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ABSTRACT

MANUFACTURING PROGRESS FUNCTIONS
BY

Dayr Ramos Americo dos Reis

The Manufacturing Progress Function (MPF) may be gener
ally defined as the relationship in which the labor input
per unit used in the manufacture of a product tends to de-
cline by a constant percentage as the cumulative quantity
produced is doubled. Where this relationship is present, it

may be represented by a straight line in a double logarithmic

scale.

The prime objective of this study is to contribute a
general symbolic-analytic model of the manufacturing progress

phenomenon .

Once the general model is established, an equally
important objective is to respond to the need for a coherent
systematic approach to be used in predicting the develop-
ments of the adaptation process in industrial concerns of

almost any kind .

The work is broadly divided into four major parts.
Chapter II contains a comprehensive review of the historical
development of the Manufacturing Progress Function and a

Summary of the more important contributions to the progress

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curve literature that are relevant to this dissertation.

The field of progress functions lacks notation unifor
mity, precise definition of the variables and functional
relationships involved, and formal mathematical proofs of
several assumed results. A coherent mathematical exposition
can be the basis for the derivation of new important results.
Towards this end an original theoretical systematization is
offered in Chapter III. Chapter IV represents a continuation
of the mathematical exposition initiated in Chapter III. Two
related topics of practical relevance are approached: the
integration of progress functions and the debatable problem

of their aggregation. Original approximations are proposed

for both problems.

A general symbolic-analytic model of the manufacturing
progress phenomenon is offered in Chapter V. In Chapter VI
the manufacturing progress model presented in Chapter V is
tested with real data from nine manufacturers representing
five different industries. A hundred and fifty-nine separate
cases of product and process startups that occurred in four
different countries and nine distinct plants are analyzed.

In addition, aggregate data was obtained for whole industries

in one country, yielding nine more startups.

The descriptive efficiency of the proposed model is
generally supported by the results of regression analysis of

the startups and startup parameters obtained from the partig

iPating industrial firms.

 

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The findings of this research and previous findings by
two other authors constitute adequate evidence to suggest
that the model can be developed into an effective means of
predicting the mathematical slope (parameter b) of a new
startup. The author believes that the parameter model
amproach presented in Chapter V has proved to be superior to
(fiber existing methods for estimating the parameters of a

rmw startup.

The dissertation is concluded (Chapter VII) with a
(fiscussion of the industrial implications of the findings
reported in Chapter VI. The importance of recognizing and
{medicting the manufacturing progress phenomenon is related
u>several decision-making functions that are encountered in
mIindustrial setting as well as in economic planning at the
Imtional level. An overall design of a computerized

bhnufacturing Progress Function (MPF) System is suggested.

MANUFACTURING PROGRESS FUNCTIONS

By

Dayr Ramos Americo dos Reis

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Management

1977

© Copyright by
DAYR RAMOS AMERICO DOS REIS

1977

TO MY FATHER

JULIO AMERICO DOS REIS, Aer.Eng.

A PIONEER OF THE BRAZILIAN
AIRFRAME INDUSTRY

(1910-1977)

IN MEMORIAM

ii

ACKNOWLEDGMENTS

The author wishes to express appreciation for the guid-
ance and encouragement provided by Dr. Richard F. Gonzalez,
as advisor throughout the thesis. The time and assistance of
Dr. Stanley E. Bryan, Dr. Dole A. Anderson and Dr. Phillip L.

Carter have also been of great value.

The author is also indebted to a large number of indi-
viduals in industry for their helpful cooperation. These
individuals, who must unfortunately remain anonymous,provided

the essential data for the dissertation.

I am grateful to Mrs. Doris Singer who so patiently

assisted in revising the manuscript.

Without my wife, Maria de Fatima, who assisted me with
editing and typing and was a source of constant inspiration,

this work might never have been completed.

DAYR R.A. DOS REIS
February, 1977

iii

iv

TABLE OF CONTENTS

page
LIST OF TABLES ........................................ xi
LIST OF FIGURES ....................................... xvi
CHAPTER I
INTRODUCTION: THE PERCEPTION OF THE PROBLEMATIC
SITUATION ............................................. l
Adaptation to Innovation and Transformation ........... 2
‘Manufacturing Progress Function Defined ............... 5
Scope ................................................. 6
GENERAL AND SPECIFIC OBJECTIVES OF THE STUDY .......... 8
A Note on Models ...................................... 8
General Statement of Objectives ....................... 9
CHAPTER II
THE EXISTING BODY OF KNOWLEDGE ........................ 11
THE WRIGHT MODEL ...................................... 11
THE FORTIES ........................................... 14
The Crawford Model .................................... 15
Bureau of Labor Statistics Studies .................... 17
Airframe Companies Publications ....................... 19
Progress Curves in Great Britain and France ........... 28

Other Contributions ................................... 31

TABLE OF CONTENTS (cont'd)

page
THE FIFTIES ........................................... 33

Extensions of the Concept Outside the Airframe

Industry .............................................. 34
Airframe Companies Publications ....................... 46
Studies Financed by the Air Force ..................... 47
Other Contributions ................................... 51
THE SIXTIES AND SEVENTIES ............................. 52
Extension to Machine-Intensive Manufacture ............ 52
Contributions by IBM Personnel ........................ 55
Other Contributions ................................... 59
SUMMARY ............................................... 65
Review of Main Hypotheses ............................. 67
CHAPTER III

