error. Five years later, Owen and Wiesen published a method

of calculating CL and PL using graphs and extended the basic model to include

the analysis of one-tail tests as well as of asymmetric settings on both the

test specification limits and the performance specification limits for

two-tail tests. * (A two-tail test has both upper and lower test specification

limits, while a one-tail test has only one of these limits.) For these

additional cases, Owen and Wiesen develop optimal decision rules for setting

test limits using the three objective functions suggested by Grubbs and Coon.

In addition, they develop test limit decision rules based on an interesting

objective function suggested by Tingey and Merrill, 2 namely to minimize the

sum of the costs of CL and PL when the cost per occurrence of CL varies in

proportion to the degree of nonconformance of the accepted product (i.e., how

far out of specifications the product is).

The general test situation being analyzed in all of the above modeling

efforts is that in which a single measurement is taken on a continuously

variable characteristic of every UUT in the process being monitored.

D. B. Owen and J. M. Wiesen, "A Method of Computing Bivariate Normal

Probabilities: With an Application to Handling Errors in Testing and

Measuring," The Bell System Technical Journal , Vol. 38 (March 1959),

pp. 553-572.

F. H. Tingey and J. A. Merrill, "Minimum Risk Specification Limits," AEC

Research and Development Report IDO-16396 (Phillips Petroleum Co., Atomic

Energy Division, Contract No. AT (10-l)-205), July 30, 1957.

10
As indicated in the previous section, the process may be manufacturing,

screening of input components prior to assembly, or periodic maintenance. The

test equipment with which the measurements are taken is subject to random

error. Because of the presence of measurement error there is a nonzero

probability that bad units will be accepted (i.e., CL) , and similarly a

nonzero probability that good units will be rejected (i.e., PL). The basic

model formulated in the above papers assumes that the distribution of the UUT

performance characteristic being measured is normal (Gaussian) with mean u u and standard deviation o u

. It is further assumed that the distribution of

the test measurement error is normal with a mean of zero and standard

deviation o £ .l The upper and lower UUT performance specification limits are

based on engineering considerations and assumed to be set at Sjj and Sl ,

respectively. 2 To adjust for the random error known to be present in the

test equipment measurements, the test specification limits are generally

assumed to be set at values different from Sy and Sl. The magnitude and

direction of this adjustment is given by TA, the tolerance adjustment of the

test equipment. 3 Thus, the upper test specification limit is set at Sjj-TA

and the lower limit is set at SL+TA. Thus, for TA>0 the test limits are

placed inside the UUT performance specification limits, which makes the test

In the formulation of Owen and Wiesen, the mean of the test measurement error distribution can assume nonzero values. However, this bias is generally

assumed to be known and corrected for in setting the test specification

limits

.

This discussion of the basic model and its extensions focuses on two-tail

tests. The computer program described in appendix A also covers upper and

lower one-tail tests.

This formulation assumes that the tolerance adjustment is equal for both

tails. The computer program in appendix A permits unequal tolerance

adjustments

.

11
limits more stringent (i.e., more likely to reject UUTs) than if there were no

tolerance adjustment. If a quality assurance manager considered measurement

errors of a particular test application to be insignificant relative to the

variance in the UUT performance distribution, TA would typically be set at

zero. In this case the test limits would equal the performance

specifications. When measurement errors cannot be ignored, a nonzero value

for TA is needed. For a given test equipment precision, the value assigned to

TA should be determined by the relative cost consequences of each occurrence

of CL and PL. The more costly is an occurrence of CL, the greater TA should

be in order to make the test more stringent and less likely to accept bad

UUTs. Conversely, the more costly is an occurrence of PL (relative to CL)

,

the smaller (perhaps even negative) TA should be in order to make the test

less stringent and less likely to reject good UUTs. Thus, when the precision

of test equipment is given, the economic trade-off between CL and PL is to be

balanced by setting the test limits with respect to the performance

specifications, that is, by choosing the appropriate value for TA. Indeed,

the primary motivation of this basic model, as developed in the 1950' s, is to

provide a rational basis for setting test limits.

The formulas for calculating CL and PL are based on the bivariate normal

probability distribution and have been worked out by Owen and Wiesen.2

The performance values of the particular UUT attribute that is being

measured and tested are denoted by u. These values are measured subject to

error, e, so that the observed measurement as taken by the test equipment is

x The very title of the article by Grubbs and Coon is, "On Setting Test Limits

Relative to Specification Limits." In the Owen and Wiesen article, the major

concluding section is entitled, "Special Criteria for Determining

Specification Limits."

^See section III, "Errors in Testing and Measuring," in Owen and Wiesen,

pp. 562-569.

12
given by m = u + e . As long as u and e are independent, the correlation

between u and m is given by p = o u

(o u

2 + o e 2 )~0 * 5 . Based on the conceptual

definitions of CL and PL, the following expressions are derived in Owen and

Wiesen:

CL = LCkj, - q2 , p) - L(kj, q x , p) - L(k2 , q 2 , P) + L(k2 , -qi , p), (2.1)

PL = CL + G(kx

) + G(k2

) - G(q x) - G(q2 ), (2.2)

where L(k,q,p) is the upper tail probability for the standardized bivariate

normal distribution,

(2tt)-1(1-P

2 )"0 * 5 / / exp [(2pum-u2 -m2 )/2(l-p 2

)] dm du, (2.3)

k q

and G(k) is the standardized univariate normal cumulative distribution

function,

k

( 2lT )-0.5

J exp [-u2 /2] du. (2. A)

—00

The arguments, kj and k2

, of the L and G functions are the standardized upper

performance specification limit and the negative of the standardized lower

performance specification limit, respectively, and are defined as follows:

k l =

( sir^u)/°u> and

(2.5)

k2 = -(SL-y u

)/a u

The arguments, qj and q2 , are the standardized upper test limit and the

negative of the standardized lower test limit, respectively, and are defined

as follows:

13
q x = (Su - TA - m u) (a u

2 + ae 2)-0.5

> and

(2.6)

Q2 = "CSL + TA - m u ) (o u

2 + o e 2)-0.5

The correctness of the formula for CL given in expression (2.1) can be

demonstrated by examining each of its four terms (i.e., the L functions).

Recall the basic definition of CL as the probability of accepting a bad UUT.

The first two terms in expression (2.1) address only those bad UUTs that

exceed the upper performance specification, S^. The first term, L(kj, -q£ , p),

is interpreted as the joint probability that the true performance value of

a UUT being tested exceeds the upper performance specification (i.e., the UUT

is bad on the high side), and that the test result is greater than the lower

test specification (i.e., the UUT is not rejected by falling below the lower

test specification). Since it is highly improbable for a UUT that exceeds Sg

to have a test result falling below the lower test limit, this first term

represents virtually all of the UUTs whose true performance values are bad on

the high side. The second term of CL, L(kj, qj , p) is the joint probability

that a UUT exceeds the upper performance specification (i.e., a bad UUT), and

that the test result exceeds the upper test limit (i.e., the UUT is rejected

by falling above the acceptable limit). In other words, the second terra in

expression (2.1) can be interpreted as the joint probability of a UUT that is

bad on the high side being correctly rejected by exceeding the upper test

limit. Thus the difference between the first two terms of CL,

L(kj, -q2 , p) - L(kj , qj , p), represents the probability that a UUT that is

bad on the high side will pass the test and be accepted. Thus, this difference

represents the upper tail portion of CL. Similarly, the remaining two terms

in the CL formula represent the probability a UUT that is bad on the low side

will pass the test and be accepted. Thus the difference given by the third

and fourth terms of expression (2.1), -L(k2, q2 , P) + L(k2, ~qi, p),

represents the lower tail portion of CL.

14
2.2 Extensions of the Model

In this subsection the basic model discussed above is extended to permit

the analysis of the effects of changes in test equipment accuracy on CL and

PL. The elements of accuracy to be analyzed include both uncorrected

measurement bias and random measurement error.

If a systematic error or bias is present in a measurement system and its

magnitude and direction are exactly known, then the value of TA can be

adjusted accordingly. This type of known and corrected bias is not of much

interest because it has no effect on CL and PL. On the other hand, a

measurement bias which is unknown and uncorrected will affect both CL and PL

because such a bias would result in different values for qj and q2 in the

formulas

.

Denote u e to be the undetected systematic measurement error (bias) in the

test equipment. Because of this bias, the mean of the observed measurement

now becomes u m

= \i u + y e

. However, the standard deviation of m is

unaffected by the bias as long as the error and the performance value are

independent. That is, a m

= (a u

2 + a e 2)0.5 f as was the case in the absence of

bias. Thus, p, which is the ratio of au

to om , is also unaffected by the

introduction of the bias.*

The basic model must be modified to take into account explicitly the

presence of undetected measurement bias with its consequent change in the mean

of the observed measurement, u m. The arguments, kj and k2 , given in expression (2.5) were derived by standardizing the upper and lower performance

specification limits and taking the negative of the lower standardized limit.

The procedure for standardizing involves subtracting the mean and dividing by

the standard deviation of the relevant distribution. Since kj and k2 represent

i 0wen and Wiesen, p. 562.

15
limits on the UUT distribution, standardizing involves subtracting y u and

dividing by a u

. The introduction of undetected measurement bias has no effect

on either of these parameters of the UUT distribution, so that the arguments,

kj and k£ , are unaffected by such bias. Consequently, the definitions of kj

and k2 found in expression (2.5) are to be used in both the basic model (with

no measurement bias) and the extended model (with measurement bias).

In contrast, the introduction of undetected measurement bias has a

decided effect on the arguments, qj and q2 , defined for the basic model in

expression (2.6). These arguments represent limits on the observed

measurement distribution (i.e., the distribution of m) . Consequently,

standardizing the test limits, Sy+TA and SL~TA, involves subtracting the mean

of m, which has changed from y u

in the basic model to U u

+jJ e in the extended

model because of the presence of nonzero measurement bias (p e ). As noted above,

the standard deviation of the observed measurement, o m , remains unaffected by

the measurement bias. Standardizing the upper and lower test limits using the

new value of \i m

and taking the negative of the standardized lower test limit

result in the following new definitions of the arguments q^ and q2 to be used

in the extended model:

2 2 a c qj = (Sy - TA - u u

- y e )(a u + ae )~°« 5

, and

(2.7)

Q2 = "( SL + TA - y u

- p e )(a u + o e ) -0 ' 5

.

In summary, the introduction of an undetected measurement bias has the

effect of reducing qj by an amount equal to y e (o u

2 + o e 2 )""0*5 and of

increasing q2 by the sarae amount. * This explicit introduction of the bias,

A Owen and Wiesen (p. 567) suggest that an undetected bias should be embodied

in the model as a change in the UUT performance specification limits, but

clearly the test bias only affects the observed measurement result, m, and

leaves the underlying UUT distribution and performance limits unchanged.

16
y e , constitutes a generalization of the basic model. The basic model is now

seen as a special case of the more general, extended model. The basic model

represents the case in which u e , the undetected measurement bias, is zero.

Consequently, the definitions of qj and q2 in expression (2.7) can be used in

all cases. This generalization of the model permits direct calculation of the

effect of any given undetected bias on CL and PL. In addition, this extended

model can be used to measure the effect of changing (i.e., increasing,

reducing, or even eliminating) an undetected bias by any amount. In this way,

the economic effects of any improvements in test equipment accuracy related to

systematic measurement error can be determined. The origin of such biasrelated accuracy improvements could be better equipment, better operation and

maintenance personnel or procedures, or an improved system of calibration, such

as is expected if the DTS system is implemented for ATE.