MANUFACTURING PROGRESS FUNCTIONS: A MATHEMATICAL

EXPOSITION ............................................ 69
FOUR TYPES OF PROGRESS FUNCTIONS ...................... 70
Type 1: The Unit Progress Function .................... 70
Type 2: The Cumulative Average Progress Function ...... 73
Type 3: The Cumulative Total Progress Function ........ 74
Type 4: The Lot Average Progress Function ............. 75
FUNDAMENTAL PROBLEMS .................................. 78
Statement of Fundamental Problems ..................... 78
Solution of Problem I ................................. 79
Solution of Problem II ................................ 82

V

TABLE OF CONTENTS (cont'd)

Solution of Problem III ............................... pig?
Solution of Problem IV ................................ 86
PARAMETER CALCULATION ................................. 90
Statement of Problem Al ............................... 91
Solution of Problem A1 ................................ 91
Statement of Problem A2 ............................... 92
Solution of Problem A2 ................................ 92
Statement of Problem A3 ............................... 92
Solution of Problem A3 ................................ 93
Statement of Problem A4 ............................... 93
Solution of Problem A4 ................................ 93
Statement of Problem B1 ............................... 94
Solution of Problem B1 ................................ 94
Statement of Problem B2 ............................... 95
Solution of Problem BZ ................................ 96
Statement of Problem B3 ............................... 97
Solution of Problem BB ................................ 97
Statement of Problem B4 ............................... 100
Solution of Problem B4 ................................ 100
SUMMARY ............................................... 103
CHAPTER IV

QUADRATURE AND SUMMATION OF PROGRESS FUNCTIONS ........ 104
QUADRATURE OF PROGRESS FUNCTIONS ...................... 104
Cumulative Total Hours Calculation .................... 104

vi

TABLE OF CONTENTS (cont'd)

Manufacturing Progress Function Hours Calculation ..... USE?
Accuracy of the Proposed Formulas ..................... 113
THE AGGREGATION PROBLEM ............................... 113
Statement of the Problem .............................. 116
Solution of the Aggregation Problem ................... 117
Accuracy of the Proposed Aggregation Method ........... 126
Accuracy in the Quadrature of Approximate Aggregate
Functions ............................................. 133
CHAPTER V

A MODEL OF THE PROGRESS PHENOMENON IN MANUFACTURING
INDUSTRIES ............................................ 136
A CONCEPT OF MANUFACTURING PROGRESS ................... 136
Factors in Manufacturing Progres ...................... 137
Hypotheses About the Progress Phenomenon -

A Reexamination ....................................... 141
MANUFACTURING PROGRESS - - A MODEL .................... 145
The Traditional Model: Some Qualifications ............ 146
Estimating the Ultimate Point (xu, yu) ................ 147
Estimating Parameter b ............................... 149
A Symbolic-Analytic Model of the Manufacturing Progress
Phenomenon ............................................. 149
GENERAL METHODOLOGY OF THE EMPIRICAL RESEARCH .......... 151
Criteria in Sample Selection .................. ' ......... 151
The Sample ............................................. 152

vii

TABLE OF CONTENTS (cont'd)

Measuring the Progress Phenomenon .....................

CHAPTER VI

EMPIRICAL FINDINGS ....................................
STARTUP ANALYSIS IN THE MANUFACTURING OF ELECTRONIC
DATA PROCESSING SYSTEMS AND COMPONENTS ................
Characteristics of the Products and Manufacturing
Processes Studied .....................................
Startup Measurement and Data ..........................
Startup Analysis ......................................
Parameter Analysis ....................................
STARTUP ANALYSIS IN THE MANUFACTURING OF BINDERY
'MACHINES AND PRINTING PRESSES .........................
Characteristics of the Products and Manufacturing
Processes Studied .....................................
Startup Measurement and Data ..........................
Startup Analysis ......................................
Parameter Analysis ....................................
STARTUP ANALYSIS IN SHIPBUILDING ......................
Characteristics of the Product and Manufacturing
Processes Studied .....................................
Startup Measurement and Data ..........................
Startup Analysis ......................................
Parameter Analysis ....................................

viii

158

158

158
163
165
167

178

178
178
180
180
188

188
189
191
192

TABLE OF CONTENTS (Cont'd)

STARTUP ANALYSIS IN THE MANUFACTURING OF OIL page
DRILLING AND PRODUCTION EQUIPMENT ..................... 194
Characteristics of the Products and Manufacturing

Processes Studied ..................................... 194
Startup Measurement and Data .......................... 195
Startup Analysis ...................................... 196
Parameter Analysis .................................... 196
STARTUP ANALYSIS IN THE AIRFRAME INDUSTRY ............. 198
Characteristics of the Products and Manufacturing

Processes Studied ..................................... 198
Startup Measurement and Data .......................... 199
Startup Analysis ...................................... 201
Parameter Analysis .................................... 201
STARTUP ANALYSIS IN THE MECHANICAL AND ELECTRICAL

INDUSTRY .............................................. 204
The Product Groups Studied ............................ 204
Startup Measurement and Data .......................... 206
Startup Analysis ...................................... 207
Parameter Analysis .................................... 207
SUMMARY ............................................... 209
CHAPTER VII

IMPLICATIONS OF THE FINDINGS .......................... 212
IMPLICATIONS AT FIRM LEVEL ............................ 212
Problem Statement ..................................... 214

ix

 