If no measurement bias is present, the basic model could be used to

analyze the effects of imprecision (i.e., o £

). On the other hand,

if changes of test equipment precision are to be analyzed in the presence of

bias , then the extended model that explicitly incorporates a term for

measurement bias in the formulas for CL and PL must be used

.

Indeed , both

analyses can be conducted using the extended model , because the bias term u e

can easily be set equal to zero in the q^ and q2 formulas.

In the CL and PL formulas of the extended model, o £ appears as part of the

denominators of qj , q£ and p. Thus, the determination of the effect on CL and

PL of an improvement in precision (i.e., a reduction in oe ) is accomplished by

first computing CL and PL using the original (lower precision) value of a e and

then recomputing CL and PL using the improved precision value of o e

. The

difference between the two computations represents the reduction in the

17
probabilities of occurrence of the CL and PL errors that could result from

improving the precision (lowering a £

) of the test equipment. The improved

precision could be realized by (1) more advanced test equipment, (2) better

operation and maintenance procedures that reduce the random measurement error

arising from such sources as electromagnetic interference, temperature,

humidity, and pressure, or (3) better calibration support systems.

Conversely, the model could determine the cost consequences, in terms of

increased values of CL and PL, of using less precise test equipment.

1

This concludes the discussion of the basic model and the extended version

of the model which permits the analysis of the effects on CL and PL of changes

in the systematic and/or random measurement errors of test equipment. The

next section summarizes the results of applying the extended model to

determine the magnitude, direction, and general pattern of these effects on CL

and PL for a broad range of test situations.

There are some circumstances in which reducing the measurement bias could

actually lead to increases in CL and PL. Similarly, it is possible that a decrease in ae may result in an increase in CL or PL. These cases are discussed in section 3.

18
3. Results

An interactive computer program has been developed which calculates the

values of the probabilities, CL and PL, as defined in the extended model

described in section 2. Complete documentation and a listing for this

program, BINORMAL. CL-PL, are provided in appendix A of this report. The

program was used to calculate CL and PL for a wide range of assumed model

parameter values. The objective of this section is to use the results of

these calculations to indicate how changes in systematic measurement error

(ue ) and in the standard deviation of measurement error (o e ) affect CL and PL.

The results reported here are all based on the case of two-tail tests with

performance specifications and test limits both symmetric around the mean

performance, u u

. So that the results can be easily generalized to other

cases, the performance distribution is standardized (i.e., P u=0, and o u=l).

3.1 Effects of Systematic Measurement Error

The analysis of the effects of systematic measurement error on CL and PL

is conducted on two levels. The first level of analysis focuses on the

direction of the effects of changing systematic error on CL and PL. The

question is whether an increase in systematic measurement error would increase

or decrease CL and PL. The second level of analysis concerns the general

pattern and order of magnitude of these effects. The questions here are

whether the effects increase with increases in systematic error and how large

are the effects. The next subsection deals with the first level of analysis.

3.1.1 Direction of Effects

The directional effects of systematic measurement error on CL and PL were

initially analyzed for 243 cases, each case being one combination of values

19
for the three parameters: performance specification limits (kj=k2); standard

deviation of measurement error (a e ); and test tolerance adjustment (TA). The

values used for these three parameters are given in table 3.1, along with the

four systematic error values used to analyze each case.

Because an increase in systematic measurement error represents a

deterioration in the accuracy performance of the test equipment, one's

intuitive expectation is that greater probabilities of wrong decisions, CL and

PL, would result. Indeed, PL behaved as expected for all of the cases

analyzed, while CL did so for most cases. The only exceptions occurred for

large negative values of TA (i.e., when the test limits lie considerably

outside the UUT performance specification limits) , where CL decreased as a

result of increasing the systematic measurement error. To indicate the bounds

for these exceptions, it is useful to examine the critical values of the ratio

of TA divided by the standard deviation of measurement error (i.e., TA/a £ ).

Critical values are to be interpreted as those values of the ratio, TA/a e ,

below which CL began to decrease with increasing systematic error. These

critical ratios are presented in table 3.2 for the cases analyzed. That is,

for values of TA/o e below (i.e., larger in a negative direction than) the

critical value given in the table, CL decreased when the systematic error was

increased. The data in table 3.2 suggest two general patterns. First, the

critical value of the ratio appears to increase (i.e., assumes a smaller

negative value) for larger performance limits. Secondly, the critical ratio

tends to decrease in value (i.e., assume a larger negative value) for smaller

random errors.

20
Table 3.1 Parameter Values and Systematic

Error Values Used to Analyze the Direction of

the Effect of Systematic Error on CL and PL

Characteristic Symbol Values

Performance

Specification

Standard Deviation

of Measurement

Error

k1 -k2 1,2,3

1/2,1/3,1/4,1/5,1/6,1/7,1/8,1/9,1/10

Tolerance

Adjustment

TA 0, ±0.5a e , ±1.0o e , ±1.5o e , +2.0o e

Systematic ye Measurement Error

0, 0.5o e , 1.0o e , 1.5o {

21
Table 3.2 Critical Values of TA/oe Below Which CL Decreases

with Increasing Systematic Error, for Various

Symmetric Performance Limits and Standard Deviations

of Measurement Error.

Standard Deviation Performance Limits

of Measurement

Error krk2=l k!=k2=2 k2 =k2 =3

0.500 -0.5 -0.5 0.0

0.333 -1.0 -0.5 -0.5

0.250 -1.0 -0.5 -0.5

0.200 -1.0 -1.0 -0.5

0.167 -1.0 -1.0 -1.0

0.143 -1.5 -1.0 -1.0

0.125 -1.5 -1.0 -1.0

0.111 -1.5 -1.0 -1.0

0.100 -1.5 -1.0 -1.0

22
Because of this contrary behavior of CL for large negative values of TA,

an improvement in accuracy achieved by reducing systematic measurement error

could theoretically result in an increase in the combined costs of CL and PL.

For the range of TA values defined by table 3.2, PL will definitely decrease,

but CL will increase. Thus, the net economic result could be positive or

negative, depending on the relative sizes of the offsetting changes in CL and

PL and on the relative economic costs of each occurrence of CL and PL.

Fortunately, this ambiguity concerning the overall economic effects of

reducing bias can be resolved for all the measurement cases analyzed. In the

first place, the problem does not arise very often because large negative TA

values are not commonly used in practice. This is because in most

applications the cost consequences of an occurrence of PL are less than those

of CL. For example, the rejection of a good item (i.e., PL) may merely result

in the costs of unnecessary retesting or the scrapping of a good component,

while the acceptance of a bad item (i.e., CL) could result in failure in the

field of an entire system with the associated costs of field diagnosis and

repair or shipment to and repair in a test and repair depot. The greater the

relative cost of CL versus PL, the more compelling is the need to guard

against occurrences of CL by employing a larger (in the positive direction)

value for TA.

A more definitive resolution of this ambiguity can be reached by

appealing to an explicit relationship between the value of TA and the relative

cost of CL versus PL. Grubbs and Coon derived such an explicit relationship

by assuming that the objective of test policy is to minimize the expected

23
value of the total cost of measurement error,! which is defined as follows:

TC = CCL • CL + CPL • PL , (3.1)

where

TC = the expected value of the total cost of measurement error for each

unit tested;

^CL 53 tne economic cost of one occurrence of CL;

CpL - the economic cost of one occurrence of PL; and

CL,PL are as previously defined.

Since CL and PL are both functions of TA, Grubbs and Coon partially

differentiated this expression for TC with respect to TA. This procedure led

to the following equation that determines the value of TA which minimizes TC:

G( Zl ) + G(z2

) - CPL/(CCL + CPL ), (3.2)

where

zi = -(1-P

2

) 0 - 5 (k + a uTA/ae

2

);

z 2 = -(l-p

2 )°- 5 [k+2k(a u

/ae ) 2 -0

uTA/o e

2

];

k = kj k2 , standardized performance

in expression (2.5) above; and

G(z) = standardized univariate normal

as defined in expression (2.4)

specification limits as defined

cumulative distribution function,

above.

Frank E. Grubbs and Helen J. Coon, "On Setting Test Limits Relative to

Specification Limits," p. 20.

24
One application of this equation is to select the cost-minimizing value

of TA for a given set of values for all the parameters: CCL , CpL, o u , o e , and

k. Another application, one more useful in resolving the ambiguity concerning

the economic effects of reducing bias, is to solve for that value of the

relative cost ratio, Cq^/Cpl, which satisfies the equation when the parameter

values listed in table 3.1 are used for k, o £ , o u , and TA.* The resulting

cost ratio gives the relative cost consequences of a unit of CL and PL that

would have to prevail in order for the TA value used in each case studied to

be optimally selected. In other words, if it is assumed that the TA values

for each case are selected with a view to minimizing TC, then the equation

permits us to specify the underlying relative cost ratio, Cci/CpL*

Following this procedure, one evaluates the two G functions for the

values of zj and Z2 specified by the parameter values for k, a e , a u , and TA,

and then sums them. This sum equals the righthand side of the equation,

Cpl/(Ccl+Cpl) ' By rearranging terms, one obtains the following expression for

the relative cost of CL versus PL:

CCL/ CPL " [G<zi) + G(z2 )]- 1 - 1. (3.3)

Investigating the direction of the effect that reducing bias has on total

measurement error costs is properly accomplished by determining the sign of

the partial derivative of costs with respect to bias. Differentiating TC as

given in expression (3.1), one obtains:

9TC = CCL • _3CL + CPL • 9PL . (3.4)

3ye 3ye 3ue

Though not explicitly given in table 3.1, <J U

=1 for all the cases tested.

25
The natural expectation is that a reduction in the bias would lead to lower

total measurement error costs. This expectation is fulfilled provided the

partial derivative expression is positive. Except for the case of large

negative TA, when 3CL/3u e becomes negative, both terms of the expression are

positive, so that the derivative is positive and expectations are fulfilled.

For those cases in which 3CL/3u£ is negative, the derivative will be

positive if and only if the absolute value of first product, Cq^ • 3CL/3y£ , i

smaller than the second product, Cp^ • 3PL/3y£

. This requirement is

equivalent to the condition:

ccl/ cpl 3PL/3ye 3CL/3ue(3.5)

An analysis was conducted for all those cases for which 3CL/3y£ turned

out to be negative. In every case, the condition (3.5) was fulfilled, and by

a substantial margin. To illustrate these results, the following three

figures present graphical comparisons of the two ratios of expression (3.5)

for the case of TA/o £ = -2.0. 1 For the figures 3.1, 3.2, and 3.3, the values

of the symmetric performance specification limits are set at one (k=l), two

(k=2), and three (k=3) standard deviations, respectively, above and below y u .

The results are similar for the other three negative values used for TA/o £

(i.e., -1.5, -1.0, and -0.5). In each figure, the solid line indicates the

These figures were derived using DATAPLOT, the graphical analysis computer

software package developed by James J. Filliben of the National Bureau of

Standards. For a summary description of this powerful package, see

J. J. Filliben, "DATAPLOT — An Interactive High Level Language for Graphics

Non-Linear Fitting, Data Analysis, and Mathematics," Computer Graphics , Vol. 15, No. 3 (August 1981), pp 199-213.

26
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29
value of the relative cost ratio, Cci/Cpi,, as calculated from equation (3.3).

The data points (designated by dots) are absolute values of the relative

change ratio, A PL/ACL, where

APL = (PL »0 .5a > ~ < PLu =0> » and

e e e

ACL = (CLU =o.5o ) ~ ( CLu =o)

•

e e e

The horizontal axis represents the relative measurement precision of each case

analyzed as indicated by the ratio, o u/o e

. As each figure demonstrates, the

relative cost ratio, C^/CpL, is consistently less than the absolute value of

the change in PL relative to CL for the entire range of measurement precision

analyzed. Thus, for these cases the condition as stated in expression (3.5)

is fulfilled, which demonstrates that an increase in ye leads to greater

measurement error costs, and conversely, that a decrease in \i e leads to lower

error costs.

3.1.2 Pattern and Magnitude of Effects

Besides the direction of the effects that changing measurement bias has

on CL and PL, the general pattern and order of magnitude of these effects need

to be analyzed. For this purpose, the computer program BINORMAL.CL-PL was run

to test additional cases which permit smaller increments in the values for

systematic measurement error. There are 60 of these cases, each case being

one combination of the three parameters: performance specification limits (kj

* ^2^5 standard deviation of measurement error (o e ); and test tolerance

adjustment (TA) . The values used for these parameters are presented in table

3.3, along with the set of nine systematic measurement error (u e

) values used

to analyze each case. As was true for the computer runs listed in table 3.1,

the value of o u for all these cases is also unity. Partial results of these

30
Table 3.3 Parameter Values and Systematic Error

Values Used to Analyze the Pattern and

Magnitude of the Effect of Systematic

Error on CL and PL.

Characteristic Symbol

Performance ^1 =^2

Specification

Standard Deviation o £ of Measurement

Error

Tolerance TA

Adjustment

Values

1, 2, 3

0.1, 0.25, 0.5, 1.0

0, 0.5o e , 1.0o e , 1.5a e , 2.0o e

Systematic

Measurement

Error

0, 0.25o e , 0.5o e , 0.75o e , 1.0c € , 1.25o e , 1.5o e , 1.75o e , 2.0o e

31
runs are presented in the figures B.l through B.6 of appendix B for the cases

in which the performance specification limits equal two standard deviations of

u (i.e., kj=lc2=2). The pattern and magnitude of the measurement bias effects

on CL and PL can be illustrated here with one set of these figures.

Figure 3.4 presents plots of the value of CL as a function of systematic

measurement error for the case of ae =0.25a u

. In order to examine the effect

of TA, five plots are given in the figure, one for each of the five values of

TA as specified in table 3.3. As noted in section 3.1.1, restricting TA to

nonnegative values assures that there is a positively sloped relationship

between systematic measurement error and CL. Figure 3.4 confirms this relationship, since all the plots indicate that CL increases as systematic error

increases.

Figure 3.5 presents five plots of PL as a function of systematic error

for the case of 0 e =O.25a u , each plot corresponding to a different positive

value of TA, as specified in table 3.3. The consistently positive relationship between PL and systematic error that was noted in section 3.1.1 is confirmed in all these plots of figure 3.5: larger systematic measurement errors

result in greater values of PL.

Throughout the entire range of values for systematic error, all of the PL

plots and most of the CL plots (i.e., those for TA > 1.0a e ) appear to be

concave upward (i.e., positive second derivative of CL and PL with respect to

systematic error). This upward concavity means that the impacts of increases

in systematic error on CL and PL tend to increase with greater errors. For

32
0.0O"e

0.25CTe 0.50ae0.75O"e 1.00(7e 1.25CTe 1.50ae 1.75ae 2.00ae SYSTEMATIC MEASUREMENT ERROR (BIAS), \x t

Figure 3.4. Consumer's Loss as a Function of Systematic Measurement Error

(Bias), for Characteristic Values of Test Tolerance

33
O.OCTe 0.25<7e 0.50ae 0.75O"e 1.00(7e 1.25(7e 1.50O~e 1.75ae 2.00ae SYSTEMATIC MEASUREMENT ERROR (BIAS), /u f

Figure 3.5. Producer's Loss as a Function of Systematic Measurement Error

(Bias), for Characteristic Values of Test Tolerance

34
example, the computed values on which figure 3.4 is based indicate that, for a

TA value of 1.0o e , CL increases from 0.001961 to 0.002403 to 0.003125 when

systematic error increases from 0.25a e to 0.5o e to 0.75o e , respectively.

Thus, the first increment is only 0.000442 (i.e., 0.002403-0.001961), while

the second increment is nearly twice as much, 0.000722 (i.e., 0.003125-

0.002403).

In spite of this general pattern of upward concavity for the plots as

shown, one can observe virtual linearity and even inflection points (reversal

to downward concavity) in two of the CL plots in figure 3.4 (i.e., those whose

TA values are 0.5a e and 0). Indeed, even for the other data plots inflection

points are eventually reached, but outside the range of systematic error

values shown in the figures.

The reason for the eventual decrease in slope is that for any fixed value

of the performance specification (kj=k2), a certain percentage of all the UUTs

will be good (within specifications) and the remaining percentage will be bad

(outside specifications). These percentages represent theoretical limits on

PL and CL, respectively. The actual limits on CL and PL for any particular

test situation are often considerably lower than these theoretical

percentages.! The plots of CL and PL approach these actual limits

asymptotically. The decrease in slope is not apparent in the figures for PL

because the systematic measurement errors used in these computations are not

For example, when jj £ is positive and very large, the test will accept almost

all of the bad UUTs that are in the lower rejection region, and thus

contribute to CL. All of the bad UUTs in the upper rejection region,

however, appear much worse than they are and so will be correctly rejected

and will not contribute to CL. Therefore, in this case the actual limit on CL is only half of the theoretical limit (i.e. , half of the total percentage

of UUTs that are bad).

35
large enough for the plots to reach their points of inflection. For some of

the CL plots the points of inflection can be observed because they tend to

occur at lower error values for CL than for PL.

Comparison of figures 3.4 and 3.5 reveals that for a given size

systematic error, the value of TA has a significant but opposite effect on CL

and PL. For example, a decrease in TA increases CL but decreases PL. The

relationship between CL and TA makes sense if one remembers that TA affects

how strict the test is; the smaller TA is, the wider (i.e., more accepting)

the test specifications become. The reason for the increase in CL is as

follows. A decrease in TA leaves the number of bad products unchanged, but

increases the likelihood of a product's being accepted. Thus, more bad

products will be accepted, so CL increases.

The converse is true for the relationship between PL and TA. A decrease

in TA decreases the likelihood of a product's being rejected by widening the

test specifications, and yet has no effect on the actual number of good units

available to be rejected. As a result, fewer good products are rejected, so

that PL decreases. In effect, the selection of a value for TA constitutes a

choice in the trade-off between CL and PL. Indeed, the optimal selection of

TA to minimize the total economic consequences of CL and PL is the basis of

equation (3.2), developed above in section 3.1.1.

36
3.2. Effects of Random Measurement Error

The analysis of the effects of random measurement error on CL and PL

parallels the analysis of systematic error followed in subsection 3.1. The

first level of analysis concerns the direction of the effects, while the

second level focuses on their general pattern and magnitude.

3.2.1. Direction of Effects

The directional effects of random measurement error on CL and PL were

analyzed for 108 cases, each case being one combination of values for the

three parameters: performance specification limits (kj=k2); systematic

measurement error (u e ); and test tolerance adjustment (TA). The values used

for these parameters are shown in table 3.4, along with the nine values of the

standard deviation of measurement error (o e

) used to analyze each case.

Because o £ is a variable in each case of this analysis, the values of TA and

ue are expressed in absolute terms rather than in units of a e as was done when

the effects of systematic error were analyzed. For all these cases, the value

of a u

is assumed to be unity.

One would expect an increase in a e to lead to greater probabilities of

incorrect decisions, in terms of both greater CL and greater PL. Indeed, for

those cases most likely to occur in test situations, CL increased as o e

increased. This expectation is fulfilled as long as the systematic error is

less than or equal to the test tolerance adjustment. This finding can be

stated in the form of a sufficient 1 condition. That is,

There are a few cases in which u e > TA and still 3CL > 0.

3ae

37
Table 3.4 Parameter Values and Standard Deviation of

Measurement Error Values Used to Analyze the

Direction of the Effect of the Standard

Deviation of Measurement Error on CL and PL.

Characteristic Symbol

Performance ^1 =^2

Specification

Systematic u e Measurement Error

Tolerance TA

Adjustment

Values

1, 2, 3

0, 0.5, 1.0, 1.5

0, +0.25, +0.5, +0.75, +1.0

Standard o e 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8,

Deviation of 1/9, 1/10

Measurement

Error

38
3CL > o if He < TA. 1 (3.8)

3ae

In those actual test situations in which the order of magnitude of u e is

known, it is likely that TA will be set larger than u e in order to control the

size of CL, provided the economic cost of each CL occurrence is significant.

If, however, the cost of CL is relatively low compared with that of PL, then

the condition given in expression (3.8) is not likely to be fulfilled. That

is, values smaller than u e or even negative values will likely be assigned to

TA. For such cases, the total cost of measurement error as specified in

equation (3.1) should be analyzed along the lines followed in subsection

3.1.1.2

An increase in o e is also expected to cause an increase in PL. Once

again, for the most likely test situations these expectations are fulfilled.

For all the cases analyzed, PL consistently increased as a result of

increasing o £ as long as the performance specification exceeded the sum of the

tolerance adjustment and the systematic measurement error. This finding can

be summarized in terms of the condition statement:

3 PL > o if k*o u

> TA+p e , (3.9)

3oe

where k is the number of standard deviations that the performance limits

lie above and below the mean for a symmetric test. That is,

k-o u

= sU

_^u = H u-Sl = 0.5(Su-SL ).

*The partial derivative could theoretically be zero for the unlikely case in

which the tolerance adjustment equals the performance specification. In

this case, CL remains zero because there is no acceptance region and the

only error possible is PL, which remains at its maximum value.

2This analysis is proposed as the subject of future research.

39
The condition stated in expression (3.9) will almost always be fulfilled

in realistic test situations because the value k*o u represents one half of the

entire range enclosed within the upper and lower performance specification

limits. The sizes of TA and y e derive from the distribution of the measurement error, e, which typically has a narrower range, by an order of magnitude,

than that of the UUT performance characteristic, u. Consequently, even the

sum, TA + y e , will be considerably smaller than half the performance range

under normal circumstances. It is interesting to note, however, that it is

theoretically possible for an increase in o e caused by a decrease in precision

to lead to a decrease in the probability of rejecting good units (PL). An

important objective for future research is to establish more exactly the

conditions for assuring that PL increases as a £ increases, either through more

detailed simulations or the application of calculus.

3.2.2. Pattern and Magnitude of Effects

The pattern and magnitude of the effects on CL and PL of changes in

measurement precision is established by running the computer program

BINORMAL.CL-PL for a wide range of cases in which a £ is assigned steadily

increasing values. Forty-five such cases were analyzed, each case being one

combination of the three parameters: symmetric performance specification

limits (kj=k2); systematic measurement error (y e ); and test tolerance adjustment (TA). The values used for these parameters ace presented in table 3.5,

along with the set of nine o e values used to analyze each case. As for all of

the cases analyzed in this section, the value of a u

is assumed to be unity.

40
Partial results of this analysis are shown in the plots of figures B.7 through

B.12 of appendix B for those cases in which the performance specification

limits equal two standard deviations of u (i.e., kj=k2=2). Both the general

pattern and the size of the effects of a e on CL and PL are illustrated

here with one set of these plots.

Table 3.5 Parameter Values and Standard Deviation of

Measurement Error Values Used to Analyze the

Pattern and Magnitude of the Effect of the

Standard Deviation of Measurement Error on CL and PL.