TABLE OF CONTENTS (cont'd)

page
Solution .............................................. 216
MULTINATIONALS AND THE COST OF LEARNING ............... 237
IMPLICATIONS AT NATIONAL LEVEL ........................ 239
Determining the Magnitude of Contracts ................ 240
A COMPUTERIZEDMPF SYSTEM ............................. 242
CHAPTER VIII
SUMMARY AND CONCLUSIONS ............................... 248
APPENDICES
APPENDIX A: MATHEMATICAL PROOFS ....................... 256
APPENDIX B: EQUATION OF THE ASYMPTOTE TO THE CURVE
REPRESENTING THE LOGARITHM OF THE UNIT PROGRESS
FUNCTION .............................................. 258
APPENDIX C: PARAMETER FORMULAS DERIVATION ............. 262
APPENDIX D: STABILITY OF COEFFICIENTS :_n_ AND E
OF THE PARAMETER MODEL ................................ 265
APPENDIX E: A CODE FOR CALCULATING a x29 x’b WITH
AN HP-25 SCIENTIFIC PROGRAMMABLE POCKE]T CALCULATOR. . .. 292
APPENDIX F: DATA TABLES (CHAPTER VI) .................. 291:
APPENDIX G: REGRESSION ANALYSIS OF LOG-
TRANSFORMED DATA - - A FORTRAN IV PROGRAM ............. 317
BIBLIOGRAPHICAL NOTES ................................. 319
BIBLIOGRAPHY .......................................... 339

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LIST OF TABLES

Table page
2.1 PARAMETERS OF THE PROGRES CURVE ................. 22

2.2. INFLUENCE OF THE NEWNESS OF MODELS AND

FACILITIES ON THE RELATIVE POSITION OF

PROGRESS CURVES ................................. 25
4.1 EXACT AND APPROXIMATE CUMULATIVE TOTAL

HOURS; a=100, b=0.152003 (90% curve) ............ 114
4.2 EXACT AND APPROXIMATE AGGREGATE PROGRESS

FUNCTION - INTERVAL [:1, 1000:] ................. 127
4.3 EXACT AND APPROXIMATE AGGREGATE PROGRESS

FUNCTION - INTERVAL [:1, 20:] ................... 131
6.1 STARTUPS ANALYZED IN FIRMS A, B AND C ........... 161
6.2 STARTUPS ANALYZED IN FIRMS D AND E .............. 162

6.3 FINAL ASSEMBLY OF THE CPU OF A THIRD
GENERATION COMPUTER (FIRM A) .................... 294
6.4 MANUFACTURING OF CARD PUNCH X (FIRM A) .......... 294
6.5 ASSEMBLY OF MAJOR UNITS OF CARD
PUNCH Y (FIRM A) ................................ 295
6-6. FINAL ASSEMBLY 6: TESTING OF CARD
PUNCH X (FIRM B) ................................ 295
6.7 FINAL ASSEMBLY OF CARD PUNCH Y (FIRM C) ......... 295
6.8. ASSEMBLY OF 2nd GENERATION COMPUTER
' UNITS (FIRM D, PROGRAM .4; 1) ..................... 296

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.14

.15

.16

.17

.18

.19

.20

.21

.22

.23
.24

LIST OF TABLES (cont'd)

ASSEMBLY OF SMALL COMPUTER COMPONENTS

(FIRM D, PROGRAM # 2) ...........................

ASSEMBLY OF COMPUTER COMPONENTS AND DATA

STORAGE UNITS (FIRM E) ..........................
MPF REGRESSION RESULTS ..........................
FIRM A PARAMETERS 3 AND B .......................

PARAMETER MODEL REGRESSION RESULTS

(FIRM A, CPU OF A 3rd.GENERATION COMPUTER) ......

PARAMETER MODEL REGRESSION RESULTS

(FIRM A, CARD PUNCH X) ..........................
FIRM D PARAMETERS g.AND g .......................
FIRM E PARAMETERS a AND E .......................

PARAMETER MODEL REGRESSION RESULTS (FIRM D) .....
PARAMETER MODEL REGRESSION RESULTS (FIRM E) .....

PARAMETER MODEL REGRESSION RESULTS

(FIRM D, STARTUPS D12, D1, D4 AND D3 EXCLUDED)..

PARAMETER MODEL REGRESSION RESULTS (FIRM D,

STARRED STARTUPS IN TABLE 6.15 EXCLUDED) ........

PARAMETER MODEL REGRESSION RESULTS (FIRM E,

STARTUPS E11 AND E10 EXCLUDED) ..................

PARAMETER MODEL REGRESSION RESULTS (FIRM E,

STARRED STARTUPS IN TABLE 6.15 EXCLUDED) ........
STARTUPS ANALYZED IN FIRM F (U.S.) ..............

MANUFACTURING OF BINDING MACHINES

' AND PRINTING PRESSES, FIRM F (U.S) ..............

xii

297

298

299

168

169

169

172

173

174

174

175

176

177

177
179

303

 

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680

LIST OF TABLES (cont'd)

FIRM F PARAMETERS _c_: AND B ......................
PARAMETER MODEL REGRESSION RESULTS (FIRM F). . ..

PARAMETER MODEL REGRESSION RESULTS

(FIRM F, STARTUP F4 EXCLUDED) ..................

PARAMETER MODEL REGRESSION RESULTS

(FIRM F, STARTUPS F4 AND F5 EXCLUDED) ..........
PARAMETERS £1 AND t_> (FIRM F,BINDING MACHINES). . . .
PARAMETERS a AND E (FIRM F,PRINTING PRESSES)....