Characteristic

Performance

Specification

Symbol

ki=k2

Values

1, 2, 3

Systematic u e Measurement Error

0, 0.25, 0.50

Tolerance

Adjustment

TA 0, 0.25, 0.50, 0.75, 1.0

Standard ae 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7,

Deviation of 0.8, 0.9

Measurement

Error

41
Figure 3.6 presents plots of the value of CL as a function of the

standard deviation of measurement error for the case of systematic error equal

to 0.25. To analyze the effect that TA has on CL, four separate plots are

given, based on the values specified for TA in table 3.5.1 F0r an Qf these

plots in figure 3.6 the condition in expression (3.8) is fulfilled, so that CL

is a positively sloped function of o e

.

The curvature of all the plots in figure 3.6 varies depending on the

value of TA and the range of interest for a e

. For high values of TA (i.e., ^>

0.5) the plots are somewhat concave upward initially and then reach inflection

points and become concave downward as a £ increases. The points of inflection

occur at lower values of a e , the lower the value of TA used. Indeed, for the

lowest value of TA (i.e., 0.25), the inflection point appears to occur for

values of o e less than 0.1. The significance of upward concavity is that the

impacts of increases in o e on CL tend to increase with greater o e

. For

example, the computed values underlying the plot for a TA value of 1.0

indicate that CL increases from 0.000200 to 0.000595 to 0.001200 when o

£

increases from 0.4 to 0.5 to 0.6, respectively. Thus, the first increment is

only 0.000395 (i.e., 0.000595 -0.000200), while the second increment is

0.000605.

In contrast to this observed upward curvature, for a £ values in excess of

0.6, a general pattern of approximate linearity is observed for all of the

plots. In this range, therefore, the impact on CL of increasing a e is

relatively constant.

*0nly four of the five values of TA given in table 3.5 are used in these plots

because for TA=0 the condition given in expression (3.8) is violated (i.e.,

Me »0.25>TA=0) . The plot of CL for this value of TA would have a negative

slope.

42
b

co CO

GO

OJ

CJc«M

0)o Ou

CO>

OJ CO O QJ3

TJ iH

»-i 03

03 > C O

03 tH

4-i 4-i C/3 CO •H

H-l l-i O 0J u c u O 03 •H Ui

4-> 03 o x: c u a o

03 U-l

CO -

CO J-i O

03 l-i CO P o w J

4-1 co c - oj P E

OJ OJ 6D D

CO CO C 03 O OJ

CO

OJ

l-l 3

bO H

CO CM

SS01 M3IAinSN00

43
Figure 3.7 presents the four plots of PL as a function of the standard

deviation of measurement error for the same values of TA as used in figure 3.6

for CL. Because the condition specified in expression (3.9) is fulfilled for

all the cases represented by these plots, they are positively sloped throughout their range.

The curvature of all the plots in figure 3.7 is such that they are

concave upward over the entire range of a£ values analyzed. Since it is

unlikely that a£ would exceed unity (the value of a u ) , this concavity can be

said to prevail for most realistic test situations. * As is true for the

effects on CL discussed above, the upward concavity means that the impacts of

increases in o£ on CL tend to increase with greater a e

. For example, the

computed values underlying figure 3.7 indicate that for a TA value of 0.25, CL

increases from 0.047473 to 0.053500 to 0.062408 when a £ increases from 0.1 to

0.2 to 0.3, respectively. Thus, the first increment is only 0.006027, while

the next increment is 0.008908.

The curvature of these plots where o e <0.1o u was not analyzed here.

Extrapolation of these plots back toward the origin would suggest that the

curvature decreases with decreasing slope.

44
i—i—i—

r

o

co 00

LO

CO

CM

00

J_l I L J I I I L

ID

CM

mom r— m cm

CO CO CO

ID Oirs CM CM

IfJ CM

CM

g in o in

g ^ « N

ID O in

CM

SS01 dBOflQOdd

45
4. Summary and Application

In this report an economic model has been presented which traces the

effects of alternative levels of accuracy in testing and measurement on the

correctness of acceptance and rejection decisions. The model expresses two

probabilities, that of accepting a bad unit (CL) , and that of rejecting a good

unit (PL), as explicit functions of systematic measurement error (y e ) and the

standard deviation of random measurement error (o £ ). These functions can be

used to calculate directly the benefits, as measured by lower probabilities of

errors, from reduced systematic and/or random errors. When applied to a case

study, an economic value, in terms of dollars saved per time period, can be

established for these reduced probabilities. The economic value depends on

the number of tests conducted in a time period and on the particular test

situation being analyzed. Multiplication of each probability reduction by the

number of items to be tested gives the expected reduction in the number of

occurrences of each type of test error. The economic value of each reduced

error occurrence can then be determined by identifying and categorizing the

cost consequences of single CL and PL occurrences. Separate estimates of the

value of each cost consequence have to be derived. Examples of such cost consequences for a CL occurrence are field repair costs, accidents due to equipment failure, and possible effects on readiness and mission success rates in

military applications. Examples of cost implications for a PL occurrence are

unnecessary diagnostic and repair costs, the costs of unnecessary retesting

(including packing and shipping to higher echelon laboratories), excessive

inventory of parts and UUTs because of downtime, and reduced production

yields.

46
The applicability as well as the dollar value of each of these and the

many other cost implications resulting from occurrences of CL and PL can vary

greatly depending on the particular test situation being analyzed. Examples

of the many test applications possible are production testing both at

intermediate stages and at the final product stage (i.e., outgoing tests),

incoming testing of components to be used in manufacturing, and maintenance

testing. The economic model presented in this report can be used to analyze

all of these test applications, provided the basic nature of the measurement

process is analog rather than digital. That is, the principal characteristic

of the UUT that is being measured must be a continuous variable, rather than

a binomial or other type of discrete variable. This requirement is not too

restrictive, however, because even when semiconductor devices that are

primarily digital in design are being tested, some critical analog

measurements must be made.

This economic model can be applied to a wide variety of fields, including

metrology, research planning, maintenance and procurement policy, and product

quality assurance functions. To illustrate metrology research applications,

the magnitudes of the reductions in CL and PL that would result from

alternative calibration support strategies can be computed by the model and

then used to assist program managers in selecting test methods and calibration

support systems that have the highest potential payoff. In the maintenance

area, the model can be used to optimize the frequency and quality of

calibration procedures with existing test equipment. In the area of

procurement policy, the model can be used to select the best new test

equipment through a comparison of additional costs with the economic benefits

of expected accuracy improvements. In quality assurance, the model can be

applied to determine, for a particular product and test environment, which

test specifications optimize the economic tradeoff between CL and PL.

47
Bibliography

1. Bell, Barry A. and Petersons, Oskars , "ATE Calibration by Means of Dynamic

Transport Standards," Proceedings of Autotestcon 81 : International Automatic Testing Conference (Orlando, FL, October 19-21, 1981), Institute of

Electrical and Electronic Engineers (IEEE Catalogue No. 81 CH 1716-0), 1981,

pp. 280-287.

2. Brady, Dale D. and Odorizzi, Dominick J., "Statistical Tolerancing of

Complex Systems," Electronic Design , January 4, 1978, pp. 134-138.

3. Burr, I. W. , "Specifying the Desired Distribution Rather than Maximum and

Minimum Limits," Industrial Quality Control , Vol. 24, No. 2, August 1967,

pp. 94ff.

4. Crow, E. L. , "Optimum Allocation of Calibration Errors," Industrial Quality

Control , Vol. 23, No. 5, November 1966, pp. 215-218.

5. Eagle, A. R. , "A Method for Handling Errors in Testing and Measuring,"

Industrial Quality Control , Vol. 10, No. 5, March 1954, pp. 10-15.

6. Eisenhart, Churchill, "Realistic Evaluation of the Precision and Accuracy

of Instrument Calibration Systems," Journal of Research of the National

Bureau of Standards , Vol. 67C, No. 2, April-June 1963.

7. Evans, D. H. , "Optimum Tolerance Assignment to Yield Minimum Manufacturing

Cost," Bell System Technical Journal , Vol. 37, No. 2, 1958.

8. Evans, D. H. , "A Statistical Tolerancing Formulation," Journal of Quality

Technology , Vol. 2, No. 4, October 1970, pp. 226ff.

9. Evans, D. H. , "An Application of Numerical Integration Techniques to

Statistical Tolerancing, II—A Note on the Error," Technometrics , Vol. 13,

No. 2, May, 1971 , pp. 315ff.

10. Evans, D. H. , "An Application of Numerical Integration Techniques to

Statistical Tolerancing, III—General Distributions," Technometrics , Vol.

14, No. 1, February 1972, pp. 23ff.

11. Evans, D. H. , "Statistical Tolerancing: The State of the Art (Part I.

Background)," Journal of Quality Technology , Vol. 6, No. 4, October 1974,

pp. 188-195.

12. Evans, D. H. , "Statistical Tolerancing: The State of the Art (Part II.

Methods for Estimating Moments)," Journal of Quality Technology , Vol. 7,

No. 1, January 1975, pp. 1-12.

13. Evans, D. H. , "Statistical Tolerancing: The State of the Art (Part III.

Shifts and Drifts)," Journal of Quality Technology , Vol. 7, No. 2, April

1975, pp. 72-76.

48
14. Glassman, Jerry and Rodriduez, Robert N. , "Treatment of Errors in

Measurement Systems," (Miraeo, Navy Metrology Engineering Center, Pomona,

CA).

15. Grubbs, F. E. , "Errors of Measurement, Precision, Accuracy, and the

Statistical Comparison of Measuring Instruments," Technometrics , Vol. 15,

No. 1, February 1973, pp. 53-66.

16. Grubbs, F. E. , "On Estimating Precision of Measuring Instruments and

Product Variability," Journal of the American Statistical Association , Vol. 43, No. 242, June 1948, pp. 243-264.

17. Grubbs, Frank E. and Coon, Helen J., "On Setting Test Limits Relative to

Specification Limits," Industrial Quality Control , March 1954, pp. 15-20.

18. Hilliard, Jimmy E. and Miller, J. R. , III, "The Effect of Calibration and

End Item Performance in Echelon Systems," Journal of Quality Technology , Vol. 12, No. 2, April 1980, pp. 61-70.

19. Juran, J. M. , Editor, Quality Control Handbook , 2nd Edition, McGraw Hill,

New York, New York, 1962.

20. Ku, Harry H. , "Notes on the Use of Propagation of Error Formulas," Journal

of Research of the National Bureau of Standards , Vol. 70C, No. 4,

October-December 1966.

21. Mansoor, E. M. , "The Application of Probability to Tolerances Used in

Engineering Design," Proceedings of the Institution of Mechanical

Engineers , Vol. 1781, No. 1, 1963, pp. 29ff.

22. McNish, A. G. , and Cameron, J. M. , "Propagation of Errors in a Chain of

Standards," IRE Transactions on Instrumentation , Vol. 1-9, No. 2,

September 1960, pp. 101-104.

23. Miller, J. R. , III, et al. , "Effect of Calibration on End Item

Performance," USAMCC-MM-75-15. U.S. Army Metrology and Calibration

Center, Redstone Arsenal, Alabama, November 25, 1975.

24. Owen, D. B., "Applications of the Tables," Chapter II of Tables of the

Bivariate Normal Distribution Function and Related Functions , National

Bureau of Standards Applied Mathematics Series 50, U.S. Government

Printing Office, Washington, DC, June 1959.

25. Owen, D. B. , and Wiesen, J. M. , "A Method of Computing Bivariate Normal

Probabilities with an Application to Handling Errors in Testing and

Measuring," Bell System Technical Journal , March 1959, pp. 553-572.

26. Pike, E. W. , and Silverberg, T. R. , "Assigning Tolerances for Maximum

Economy," Machine Design , September 1953, pp. 139-146.

49
27. Rajaraman, M. K. , "Allocation of Errors in Hierarchical Calibration or Assembly," Journal of Quality Technology , Vol. 6, No. 1, January 1974, pp.

42-45.

28. Tingey, F. H. and Merrill, J. A., "Minimum Risk Specification Limits," AEC

Research and Development Report IDO-16396 (Phillips Petroleum Co., Atomic

Energy Division, Contract No. AT (10-l)-205), July 30, 1957.

29. Woods, W. M. and Zerna, P. W. , "Cumulative Effect of Calibration Errors,"

Industrial Quality Control , Vol. 22, No. 8, February 1966, pp. 411-412.

50
Appendix A. Binormal.CL-PL, an Interactive Computer Program for Calculating

Consumer's Loss and Producer's Loss in Testing

A.l Purpose

BINORMAL.CL-PL is an interactive FORTRAN program developed for use in

measuring the benefits of improved accuracy of test equipment.! Two types of

accuracy improvements can be analyzed: (1) a reduction of the systematic

measurement error of test equipment (i.e., a reduction of the bias present in

the measurement results); and (2) a reduction of the standard deviation of the

measurement error of the test equipment (i.e., an increase in measurement

precision). These improvements might be achieved, for example, through better

test operating procedures, more controlled environmental conditions during

testing, more frequent calibration procedures, as well as through the

acquisition of more accurate test equipment. When provided with parameters

that fully characterize the distribution of the UUT performance2 and the

accuracy of the test equipment, BINORMAL.CL-PL calculates the probability of

accepting a UUT that does not conform to its performance specifications (i.e.,

Consumer's Loss or CL) and the probability of rejecting a UUT that does

conform to its performance specifications (i.e., Producer's Loss or PL). The

option is available for conducting a sensitivity analysis of the results by

The software documentation in this appendix follows the principal Federal

Information Processing Standards (FIPS) found in Guidelines for Documentation

of Computer Programs and Automated Data Systems , FIPS Publication #38, U.S.

Department of Commerce, National Bureau of Standards, February 15, 1976.

UUT performance refers to the value of the particular UUT attribute that is

being measured by the test equipment.

51
varying the parameters relating to the accuracy of the test equipment and the

setting of the test specifications.

BINORMALoCL-PL can be helpful in planning research, making maintenance

and procurement decisions, and solving certain quality control problems. The

results of the program can be used to guide decisions regarding which test

methods and calibration support strategies would have the highest payoff. The

program also could be used to optimize the frequency and quality of

calibration procedures with existing test equipment, as well as to select new

test equipment by comparing additional costs with expected accuracy

improvements. Finally, in the area of quality control, the sensitivity

analysis capability of the program can be used to establish for a particular

product the test specifications that optimize the trade-off between CL and

PL.

A. 2 Scope

BINORMAL . CL-PL can be run on any computer with a FORTRAN 77 compiler 1 and

access to routines supplied by the International Mathematical and Statistical

Libraries, Inc. (IMSL).2 The program is quite flexible in a number of ways.

In the first place, it can be applied to the three most commonly encountered

test situations: two-tail tests; upper-tail tests; and lower-tail tests.

3

In a two-tail test, the UUT performance is expected to be between established

1 American National Standards Programming Language, FORTRAN, ANSI X3. 9-1 978.

2 Documentation for the two routines used in the program, MDNOR and MDBNOR, can be found in International Mathematical and Statistical Libraries, Inc.,

The IMSL Library , 9th Edition, Vol. 3 (Houston, TX, 1982).

^The only test situations that cannot be analyzed by the program are those in

which the acceptable values are in any way separated from one another by

unacceptable values. For instance, if the extreme values (i.e., the two

tails) of the UUT performance distribution constituted acceptable regions

while the values close to the mean constituted the unacceptable region, then

this program could not be used.

52
upper and lower performance specifications in order to be acceptable. An

upper-tail test is one in which the UUT performance is expected to be below a

given upper performance specification to be acceptable, while a lower-tail

test is one in which the UUT performance is expected to be above a lower

performance specification to be acceptable. The program is also flexible in

that the test specifications, that is, the cut-off values for the measured UUT

performance, can be set anywhere in relation to the performance

specifications. Moreover, when the program is used for a two-tail test, the

performance specifications need not be symmetric around the mean UUT

performance. There is also great flexibility in the types of UUTs and test

equipment the program can analyze. The only requirement is that the UUT

performance and the test equipment measurement error both must be assumed to

follow normal distributions, which implies that they can be treated as

continuous variables. The program user is free to specify any means and

standard deviations for these normal distributions.

1

A. 3 Using the Program

Once the user has entered the proper control statements to begin

execution of the program, 2 the computer prompts the user for each input value

to be entered. Table A.l lists and defines the symbols used for the input

variables in this appendix and in the program. In addition to entering these

values, the user must indicate whether the test has two tails, an upper tail

only, or a lower tail only.

•'In practice, the standard deviation of the measurement error will typically

be less than that of the UUT.

^Since control statements are specific to the computer system on which the

program is run, they will not be discussed here.

53
The output of the program is a listing of all the input values and the

corresponding computed probabilities, CL and PL. After the results are

printed the user indicates whether or not a sensitivity analysis is desired.

If not, the program ends. If sensitivity analysis is requested, the user is

prompted to enter new values for ME, SDE, and BU and/or BL (depending on the

test situation). New values for CL and PL are calculated and the new results

are printed in the same format as before. The user can continue entering new

values as many times as desired. A sample run with one case of sensitivity

analysis is shown in figure A. 1.

Table A.l Input Variables for BINORMAL . CL-PL

Symbol Definition

MU Mean of the UUT performance distribution.

SDU Standard deviation of the UUT performance distribution.

ME Mean of the test equipment measurement error distribution, also known as undetected systematic measurement

error or bias.

SDE Standard deviation of the test equipment measurement

error distribution, also known as measurement imprecision.

SU Upper specification limit for UUT performance.

SL Lower specification limit for UUT performance.

BU Upper test equipment tolerance factor.

1

BL Lower test equipment tolerance factor.*

The tolerance factor is the number of SDEs between the UUT performance

specification value and the test specification value. It is generally used

to allow for measurement error of the test equipment. A positive factor

means that the test specification limits are more stringent (i.e., more likely to reject UUTs) than the unadjusted performance specification limits.

In many cases the tolerance factor is zero, which means that the test specification limits are set equal to the UUT performance specification limits.

54
Figure A.l. Illustrative Computer Run of BINORMAL . CL-PL

ENTER MEAN OF MEASUREMENT ERROR

>.25

ENTER STANDARD DEVIATION OF MEASUREMENT ERROR

>.5

ENTER MEAN OF TRUE PERFORMANCE

>0

ENTER STANDARD DEVIATION OF TRUE PERFORMANCE

>1

IS THIS A TWO-TAIL TEST OF PERFORMANCE?( Y/N)

>Y

ENTER UPPER PERFORMANCE SPECIFICATION

>2

ENTER LOWER PERFORMANCE SPECIFICATION

>-2

ENTER UPPER TOLERANCE FACTOR OF ATE

>1

ENTER LOWER TOLERANCE FACTOR OF ATE

>1

FOR:

MEAN OF MEASUREMENT ERROR = .25000000

STANDARD DEVIATION OF MEASUREMENT ERROR = .50000000

MEAN OF TRUE PERFORMANCE = .00000000

STANDARD DEVIATION OF TRUE PERFORMANCE = 1.0000000

UPPER PERFORMANCE SPECIFICATION = 2.0000000

LOWER PERFORMANCE SPECIFICATION = -2.0000000

UPPER TOLERANCE FACTOR OF ATE = 1.0000000

LOWER TOLERANCE FACTOR OF ATE = 1.0000000

THEN CONSUMER LOSS = .003878

AND PRODUCER LOSS = .148917

WOULD YOU LIKE TO DO A SENSITIVITY ANALYSIS?( Y/N)

>Y

ENTER NEW VALUES FOR MEAN OF MEASUREMENT ERROR

STANDARD DEVIATION OF MEASUREMENT ERROR .UPPER TOLERANCE FACTOR OF ATE, AND

LOWER TOLERANCE FACTOR OF ATE , ONE VALUE PER PROMPT.

>.125

>.5

>1

>1

FOR:

MEAN OF MEASUREMENT ERROR

STANDARD DEVIATION OF MEASUREMENT ERROR

MEAN OF TRUE PERFORMANCE

STANDARD DEVIATION OF TRUE PERFORMANCE

UPPER PERFORMANCE SPECIFICATION

LOWER PERFORMANCE SPECIFICATION

UPPER TOLERANCE FACTOR OF ATE

LOWER TOLERANCE FACTOR OF ATE

.12500000

.50000000

.00000000

1.0000000

2.0000000

-2.0000000

1.0000000

1.0000000

THEN CONSUMER LOSS = .003214

AND PRODUCER LOSS = .140143

WOULD YOU LIKE TO DO A SENSITIVITY ANALYSIS?( Y/N)

>N

55
The user should be aware that there is no capability in the program for

evaluating whether the numbers entered are reasonable. That is, there is no

edit checking in the program. Thus, the user must be sure that the values

entered represent a realistic situation. For example, if the values entered

produce a lower test specification that is greater than the upper test

specification, then the program will not fail to run, but may result in

negative values for CL and/or values for PL that are greater than unity.

A. 4 Program Characteristics

B I NORMAL. CL-PL is written in FORTRAN 77 with external routines supplied

by IMSL. These routines, MDNOR and MDBNOR, give single precision results.

The accuracy of MDNOR, which calculates univariate normal probabilities, is

dependent on the computer system being used. To compute the CL and PL values

presented as illustrations and results throughout this report, the authors

employed the UNIVAC 1100/82 system installed at NBS in Gaithersburg , Maryland.

The routine MDNOR would be accurate to the eighth digit on such a 36 bit

machine. 1 Documentation in the IMSL Library states that MDBNOR, which

calculates bivariate normal probabilities, gives values differing by less than

0.00001 from the values in Tables of the Bivariate Normal Distribution

Function and Related Functions . 2 The values in these tables have a maximum

absolute error of 0.000002,3 so that the total absolute error for each

bivariate normal probability computed by MDBNOR is less than 0.000012. Given

these accuracies for the two external routines used, it is expected that the

value of CL

^Telephone conversation with technical staff at IMSL.

^U.S. Department of Commerce, Tables of the Bivariate Normal Distribution

Function and Related Functions , National Bureau of Standards, Applied

Mathematics Series 50 (Washington, D.C., 1959).

3 Ibid . , p. vi.

56
computed by BINORMAL . CL-PL will have a maximum alsolute error of 0.000048,

while the value of PL would have a maximum absolute error of 0.00004804.

The following formulas are used to calculate CL and PL:

RHO = SDU ; where RHO is the statistical correlation

between the UUT performance values and the

/SDU^+SDE7 observed measurement values;

Kl = SU-MU ; SDU

K2 = MU-SL ; SDU

Ql = SU-MU- (BU*SDE) -ME

; and

/SDUZ+SDE*

02 = MU-SL-(BL*SDE)+ME

/SDU*+SDE*

For a two-tail test,

CL - L(-K1 ,Q2,RH0)-L(-K1 ,-Ql ,RHO)-L(-K2,-Q2,RHO)+L(-K2,01 ,RH0)

,

where L(x,y,r) is the joint probability that X < x and Y < y given that (X,Y)

is the standardized bivariate normal cumulative distribution function

with correlation RHO between X and Y, as calculated by MDBN0R; and

PL = CL+G(K1)+G(K2)-G(Q1)-G(Q2)

,

where G(x) is the probability that X < x given that X is the standardized

univariate normal cumulative distribution function, as calculated

by MDNOR.