PARAMETER MODEL REGRESSION RESULTS

(FIRM F, BINDING MACHINES) .....................

PARAMETER MODEL REGRESSION RESULTS

(FIRM F,PRINTING PRESSES) ......................

PARAMETER MODEL REGRESSION RESULTS
(FIRM F, BINDING MACHINES, F4 AND

F5 EXCLUDED) ..................................

PARAMETER MODEL REGRESSION RESULTS (FIRM F,

BINDING MACHINES; F4, F5 AND F2 EXCLUDED) ......

PARAMETER MODEL REGRESSION RESULTS (FIRM F,

PRINTING PRESSES, F28 EXCLUDED) ................

PARAMETER MODEL REGRESSION RESULTS (FIRM F,

PRINTING PRESSES; F28, F21 AND F22 EXCLUDED)....
STARTUPS ANALYZED IN FIRM G (BRAZIL) ...........
SHIPBUILDING, FIRM G (BRAZIL) ..................

FIRM G PARAMETERS a AND E ......................

xiii

page
181

183

183

183

308

309

185

185

185

186

187

187

190

310
193

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.49

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.52
.53

LIST OF TABLES (cont'd)

PARAMETER MODEL REGRESSION RESULTS (FIRM A,
FERRY-BOAT OF 1250 DWT) .........................
STARTUPS ANALYZED IN FIRM H (BRAZIL) ............
OIL DRILLING TOOLS & EQUIPMENT,

FIRM H (BRAZIL) ................................
FIRM H PARAMETERS adAND b ......................
PARAMETER MODEL REGRESSION RESULTS (FIRM H,

OIL DRILLING AND PRODUCTION EQUIPMENT) .........
STARTUPS ANALYZED IN FIRM I (BRAZIL) ...........
AIRFRAME INDUSTRY, FIRM I (BRAZIL) .............
PARAMETERS a AND E (FIRM I, SPARE PARTS) .......
FIRM I PARAMETERS _a AND 13 (PRODUCTS) ...........
PARAMETER MODEL REGRESSION RESULTS (FIRM I,
PRODUCTS, STARTUPS II THROUGH I4) ..............
PARAMETER MODEL REGRESSION RESULTS (FIRM I,
SPARE PARTS, STARTUPS I5 THROUGH I69) ..........
MECHANICAL AND ELECTRICAL INDUSTRY

(BRAZIL: 1960-1964) ............................
INDUSTRY J PARAMETERS _a AND 13 ..................
PARAMETER MODEL REGRESSION RESULTS

(INDUSTRY J, STARTUPS J1 THROUGH J9) ...........
MANUFACTURING PROGRESS FUNCTION AND

COST DATA ......................................
PRODUCTION SCHEDULE (PRODUCT P) ................

ULTIMATE HOURS AND UNITS (PRODUCT P) ...........

xiv

page

194
195

311
197

197
200
200
312
202

202

203

316
208

209

214

215
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'24. PARAMETER a VALUES CALCULATED page

ACCORDING TO FORMULA (5.6) ...................... 218
5 MPF HOURS AND COST CALCULATION (EXACT METHOD) ..220
6 MPF HOURS AND COST CALCULATION

(APPROXIMATE METHOD) ............................ 221
7 TOTAL LABOR HOURS FOR PRODUCT P

(EXACT METHOD) .................................. 223
8 LABOR AND SPACE REQUIREMENTS

(MECH. ASSY., EXACT METHOD) ..................... 226
9 LABOR AND SPACE REQUIREMENTS

ELECT. ASSY , EXACT METHOD) ..................... 227
.10 LABOR AND SPACE REQUIREMENTS

(TESTING, EXACT METHOD) ......................... 228
.11 LABOR AND SPACE REQUIREMENTS (FINAL ASSY.

AND TESTING, EXACT METHOD) ...................... 231
.12 LABOR AND SPACE REQUIREMENTS

(TESTING, APPROXIMATE METHOD) ................... 233
.13 AGGREGATE LABOR AND SPACE REQUIREMENTS

(FINAL ASSY. AND TESTING, AGGREGATE

PROGRESS FUNCTION METHOD) ....................... 236
1. SAMPLE 1, FIRM D ................................ 268
2 SAMPLE 2, FIRM D ................................ 268
3 SAMPLE 1, FIRM F ................................ 278
4- SAMPLE 2, FIRM F ................................ 279

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115 SAMPLE 1 (SIMULATES PAST DATA) ................... 286
116 SAMPLE 2 (SIMULATES NEW STARTUPS) ................ 286
117 PREDICTED PARAMETER b VALUES ..................... 291
118 PREDICTED PARAMETER a VALUES ..................... 291

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LIST OF FIGURES

Figure page
PLOTTING THE PROGRESS LINE FOR

A GIVEN CATEGORY ............................... 44
2 PROGRESS ON ARITHMETIC GRAPH ................... 46
3 PROGRESS LINE FOR A CATALYTIC CRACKING UNIT. . . . 62