For an upper-tail test,

CL = G(-K1)-L(-K1 ,-01 ,RH0) and

PL - G(-Q1)-L(-K1 ,-Ql ,RH0).

For a lower-tail test,

CL = G(-K2)-L(-K2,-Q2,RH0) and

PL = G(-Q2)-L(-K2,-Q2,RH0).

A listing of BINORMAL. CL-PL is shown in figure A. 2.

57
Figure A. 2. Computer Code Listing of BI NORMAL. CL-PL

ACID*BIN0RMAL(1).CL-PL(45)

1 C PROGRAM: B I NORMAL. CL-PL

2 C

3 C DATE OF LAST REVISION: 12/22/83

4 C

5 C DOCUMENTATION: THIS PROGRAM CALCULATES CONSUMER LOSS (CL) AND

6 C PRODUCER LOSS (PL) FOR INPUTTED LEVELS OF ACCURACY OF AUTOMATIC

7 C TEST EQUIPMENT (ATE). THE LOSSES CAN BE CALCULATED FOR BOTH

8 C ONE-TAIL AND TWO-TAIL TEST SITUATIONS. IT IS ASSUMED THAT

9 C MEASURED PERFORMANCE FOLLOWS A BIVARIATE NORMAL DISTRIBUTION

10 C WHERE MEASURED=TRUE+ERROR IN MEASUREMENT, WHERE TRUE AND

11 C ERROR ARE NORMAL DISTRIBUTIONS. THE VARIABLES USED ARE:

12 C 13 C MEANE=ME=MEAN OF MEASUREMENT ERROR

14 C SDE=STE=STANDARD DEVIATION OF MEASUREMENT ERROR

15 C MEANU=MU=MEAN OF TRUE PERFORMANCE OF UNIT UNDER TEST

16 C SDU=STU=STAMDARD DEVIATION OF TRUE PERFORMANCE OF UUT

17 C SU=UPPER PERFORMANCE SPECIFICATION

18 C SL=LOWER PERFORMANCE SPECIFICATION

19 C BU=UPPER ATE TOLERANCE FACTOR

20 C (BU*SDE=UPPER TOLERANCE, WHERE BU>0 MEANS UPPER TEST SPECIFICATION

21 C LIES BELOW THE UPPER PERFORMANCE SPECIFICATION.)

22 C BL=LOWER ATE TOLERANCE FACTOR

23 C ( BL*SDE=LOWER TOLERANCE, WHERE BL>0 MEANS LOWER TEST SPECIFICATION

24 C LIES ABOVE THE LOWER PERFORMANCE SPECIFICATION.)

25 C ^STANDARDIZATION OF SU

26 C ^STANDARDIZATION OF SL

27 C Q1=STANDARDIZATI0N OF UPPER TEST SPECIFICATION

28 C Q2=STANDARDIZATI0N OF LOWER TEST SPECIFICATION

29 C RHO=CORRELATION OF MEASURED PERFORMANCE WITH TRUE PERFORMANCE

30 C P=BIVARIATE NORMAL PROBABILITY OR UNIVARIATE NORMAL PROBABILITY

31 C IER=ERROR FLAG, MEANING RH0>=1. IN MDBNOR.

32 C STR1 TO STR8 AND SI TO S8 ARE STRINGS FOR PRINTING.

33 C YESNO AND YN ARE USER'S ANSWERS TO QUESTIONS.

34 C 35 C ALGORITHMS USED ARE:

36 C 37 C RHO=SDU/SQRT((SDU*SDU)+(SDE*SDE))

38 C K1=(SU-MU)/SDU

39 C K2=(MU-SL)/SDU

40 C Q1=(SU-MU-(BU*SDE)-ME)/SQRT((SDU*SDU)+(SDE*SDE))

41 C Q2=(MU-SL-(BL*SDE)+ME)/SQRT((SDU*SDU)+(SDE*SDE))

42 C ALGORITHMS FOR CL AND PL DEPEND ON WHAT TYPE OF TEST IS USED.

43 C 44 REAL MEANE , SDE ,MEANU , SOU

45 CHARACTER*40 STR1,STR2,STR3,STR4,STR5,STR6,STR7,STR8

46 CHARACTERS YESNO

47 DATA STR5/' UPPER PERFORMANCE SPECIFICATION'/

48 DATA STR6/' LOWER PERFORMANCE SPECIFICATION'/

49 DATA STRl/'MEAN OF MEASUREMENT ERROR'/

50 DATA STR2/ ' STANDARD DEVIATION OF MEASUREMENT ERROR'/

51 DATA STR3/'MEAN OF TRUE PERFORMANCE'/

58
52 DATA STR4/ ' STANDARD DEVIATION OF TRUE PERFORMANCE'/

53 DATA STR7/' UPPER TOLERANCE FACTOR OF ATE'/

54 DATA STR8/' LOWER TOLERANCE FACTOR OF ATE*/

55 C 56 C INPUT COMMON VALUES.

57 C 58 CALL INPUT (MEANE ,STR1)

59 CALL INPUT (SDE,STR2)

60 CALL INPUT (MEANU, STR3)

61 CALL INPUT (SDU,STR4)

62 C 63 C DETERMINE WHAT TYPE OF TEST IS DONE AND CALL APPROPRIATE

64 C SUBROUTINE. UPPER-TAIL MEANS ONLY TRUE PERFORMANCE BELOW

65 C SOME VALUE X IS ACCEPTABLE . LOWER-TAIL MEANS ONLY PERFORMANCE

65 C ABOVE VALUE Y IS ACCEPTABLE. TWO-TAIL MEANS ONLY PERFORMANCE

67 C BETWEEN VALUES X AND Y IS ACCEPTABLE.

68 C 69 WRITE(6,2000)

70 2000 FORMATC IS THIS A TWO-TAIL TEST OF PERFORMANCE?( Y/N) '

)

71 READ(5,1000) YESNO

72 1000 FORMAT(Al)

73 IF (YESNO. EQ. 'Y' ) CALL CLPL2(MEANE ,SDE, MEANU, SDU,STR1,STR2,

74 + STR3,STR4,STR5,STR6,STR7,STR8)

75 WRITE(6,2010)

76 2010 FORMAT

(

1 IS THIS AN UPPER-TAIL TEST? (Y/N) 1 ) 77 READ(5,1000) YESNO

78 IF (YESNO. EQ.

1 Y1 ) CALL CLPLUP(MEANE ,SDE ,MEANU,SDU,STR1,STR2,

79 + STR3 STR4 STR5 STR7)

80 IF (YESNo'.EQ. 1 N' ) CALL CLPLOW( MEANE ,SDE , MEANU, SDU, STR1 ,STR2,

81 + STR3,STR4,STR6,STR8)

82 1 STOP

83 END

84 C 85 C 86 C 87 SUBROUTINE CLPL2(ME,STE,MU,STU,S1,S2,S3,S4,S5,S6,S7,S8)

88 EXTERNAL MDNOR,MDBNOR

89 C 90 C MDNOR & MDBNOR ARE IMSL ROUTINES. SEE IMSL MANUAL FOR INFO..

91 C 92 REAL SU,SL,ME,MU,STE,STU,BU,BL,RH0,K1,K2,K3,K4,P1,P2,P3,P4

93 REAL Ql, 02,03, Q4.CL, PL

94 INTEGER IER

95 CHARACTER*40 S1,S2,S3,S4,S5,S6,S7,S8

96 CHARACTER"*! YN 97 C 98 C INPUT UPPER AND LOWER LIMITS AND TOLERANCE FACTORS.

99 C 100 CALL INPUT(SU,S5)

101 CALL INPHT(SL,S6)

102 CALL INPUT(BU,S7)

103 CALL INPIJT(BL,S8)

104 C 105 C CALCULATE NEEDED VALUES. RH0/STU=1/SQRT( (STU**2)+(STE**2)

)

106 C 107 2 RHO=STU/SQRT((STU*STU)+(STE*STE))

108 K1=(SU-MU)/STU

109 K2=(MU-SL)/STU

110 K3=-K1

111 K4=-K2

59
112 Q1=(SU-MU-(3U*STE)-ME)*RH0/STU

113 Q2=(MU-SL-(BL*STE)+ME)*RH0/STU

114 Q3=-Q1

115 Q4=-Q2

116 CALL MDBN0R(K3,Q2,RH0,P1 5 IER)

117 CALL MDBN0R(K3,Q3,RH0,P2,IER)

118 CALL MDBN0R(K4,Q4,RH0,P3,IER)

119 CALL MDBN0R(K4,Q1,RH0,P4,IER)

120 CL=P1-P2-P3+P4

121 CALL MDN0R(K1,P1)

122 CALL MDN0R(K2,P2)

123 CALL MDN0R(Q1,P3)

124 CALL MDN0R(Q2,P4)

125 PL=CL+P1+P2-P3-P4

126 C 127 C WRITE RESULTS.

128 C 129 WRITE(6,2100)S1,ME,S2,STE,S3,MU,S4,STU,S5,SU,S6,SL,S7,BU,

130 + S8,BL,CL,PL

131 2100 FORMAT

(

1 FOR: 1 ,/,8( 1X,A40, ' =

1 ,G15.8,/)/,5X, 'THEN \

132 +

1 CONSUMER LOSS = \F10,6,/, 5X,

' AMD PRODUCER LOSS'

133 + ' = ',F10.6)

134 C 135 C OPPORTUNITY FOR SENSITIVITY ANALYSIS.

136 C 137 WRITE(6,2U0)

138 2110 FORMAT

(

1 WOULO YOU LIKE TO 00 A SENSITIVITY ANALYSIS?( Y/N) 1 ) 139 READ(5,1100)YN

140 1100 FORMAT(Al)

141 IF (YN .NE.'Y') STOP

142 WRITE(6,2120) S1,S2,S7,S8

143 2120 FORMAT

(

1 ENTER NEW VALUES FOR ' ,A40, ',

' ,/,lX,A40, ' ,\A40,

144 + AND' ,/,lX,A40,\ ONE VALUE PER PROMPT.')

145 READ *, ME,STE,BU,BL

146 GO TO 2 147 END

148 C 149 C 150 C 151 SUBROUTINE CLPLUP (ME,STE,MU,STU,S1,S2,S3,S4,S5,S7)

152 EXTERNAL MDNOR.MDB NOR

153 REAL ME , STE ,MU , STU ,BU , SU ,RHO , Kl , K3 , PI , P2 , Ql , 03 , CL , PL

154 INTEGER IER

155 CHARACTER*40 S1,S2,S3,S4,S5,S7

156 CHARACTER*1 YN

157 C 158 C INPUT UPPER LIMIT AND TOLERANCE FACTOR.

159 C 160 CALL INPUT (SU.S5)

161 CALL INPUT(BU,S7)

162 C 163 C CALCULATE NEEDEO VALUES.

164 C 165 3 RHO=STU/SQRT((STU*STU)+(STE*STE))

166 K1=(SU-M!I)/STU

167 K3=-K1

168 Ql= ( SU-MU- ( BU*STE ) -ME ) *RHO/STU

169 Q3=-Q1

170 CALL MDBM0R(K3,Q3,RH0,P1,IER)

171 CALL MDN0R(K3,P2)

60
CL=P2-P1

CALL MDN0R(Q3,P2)

174 PL=P2-P1

175 C 176 C WRITE RESULTS.

177 C 178 WRITE(6,2200) S1,ME,S2,STE,S3,MU,S4,STU,S5, SU,S7,BU,CL,PL

179 2200 FORMATC FOR: 1 ,/,6(lX,A40, 1 = ' ,G15.8,/)/,5X, 'THEN

180 + 'CONSUMER LOSS= ' ,F10. 6,/, 5X, 'AND PRODUCER LOSS=

181 + F10.6)

182 C 183 C OPPORTUNITY FOR SENSITIVITY ANALYSIS.

184 C 185 WRITE(6,2210)

186 2210 FORMATC WOULD YOU LIKE TO 00 A SENSITIVITY ANALYSIS?( Y/N) '

)

187 READ(5,1200)YN

188 1200 FORMAT(Al)

189 IF(YN.NE. ' Y' ) STOP

190 WRITE(6,2220) S1,S2,S7

191 2220 FORMATC ENTER NEW VALUES FOR ' ,A40, ',',/, IX, A40, ', AMD ', 192 + A40,',',/,' ONE VALUE PER PROMPT.')