4 PROGRESS CURVE ULTIMATE POINT (xu, yu) AND

MANUFACTURING PROGRESS FUNCTION HOURS .......... 105
5 EXACT AND APPROXIMATE CUMULATIVE

TOTAL HOURS vs. CUMULATIVE PRODUCTION

a = 100, b = 0.152003 (90:, curve) ............. 115
6 EXACT AND APPROXIMATE AGGREGATE PROGRESS

FUNCTION - - INTERVAL [1, 1009] 129
'7 EXACT AND APPROXIMATE AGGREGATE PROGRESS

FUNCTION - - INTERVAL [1, 20:] 132
8 VISUAL TABLE OF CONTENTS OF THE PACKAGE

DESCRIBING THE MPF SYSTEM ...................... 243
9 OVERVIEW DIAGRAM NUMBERED 1.0 OF

THE MPF SYSTEM (THE HIGHEST-LEVEL DIAGRAM) ..... 244
10 OVERVIEW DIAGRAM NUMBERED 2.0 OF

THE MPF SYSTEM ................................. 245
11 OVERVIEW DIAGRAM NUMBERED 3.0 OF

THE MPF SYSTEM ................................. 246

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12 OVERVIEW DIAGRAM NUMBERED 4.0

OF THE MPF SYSTEM .............................. 247

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CHAPTER I
INTRODUCTION: THE PERCEPTION OF THE PROBLEMATIC SITUATION

Presumably, the process of economic progress involves
three levels of activity: invention, innovation and transfog
mation. Invention is a new idea, the discovery of relation
ships not before perceived. Innovation is the pioneering
application of an invention. Transformation constitutes the
substitution of existing processes and outputs by those which
innovation has already shown to be superior or preferable.

The end result of transformation is economic progress.

It is perhaps a platitude to say that the industrial
environment in the United States and abroad - including some
modern less developed countries - has been characterized by
an ever increasing rate of technological innovation and
tranSfOrmation. Some degree of innovation and transformation
is shared by virtually all manufacturing industries and their

c .
Omponent enterprises .

Given that rapid innovation and transformation proces§_
es have become the environment as well as the internal life
of a large number of industries - here and abroad - it is
appropriate and opportune to investigate the possible influ-
ence Of those processes upon the manufacturing activities of
the aft.eczted firms and their consequences in terms of

deCis ion ~making .

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Adaptation to Innovation and Transformation

Innovation or transformation imply change, and change

gmMmates the need for adaptation. In the implementation of

arww product, the following represent general decision-

mfldng areas in which the adaptation phenomenon might have

mabe taken into account:

Long-range forecasting and planning for capacities

and locations.

Selection of equipment and processes

. Production design

Job design and work measurement

. Location of the system

Facility layout

Short-range forecasting

Inventory control

Aggregate planning and scheduling
Scheduling and production control
MMintenance

Quality control

Labor and cost control

SuChalist indicates that the adaptation phenomenon

D
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8
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Productivity, System Adaptation and Decision-Making.
The development and implementation of new products or new
production processes can alter the means and methods of manu
facture. In responding to these changes the production orga-
nization can experience a period of adaptation. The efforts
Of the production system give rise to notable increases in
manufacturing productivity. In fact, it is generally known
that the hours required to produce a new product decrease as
cumulative production increases. Several studies in many
different manufacturing areas and other situations support
this assertion.1

The occurrence of this adaptation phenomenon in manu-
facturing is rarely a consequence of direct-labor learning
Of a manual task. The phenomenon usually results from an in-
tegrated adaptation effort on the part of direct-labor, indi
rect-labor and technical personnel. It relies primarily on
Cognitive rather than manual learning. Managers, engineers,
supe1""1-SOrs, machine operators, maintenance men, quality
control personnel, purchasing personnel and other indirect-
labor employees can all make important contributions toward

1 . o g o
ncreaSIng the effICIency of a manufacturing process.

This type of manufacturing progress phenomenon may have
a Significant effect on many decision-making activities in a
firm- Practically every manufacturing concern has to forecast
labor time and cost per unit of new products in order to set

S O
elllng Prices, plan delivery schedules, estimate capital .

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labor and space needs, and the like. Large, sustained incre-

mmn$ in productivity can influence a variety of planning

muicontrol functions. N.Baloff suggests the following

panflal list of decision-making areas that can be influenced

by changes in productivity:

"1.

Price-setting. The level of productivity affects
the direct manufacturing costs and, in many cases,

the allocation of overhead costs.

Delivery commitments. Delivery commitments cannot
be made reliably unless the rate of product output

can be correctly anticipated.

Production scheduling. The rate of output of a
"bottleneck" can materially affect the schedulingcflf
other processes in a sequential manufacturing opera

tion.

Purchasing and raw material inventory. These ac-
tivities must be synchronized with the productiv-

ity of the production processes.

Manpower requirements. Akin to the scheduling
‘problem, the total labor requirements in sequential
manufacturing operations may depend on the rate of

output of a new bottleneck process.

'Work standards. The determination of work stan-

dards and wage incentives is both hazardous and

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difficult under conditions of changing productivity.

7. Cost accounting, budgeting, and cost control.
These control procedures are sensitive to changing
productivity, as well as being motivationally

sensitive - a volatile combination."

However, before considering the above list, the initial
decision of whether or not to implement the new product, the
major change or the new process must be made. In order to do
this, one must count upon a valid and reliable tool for pre-
dicting the pattern and the magnitude of the adaptation
Phenomenon. Moreover, unless such productivity changes can
be Predicted at the beginning of a manufacturing startup, an

appreciable degree of costly uncertainty can result.

Manufacturing progress functions (MPFs) have been docu
mented for years in the literature as a means of describing

and sometimes predicting the pattern and the magnitude of the

adaptation phenomenon .