193 READ *, ME,STE,BU

194 GO TO 3 195 END

196 C 197 C 198 C 199 SUBROUTINE CLPL0W(ME,STE,MU,STU,S1 ,S2,S3,S4,S6,S8)

200 EXTERNAL MDNOR.MDBNOR

201 REAL ME,STE,MU,STU,BL,SL,RH0,K2,K4,Q2,Q4,P1,P2,CL,PL

202 INTEGER IER

203 CHARACTERMO S1,S2,S3,S4,S6,S8

204 CHARACTERS YN

205 C 206 C INPUT LOWER LIMIT AND TOLERANCE FACTOR.

207 C 208 CALL INPUT(SL,S6)

209 CALL INPUT(BL,S8)

210 C 211 C CALCULATE NEEDED VALUES.

212 C 213 4 RHO=STU/SQRT((STU*STU)+(STE*STE))

214 K2=(MU-SL)/STU

215 K4=-K2

216 Q2=(MU-SL-(BL*STE)+ME)*RH0/STU

217 Q4=-Q2

218 CALL MDBN0R(K4,Q4,RH0,P1,IER)

219 CALL MDN0R(K4,P2)

220 CL=P2-P1

221 CALL MDN0R(Q4,P2)

222 PL=P2-P1

223 C 224 C WRITE RESULTS.

225 C 226 WRITE(6,2350) S1,ME,S2,STE,S3,MU,S4,STU,S6,SL,S8,BL,CL,PL

227 2350 FORMATC FOR: ' ,/,6(lX,A40, ' = ' ,G15.8,/)/,5X, 'THEN ', 228 + 'CONSUMER LOSS= ' ,F10. 6,/, 5X, 'AND PRODUCER LOSS = 229 + F10.6)

230 C 231 C OPPORTUNITY FOR SENSITIVITY ANALYSIS.

61
232 C 233 WRITE(5,2310)

234 2310 FORMAT

(

1 WOULD YOU LIKE TO 00 A SENSITIVITY ANALYSIS?( Y/N) 1 ) 235 REA0(5,1350) YN 236 1350 FORMAT(Al)

237 IF (YN . NE. 'Y') STOP

238 WRITE(6,2320) S1,S2 S SR

239 2320 FORMATC ENTER NEW VALUES FOR ' ,A40, ',',/, IX, A40, ' , AND ', 240 + A40, ',',/,' ONE VALUE PER PROMPT. '

)

241 READ *, ME.STE.8L

242 GO TO 4

243 END

244 C 245 C 246 C 247 SUBROUTINE INPUT (VAR, STRING)

248 REAL VAR

249 CHARACfER*40 STRING

250 WRITE(6,4000) STRING

251 4000 FORMATC ENTER 1 ,A40)

252 READ *,VAR

253 END

END PRT

62
Appendix B. Plots of Consumer's Loss and Producer's Loss as Functions of

the Mean and Standard Deviation of Measurement Error

B.l Introduction

This appendix presents plots illustrating the effects of changes in the

mean and standard deviation of the measurement error of test equipment on

Consumer's Loss (CL) and Producer's Loss (PL) when the distribution of UUT

performance and the performance specifications are both held constant. These

effects are shown for a range of alternative values of test specification

tolerances. Examples are given of how to read and interpret the plots. These

plots have the same general applications as the computer program which

generated them, BINORMAL.CL-PL; however, they are somewhat less flexible than

the program because the range of situations they cover is limited.

Consequently, the graphs are most useful when approximations of CL and PL are

adequate or when access to a computer that has BINORMAL.CL-PL installed is

unavailable or limited. The algorithms used tc calculate CL and PL are not

discussed in this appendix, but are given in appendix A and explained in more

depth in section 2. The notation used in this appendix is the same as used in

the main body of the report and as defined in table 2.1.

B.2 Scope

Steps were taken to ensure that the values used to generate CL and PL

were reasonable and representative of values found in actual test equipment

applications; however, because of space limitations not all conceivable

applications are represented. For instance, the graphs show CL and PL for

63
two-tail tests, but many applications call for one-tail tests. Similarly, all

systematic measurement errors and tolerances used in these plots are

non-negative, although these parameters can certainly assume negative values

in actual test situations.

There are two restrictions common to all the figures in this appendix.

The first is that the underlying UUT performance distribution is standard

normal (i.e., u u 0, a u

= 1). The second restriction is that the test is

two-tailed with performance specifications and test specifications symmetric

around y u and performance specifications set at Sy = -S^ = 2o u

. It is

possible to make adjustments to any data set for which the UUT performance can

be treated as normal in order to satisfy the first restriction. The second

restriction, however, must be strictly satisfied. To illustrate how these

restrictions affect the applicability of the figures, suppose the following

parameters describe a test situation: m u

= 10, o u

= 2, y e = 4, o e = 1 , S\j =

14, SL

= 6, and TA = 4. The first step in adjusting the data to satisfy the

first restriction is to standardize the UUT performance specifications and the

mean UUT performance. Denoting the resulting adjusted parameters with an

asterisk, we have u u

* = (10-10)/2 = 0, Sv* = (14-10)/2 = 2, and SL*=(6-10)/2 =

-2, Because Sy* ""S^* 2 in this case and because the test specifications

are symmetric at TA5S 4, the second restriction is satisfied and it is possible

to use the plots to analyze the situation. If this were not the case,

BINORMAL.CL-PL would have to be used for the analysis. The next step in

adjusting the parameters is to divide all remaining values by 0 U

to convert

them to the new scale. Thus, o u

* = 2/2 = 1, a e * = 1/2 = .5, u e * « 4/2 = 2,

and TA* 4/2 2. These values should be used to identify which plots are

appropriate for analyzing the situation.

64
B.3 Interpretation of the Plots

There are three figures showing the effect of systematic measurement

error on CL (figures B.l, B.2, and B.3), and three showing its effect on PL

(figures B.4, B.5, and B.6). The assumed standard deviation of the

measurement error, a

£ , is 0.5 in figures B.l and B.4, 0.25 in figures B.2 and

B.5, and 0.1 in figures B.3 and B.6. Other than this variation, all

assumptions are the same for these six figures. There are also three figures

showing the effect of the standard deviation of the measurement error on CL

(figures B.7, B.8, and B.9) and three showing its effect on PL (figures B.10,

B.ll, and B.12). The assumed systematic measurement error is 0.5 in figures

B.7 and B.10, 0.25 in figures B.8 and B.ll, and 0.0 in figures B.9 and B.12.

Other than this variation, all assumptions for these six figures are the same.

For all twelve figures, several plots are given, each plot having a

different assumed level of TA. This arrangement permits one to observe what

effect TA has on CL and PL for the different situations. As was discussed in

section 3, the value of TA has significant effects, but in opposite

directions, on CL and PL. As can be observed in all the figures, a decrease

in TA increases CL but decreases PL.

Note that the figures showing the effects of a e (i.e., figures B.7

through B.12) do not all have the same number of plots. As stated in

expression (3.8) of section 3, when TA is smaller than the systematic

measurement error, a decrease in o e does not always cause a decrease in CL.

The plots included in the figures are restricted to those that represent

65
situations where it is clearly advantageous to decrease y e and a e .l

Consequently, no plots were included in figures B.7 through B.9 to show the

effect of a £ on CL for cases in which TA is smaller than y e . Figures B.10

through B.12 show the effect of o £ on PL for the same TA values used in

figures B.7 through B.9.

For a discussion of this phenomenon and more information about the effect of

y e and ae on CL and PL in situations not covered by the figures, see section

3.

66
o.oae o.25ae o.50ae o.75ae i.ooae i.25ae i.50ae U5ae 2.ooae SYSTEMATIC MEASUREMENT ERROR (BIAS) , Me

Figure B.l. Effect of Systematic Measurement Error on Consumer's Loss

(o e =0.5, y u-0, o u=l, Su=2, SL=-2)

67
m i i i i i i i i i i i i [

.0(j6

0.25

a

6 o.50ae

0.75

a

e

1.00

a

6 i.25ae i.50ae i.75ae

2.1 SYSTEMATIC MEASUREMENT ERROR (BIAS),/x 6

Figure B.2. Effect of Systematic Measurement Error on Consumer's Loss

(o £ =0.25, M u=0, a u=l, Su=2, SL—2) 67

68
o.oae

0.25

a

e o.50ae

0.75

a

e i.oo<7e i.2sae i.5oae 1.75

a

f 2.ooaf

SYSTEMATIC MEASUREMENT ERROR (BIAS), Me

Figure B.3. Effect of Systematic Measurement Error on Consumer's Loss

(o e -0.1, y u=0, o u=l, Su=2, SL=-2)

69
0.0<7e O.250e 0.50Oe 0.75O"e 1.00(7e 1.25ae ).50ae 175ae 2.ooae SYSTEMATIC MEASUREMENT ERROR (BIAS) , fx €

Figure B.4. Effect of Systematic Measurement Error on Producer's Loss

(0 e =O.5, u u=0, o u«l, Su=2, SL

=-2)

70
.19 -

.18 -

.17 -

.16 -

I I

I I I I I I I 0.0(7e

0.25(Je 0.50Oe 0.75(Je 1.00(7€ 1.25<7€ 150(Je 1.75(7e 2.00(7e SYSTEMATIC MEASUREMENT ERROR (BIAS),^€

Figure B.5. Effect of Systematic Measurement Error on Producer's Loss

(o

e =0.25, u u=0, o u=l, Su=2, SL—2)

71
0.0Oe 0.25a£ 0.50O~e 0.75 ae 1.00<7e 1.25ae 1.50ae

1.75 ae2.00(7e SYSTEMATIC MEASUREMENT ERROR (BIAS), ju e

Figure B.6. Effect of Systematic Measurement Error on Producer's Loss

(o

e =G.l, y u=0, o u=l, S„=2, SL=-2)

7 2
00

CO

CDE

COc

cO

co

J-l oM

w

c

cve

0»

J-l /~N 3 <N|

to I CO II Ol _3 X co O CM

II O C/5 •H

•U

CO i—i •H II > 3

o

XI o

U || CO 3

73 3. 3

CO •> w . o <4-i H O CO a

u —

'

o

0) CO

u-i co CM O

PQ

0)3

ii§§i!§§§§§§

SS01 dSNflSNOO

73
i i i r~i i I

J

I I I I I I

CO

5

g

|

cuB

3

COc

ou

co

ouW on u

C

01B

0)

(-1 3

CO

CO

cu X

CM

I

II|J

C/5

>W CN O ||!=> C C/5 O

•H *

ft) II > O

CU o « o TJ II l-i 3

CO 3. T3C -

co m *J CN

CO • o >*-l II O CO 3.