MannfaCturing Progress Function Defined

The manufacturing progress function (MPF) has been used
in the literature to describe several different proposed
relationships between the labor input involved in the opera-
tions required to manufacture a product and the volume of
plT’duetion. For the purpose of this dissertation the MPF can

b _
e generally defined as the relationship in which the labor

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input per unit used in the manufacture of a product tends to
decline by a constant percentage as the cumulative quantity
produced is doubled. Where this relationship is present, it
may be represented by a straight line in a double logarithmic

scale. Chapter III is aimed at a complete clarification of

this subject .

Scope

The following observations made by the author since
1959 have served the purpose of setting the scope for this

dissertation .

Universalization. The need for making the progress
curve model more generally applicable in labor-intensive 3
manufacture has been stressed for years in the literature.
To date, however, its importance has been mainly recognized
by the aerospace industry apart some few extensions of the

model to other labor and machine-intensive production

SYStems .

Form of the Model. The adequate form of the manu-
facturing progress function model has also been discussed
for Years. In the following work empirical support has been
found for some of the forms proposed in the literature.5 The
moSt POPUIar are the cumulative-average curve and the unit
Cturve.

Parameters Estimation. The estimation of the paramg

te '
rs of the progress function for different products and

 

 

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u. .

 

 

 

7
processes has been of significant concern, although few
authors have tried the empirical approach. It must be undeg
stood that without a valid and reliable method for estimating
the parameters of the manufacturing progress function before
the inception of the production process, its usefulness as a

predicting tool can be virtually dismissed.

When Does Learning End? The occurrence and predict-
flfility of steady-state plateaux in the curve have also been
flMasubject of much argument and little empirical research
effort.7

Comparative Studies. Comparative studies of produc-
thfity values across facilities located in different coun-
tnies are missing in the literature of the progress curve.

Such an approach might provide thoughtful insights in theo-

rizing about the adaptation phenomenon.

Better Theoretical Systematization. The field of
prOgress functions lacks notation uniformity, precise defini
tion 055 the variables and relationships involved, and proofs
of s"Wei-"a1 results assumed. A coherent mathematical exposi-

tion can be the basis for the derivation of new theoretical

results,

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8
GENERAL AND SPECIFIC OBJECTIVES OF THE STUDY

A Note on Models

Models are taken to be analogues of existing or con-
ceivable systems, resembling their referent systems in form
but not necessarily in content.8 They exhibit structural re-
lationships among elements found in the referent system. At
the same time they are abstractions, omitting some aspects

Of the referent systems and duplicating only those that are

of interest for the purposes at hand.

A distinction can be drawn between representation and
explanation and hence between models and theory. Models need
9111? represent the referent system; explanation is the role
0f theory. While many theories may, in fact, be models, not
all are, for as A. Kaplan suggests, theories need not actu-
ally exhibit the structure they assert the referent system to
possess. On the other hand, models that do not purport to ex_
plain are not theories. Models can also be used E9 predict

phenomena without necessarily explaining them.

Models have long been considered the central necessity
of seientific procedure. Models can increase understanding
of the referent systems, aid in the development of theory,

an . .
d serve as a framework for experimentation.

The disadvantages of modeling in general, stem from the

model‘s artificiality, simplification, and idealization, and

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the consequent difficulties and dangers in making inferentflfl.
leaps from a model to the real world. As a representation of
a real system, the model's representativeness of that system

is a crucial issue.

Symbolic models are of special interest in social and
administrative science and may take various forms, such as
verbal, analytic (in the mathematical sense), or numerical.
In addition, flow charts and similar pictorial models are bg

coming increasingly popular.

General Statement of Objectives

The overall end purpose is to advance knowledge on the

subject matter of Manufacturing Progress Functions.

The prime objective of the study is to contribute a
general symbolic-analytic model of the manufacturing progress

phenomenon.

Once the general model is established, an equally impog
tant objective is to respond to the need for a coherent sys-
tematic approach to be used in predicting the developments
of the adaptation process in industrial concerns of almost

any kind.

Subobjectives. The foregoing general statement of
purpose can be broken down into a number of layers of inves-
tigation leading to the following more specific subobjec-

tives:

10
A review of the literature that is of relevance to

the objectives of the dissertation.

An investigation of the theory of the manufacturing
progress function aiming at a systematization of
the existing body of knowledge, and at the deriva-
tion of new theoretical results that will settle
the question of the parameters estimation of the

general model.

The conception of a general symbolic-analytic model
of the system adaptation phenomenon, including the
development of a method for using the model to

predict the course of future startups.

The testing of the model in a number of real world
situations by using data from.diverse industrial

operations.

The possibilities of implementing the model into
practical use at firm level and national leve1.This
signifies that the model can be practically inte-
grated into the strategical planning of an industri
a1 concern as well as used for macroeconomic

decision-making.

CHAPTER II
THE EXISTING BODY OF KNOWLEDGE

The present chapter contains a review of the histori-
cal development of the manufacturing progress function, as
well as a summary of the more important contributions to
the progress curve literature which are of interest to the

purpose of this dissertation.
THE WRIGHT MODEL

Historically this model is the first published attempt
to relate labor hours and cumulative production.1 After
fifteen years of empirical studies in the airframe industry
Dr. T.P. Wright discovered the following hyperbolic func-

tional relation:

§'= ax.b (2.1)

where,

V = the cumulative average number of direct
labor hours (or related cost) required
per unit of output;

x = the cumulative units of output;

a = a parameter of the model, i.e., the labor
hours required to produce the initial
unit of production; (note that for x = 1,

ll

12
§= 6(1)“b = a)

b = a parameter of the model, i.e., a constant
dependent upon the rate of progress.It is
an index of the rate of decrease in labor
hours during the start up (usually ,

0<<b<<1).

The so-called learning curve equation (2.1) is such

that the logarithms of y and x are in a linear relationship,

given by:
log § = log a - b log x (2.2)

Such relationship exhibits the convenient property
that every time E doubles, successive values of y are a
constant multiple (some fraction between 0 and l) of the
preceding value. This fraction of y, expressed as a percent-
age, is commonly called the learning factor, or the learnéng

curve "slope", according to Wright's unusual terminology.

Thus :

é __(2X)-b = 2'13 (2.3)

-b
x

Wright's "slope"

Learning curves are often identified by this percent-

age. The most popular learning curve from 1930-1950 was the

80 per cent curve.

Factors in Labor Reduction. Wright attributed the

reduction of labor time per unit as production quantity is

 

II
a.

 

 

13
increased to four factors: (i) The improvement in proficien-
cy of a workman. (ii) The greater spread of machine and
fixture set up time in large quantity production. (iii) The
economy in labor which greater tooling can give as the
quantity increases. And finally, (iv) as more and more
tooling and standardization of procedure is introduced, it
is possible to use less skilled labor.3 Although he did not
Offer a theoretical explanation of the relationship, Wright
suggested that the improvements in worker proficiency that
occur wizh increased experience are the main causative

factors. He argued that this effect was particularly noticg

able in assembly operations.

 

Material, Labor and Overhead Versus Quantity. Wright
holds that the three major costs of production vary with the
cumulative quantity of production. It is worthy to note that
although it has been a widely acknowledged fact that labor
and overhead vary as quantity produced is increased, there
has been a surprising lack of development on the direct
material curve which Wright originally suggested.5 He
hypothesized that in addition to the reduction of waste,
greater cutting efficiency and more economical purchasing
partially explain the reduction in material cost. Other pos-

sible factors are reductions in price of materials because

Of quantity purchases and greater vendor efficiency.

The Value of Wright's Model. Many magazine articles

were written about the learning curve during Worl War II

 

"a.

,9.

 

 

 

14
By that time, acceptance of its theory had increased as

rapidly as the aviation industry itself.

By the end of World War II, the major aircraft compa-
nies had generally recognized the value of the curve by its
application to their specific production data. The smaller
firms, however, comprising most of the subcontractors, had

little knowledge of its use.

As to the learning-curve equation, it should be noted
that y approaches zero as g increases. This feature as well
as the linear logarithmic relationship were sometimes criti-
cized. A number of models have been proposed since wright's
initial work. However, the Wright formulation continues in
almost universal use in the airframe industry.6 H. Asher
notes that most airframe producers, and a large number of

7
Air Force personnel, accept and use the Wright model.

THE FORTIES

After Wright's pioneering contribution a number of mod;
fications of the original model have been proposed. Some are
similar to Wright's hyperbolic function formulation. Others
include additional parameters so as to fit certain World
War II production data that were not well explained by the
early models. Empirical verification of the modified models
has been fragmentary, and they have found little acceptance
in the airframe industry. A discussion of some of the ap-

proaches to the progress curve that appeared in the forties

is given below.

The Crawford Model

15

J. R. Crawford has contributed much to the literature

of progress functions. He was associated with the Stanford

Research Institute

following WOrld war
Of two hundred jobs
which resulted in a
Crawford's equation

8
as follows:

yL

where:

‘<€
II

x =

a and p =

and the Air Materiel Command immediately
II. His chief contribution was a study
in the airframe-manufacturing process ,
new formulation of the progress curve.

for the learning curve can be expressed

ax (2.4)

lot-average direct-labor hours per unit

of output
cumulative units of output

parameters with the same meaning as in

9
Wright's model.

Clearly, the difference from the Wright formulation

resides in the definition of the dependent variable. In the

wright model y is calculated as a cumulative-average manhour

 

expenditure, whereas in the Crawford model y’is calculated

as the average manhour expenditure for the "lot" produced

in a single month.

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16
In reality, Crawford's model cannot be considered a
truly 2215 curve, since Vi does not refer to the labor hours
consumed in the xth. unit but simply has to do with the ave;
age number of direct-labor hours per unit, expended for the
"lot" produced in the kth. month. In Chapter III, this form
is further discussed as well as its relationship with other

forms proposed in the literature.

According to H. Asher, a number of Crawford's attempts
to develop a progress curve of simpler equations did not
yield forms acceptable to the airframe industry or to the
Air Force.10

Crawford makes a relevant observation that will be of
interest at a later point, specifically, that the shape of
an individual operator's progress curve is a function of the
mental effort required of the operator to perform the task.
For Crawford, mental effort expended is also a function of
the complexity of the job and of the lack of experience of
the worker. Thus, when the work requires more (less) mental
effort, the ratio of improvement increases (decreases). This
is true when the job is complex (simple) requiring great
(little) mental effort, and also when the operator is less
(more) experienced and requires more (less) mental effort

to learn the technique involved.

It seems that Crawford was the first author to suspect
a link between the rate of improvement in a job and its

degree of complexity and novelty. This relationship would

17
come to be investigated by Hirsch, Conway, Schultz, and

others, some years later.