AJ wCJ

CU CO

CO

«+-( o

00

CO

CUu

3

g § i i § 1 i i § i 1 s

SS01 dBNDSNOO

74
_ «? 00^ 3 3 § C oowcB

0)

a|O

»—GO 3W --v & 1 E ll oC csi 0 II •H D

co> -<

01 II O 3o

T3

(-1 -

CO

O

T5 II C 3

co

a

•uCO « o U-l II o

4-1 o u3

bO

lO ^ CO 5 5 5

ssoi ounsNoa

75
ssoi MonaoMd
i i r t—i—i—i—i r

in

S

2 &

CM

J I

I I I I I I I I I I I I 9 irt a in

5 2 1 £

77
o D c e ° £ u

"J £u — w

CO ,9c

<u6

0)

Ij3

U3 o<

<U CM

a> i C CM O II •H 3 4-> C/0 •H •> > -H

a> li Q 3D

-Ou " « o

-O II C 3

(0 3. u

a> « o U-l || u

0) CO

«-J to

<4-l

O 3

78
B.4 Illustrative Applications of the Figures

In this subsection several examples are presented to illustrate how one

can apply the figures to actual test situations. Suppose ATE is used in an

electronics manufacturing plant to test a resistor that is a critical part of

a calculator produced there. The resistor is known to have a normal

performance distribution with mean 100.0 ohms and standard deviation 0.4 ohm.

Historical records show that if the resistor performance is not within two

standard deviations of the mean, then the calculator is very likely to fail

during the one-year warranty period. On the basis of these records, the

limits of acceptable performance for this resistor have been set at y u±2o u

ohms. The plant quality assurance engineers also know that, for this range of

resistance values and for the current system and frequency of calibration, the

ATE readings are normally distributed with a typical systematic measurement

error equal to 0.1 ohm and standard deviation of the measurement error equal

to 0.1 ohm. In the past, the engineers have always set the test tolerance at

0.1 ohm to adjust for the measurement errors, but the resulting PL has been

determined to be too high (i.e., the yield is too low because of unnecessary

rejections). The plant managers want the engineers to find out what can be

done to decrease PL and how CL will be affected as a result.

Case I ; The first step in applying the figures to analyze this problem is to

make sure the restrictions discussed in section B.2 are satisfied. This means

adjusting the parameters as described in section B.2 and observing whether Sy*

= -Sl* = 2. The values given for the parameters were u u

=» 100.0, a u 0.4, Su

= y u+2au

= 100.8, SL- U u

-2a u

* 99.2, u e -0.1, o e =0.1, and TA - 0.1.

79
Standardizing u u , Sjj, and Sl, one obtains y u

* = (100.0-100.0)/0.4 = 0, Sy* =

(100.8-100.0)/0.4 = 2.0, and Sl* = (99.2-100.0)/0.4 = -2.0. Because Su* =

-Sl* = 2, the figures can be used to analyze this problem. Dividing the rest

of the values by o u

to convert to the new scale yields o u

* = 0.4/0.4 = 1.0,

u e * - 0.1/0.4 - 0.25, o e * = 0.1/0.4 = 0.25, and TA* = 0.1/0.4 = 0.25. Since

both u e * and a e * equal 0.25, the relevant plots are in figures B.2, B.5, B.8,

and B.ll. Reading from figures B.2 and B.5 we observe CL = 0.0041 and PL =

0.058 when y e * - 0.25 (=1.0a £ *) and TA* = 0.25 (=1.0a e *). Figures B.8 and

B.ll give the same values, as expected. Table B.l summarizes the parameter

values used and the resulting values of CL and PL for this base case and all

remaining case illustrations in this section.

Case II : There are three variables that could possibly be changed: y e , oe ,

and TA. Suppose improved or more frequent calibration of the ATE completely

eliminated the systematic measurement error (i.e., M e *=0). From figures B.2

and B.5 it is seen that if TA is held constant, then CL will fall to 0.0018

and PL to 0.046. Elimination of the systematic error causes a 0.0023

reduction in CL and a 0.012 reduction in PL. Thus, out of every lot of 10,000

resistors tested, 23 fewer bad UUTs will be incorrectly accepted and 120 fewer

good UUTs will be unnecessarily rejected.

Case III ; Now suppose that instead of improving ATE calibration, the company

buys more precise ATE, which reduces o £ to 0.04 ohm. Then o e * 0.1 (=

0.1ou

*) and it is seen from figures B.8 and B.ll that CL falls to 0.0019 and

PL falls to 0.0475. The drop in o e causes a 0.0022 reduction in CL and a

0.0105 reduction in PL. Thus, out of every lot of 10,000 resistors tested, 22

80
Table B.l Parameter Values and Resulting Values of CL and PL for Case Examples.

Parameter Parameter 'Values 3 Resulting Values Change from Base

Case Changed o e * TA* CL PL CL PL

I Base 0.25 0.25 0.250 0.0041 0.0580

II 0.00 0.25 0.250 0.0018 0.0460 -0.0023 -0.0120

III 0.25 0.10 0.250 0.0019 0.0475 -0.0022 -0.0105

IV TA 0.25 0.25 0.125 0.0070 0.0385 +0.0029 -0.0195

V He »?£ »TA 0.00 0.10 0.050 0.0019 0.0088 -0.0022 -0.0492

VI y e ,TA 0.00 0.25 0.125 0.0041 0.0276 0.0000 -0.0304

aSo that the figures can be applied to these case examples, each parameter value is given

in units of the standard deviation of the UUT distribution. That is, as indicated by

the asterisk, each has been divided by o u

= 0.4 ohm. Note further that to interpret the

figures correctly the values of y e * and TA* must both be read in units of <J e *.

81
fewer bad UUTs will now be incorrectly accepted and 105 fewer good UUTs will

now be unnecessarily rejected. Clearly, in this case reducing u e is more

advantageous than reducing o £

. Of course, tbe decision between improving

calibration procedures to remove systematic error or purchasing more precise

ATE would also have to take account of the costs of implementing each policy.

Case IV : Suppose no changes are made in the measurement errors, but the test

specifications are loosened by decreasing TA to 0.05 ohm (TA* = 0.125o u

* =

0.5oe *). From figures B.2 and B.5, it is seen that this reduces PL to 0.0385

but increases CL to 0.0070. The decrease in TA causes a 0.0195 reduction in

PL and a 0.0029 increase in CL. The benefits of the large decrease in PL

would have to be compared to the disadvantages of the increase in CL in order

for the company to determine whether these changes in CL and PL, considered

together, are indeed advantageous.

Case V : Changes in u e , a e , and TA can also occur in combination. For

instance, suppose the bias is eliminated, o £ is reduced to 0.04 ohm (o

e * =

0.1a u*), and TA is reduced to 0.02 ohm (TA* = 0.05a u

* = 0.5a e *). Then it can

be seen from the TA = 0.5c e plots of figures B.3 and B.6 (where a e * = 0.1)

that CL falls to 0.0019 and PL falls to 0.0088. The combined changes in y e ,

o e , and TA cause a 0.0022 decrease in CL and a 0.0492 decrease in PL from the

original values. Thus, out of every lot of 10,000 resistors tested, 22 fewer

bad UUTs will be incorrectly accepted and 492 fewer good UUTs will be

unnecessarily rejected. The only combination of changes that cannot be

analyzed with these figures is a decrease in a

£ combined with a decrease in TA

to a value smaller than the assumed level of u e

. This combination represents

the situation discussed in section B.3. It should also be noted that not all

possible values for the three variables are represented, so interpolation must

sometimes be used to estimate the correct result.

82
Case VI ; Another application of these figures involves converting the effect

of changes in u e or a e into a single type of loss, holding the other loss

value constant. This conversion would be useful when, for example, a purchase

contract includes a specification limiting the allowable size of CL for a

product, and the supplier wishes to hold CL at that limit while reducing PL as

much as possible. It is also advantageous to convert a change in CL into an

equivalent change in PL whenever it is difficult or impossible to establish a

dollar value for a one-unit reduction in CL (e.g., in the case of health and

safety hazards). To illustrate this application, consider again the resistor

example. When the systematic measurement error was eliminated, it was

determined that CL fell by 0.0023 and PL by 0.012. But suppose the company

management wished to keep CL at or near the original value of 0.0041. From

figure B.2 it is seen that this can be accomplished by lowering TA to 0.05 ohm

(TA* = 0.125 = 0.5o e *) when the systematic error is eliminated (i.e., y e 0).

It is seen from figure B.5 that PL falls to 0.0276 when the systematic error

is eliminated and TA is reduced just enough to keep CL at its original level.

Of this total reduction in PL, 0.012 is directly due to the elimination of the

systematic error, and the additional 0.0184 can be attributed to the effect of

reducing TA enough to leave CL unchanged. In this way, the 0.0023 reduction

in CL that is made possible by eliminating the systematic error has been

converted to an equivalent 0.0184 reduction in PL.

One can apply the same method to make CL as low as possible while holding

PL constant. Quality control policy might call for this type of conversion if

the primary objective were to reduce reported failure rates. The extra

83
reduction in CL would be accomplished by increasing TA. In a similar manner,

it is possible to achieve any desired ratio of CL to PL by changing TA. This

possibility exists even when ue and o e both remain unchanged. The figures

presented here may require interpolation of TA in order to estimate the

effects on CL and PL of changing TA. More accurate results can be obtained by

using the program BINORMAL . CL-PL , described in appendix A, and using the

sensitivity analysis capability to vary TA until acceptably accurate results

are obtained. The ability to vary TA to achieve a desired ratio of CL and PL

is useful in establishing the economically optimal trade-off between CL and

PL, provided data are available on the relative cost of each unit of CL and

PL.

•frU.S. GOVERNMENT PRINTING OFFICE: 1 9 8<f 420 997 99

84
NBS-114A (REV. 2-8C) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NBS SP 673

2. Performing Organ. Report No. 3. Publication Date

April 1984

4. TITLE AND SUBTITLE

Economic Model of Calibration Improvements for Automatic Test Equipment

5. AUTHOR(S)

Stephen F. Weber and Anne P. Hillstrom

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions) 7. Contract/Grant No. NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered WASHINGTON, D.C. 20234 Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City. State, ZIP)

Center for Electronics and Electrical Engineering

Center for Applied Mathematics

National Bureau of Standards

Washington, DC 20234

10. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 84-601027

| | Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most si gnificant information. If document includes a significant

bibliography or literature survey, mention it here) This paper presents a model for measuring the benefits of improved accuracy

in test equipment. The model permits calculation of the probability of accepting

a unit under test (UUT) that is out of specifications (i.e., Consumer's Loss) and

the probability of rejecting a UUT that is within specifications (i.e., Producer's

Loss) for alternative levels of test equipment accuracy. Accuracy is defined in

terms of both the systematic and the random measurement error of the equipment.

Other parameters that are taken into account by the model are the mean and variance of the UUT attribute of interest, the performance specifications of the UUT,

and the test specifications which define acceptance and rejection in terms of

test measurement results. A discussion of the economic consequences of Consumer's

Loss and Producer's Loss is included. The model may be used to optimize both

procurement policy for new test equipment as well as maintenance and calibration

policy for existing test equipment.

12. KEY WORDS (Six to twelve entries; alfihobetico/ order; capitalize only proper names; and separate key words by semicolon s) accuracy; automatic test equipment; calibration; Consumer's Loss; economics;

error analysis; measurement; Producer's Loss; quality assurance; random error; systematic error; testing.

13. AVAILABILITY

Unl imited

| | For Official Distribution. Do Not Release to NTIS

Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

J] Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

94

15. Price

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