In addition to proposing his modified learning curve,
Crawford also noticed that different slopes might exist for
different airframes - a result that was largely ignored by

12
the rest of the industry.

Bureau of Labor Statistics Studies

The Productivity and Technological Development undeion
Of the Bureau of Labor Statistics, Department of Commerce,has
prepared a number of studies that deal with progress-curve

data, and their content will be summarized below.

The Liberty Vessel Study. The first progress curve
data on ship construction was reported by Montgomery in
1943.13 The study deals with the EC2 Liberty ship of 10,800
dead weight tons (TDW). The man-hour requirements data
represented both direct and indirect labor hours. The
direct man-hours had to be estimated from actual man-hours
required for a group of ships; the total was then averaged
for the number of units 'within the group. Between December
1941, when the first two ships were completed, and April
1943, when nine hundred ships where delivered, the average
man-hours required per vessel decreased by more than one-
half.

Wartime Shipbuilding Study. Another relevant study

. 14
was done at the Bureau of Labor Statistics by A.D. Searle.

 

18

The study is similar to that made by F.J. Montgomery except
that it covered Liberty and Victory ships, tankers and
standard cargo vessels. The labor data used to determine
labor requirements are E9£§l_man-hours defined in the same
way as in the Montgomery study, already described.The data
covers the period between December 1941 and December 1944
with emphasis on the interval between April 1943 and December
1944. A.D. Searle notes that each individual shipyard exhib-
ited a different learning curve as to the Wright "slope".
waever, the "slopes" were almost identical for the different
types of vessels considered, i.e., about 80 per cent decline
finumnrhour requirements between doubled quantities.15 The
Study concludes that differences in the types of vessel are
lggg significant in determining the rate of man-hour requirg
IDent reduction, than the production function differences
batween individual yards. The above two studies demonstrate
true existence of the progress phenomenon in ship construction
Pregrams. Moreover, they constitute a pioneering effort

towards studying learning curves outside the airframe

industry .

Wartime Productivity Changes in the Airframe Industry.
UFIiiezis the third in a series of studies which were carried
C>Iit: by the Bureau of Labor Statistics. Conducted by Kenneth

15.. Ididdleton it deals with productivity changes.16 He noticed
tiflzat: there was approximately a tripling of production per

Unéiri-éhour during World War II and he attempts to identify the

E<‘='=1c tors responsible for the observed productivity increment.

 

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l9
Middleton holds that the remarkable reduction of labor cost
per unit is the result of technological changes such as
increasing standardization of models, introduction of more
sophisticated hand tools, and gauges. The rate of adoption
of the new methods and new production techniques depends on

management willingness to break with traditional methods.

Kenneth A. Middleton also attempted to construct an
industry wide production index by relating man-hours per

pound of airframe and the cumulative production pounds in
17

the particular facility. He found that the decrease in

Immrhours required per pound is similar for all types of air
planes and that this index is useful for comparative purposes,

although somewhat more man—hours per pound were required for

a one engine fighter than for standard four engine bombers.

ITuJ , using the above indicator of productivity, Middleton
Concludes that the 70 per cent progress curve is more repre-

Sentative of the aircraft industry during World War II than
18

the normally assumed 80 per-cent curve. It is worthy to

Ilcrte that Middleton also has found considerable variation in

the "slope" of the curve from one plane to another.

Airframe Companies Publications

Besides J.R. Crawford's contribution while working at

the Lockheed Aircraft Corporation several progress-curve

StImidies had also been prepared by other airframes companies

- 19
In the forties, like Chance Vought Aircraft, Inc., and the

20
20
Boeing Airplane Company.

The Chance Vought study was designed to instruct compa
ny personnel in the application of the cumulative average
curve, being similar to the Wright presentation, and hence
will not be discussed herein. The study prepared at Boeing

has some features relevant to the scope of this dissertation

and will be briefly reviewed.

The Experience Curve. This short book - one of the
earliest company publications available - presents a concise
statement of the unit curve, the cumulative average curve
and the total curve as they were used by Boeing Cost Accoung
ing Department. The pamphlet provides a mathematical preseng
ation of the progress curve. From the equation for the unit

curve, i.e.,
y = ax (2.5)

time exact expression for the total labor hours expended in
tile manufacturing of units one up to and including the nth.

unit is given by:

n
y = a Z x. (2.6)

SinCe equation (2.6) requires calculating every unit value
iEITCHH l to p, an approximation was suggested by the authors,
:Lt1 ‘which the limits of integration were formed by extending

‘t11ee range of the summation index so as to reduce the error

21

inherent in approximating a discrete function by a continuous

function. Thus:

n xi+0.5
yT = a Z xtb E a f x.b dx —
n i=1 1 0.5
l-b l-
= 1%,; [(xi + 0.5) - 0.5 b] (2.7)

Studies Financed by the Air Force
The Source Book of World War II Basic Data. This Air

Force study of the progress curve published in 1947 has been

the principal source of data for several empirical progress-

21
It contains data from every facility engaged

curve studies.
in the production of military models from 1940 through mid-

1945, presenting unit progress curves for each type of air-

craft and one progress curve for £11 aircraft produced durfin;
tflne war period. The curves obtained by using the least-

Squares method represent direct labor hours per pound of ai;
frame weight versus the cumulative number of airframes pro-

duCed. The following table presents the parameters of the

pIT’gress curve obtained in the Source Book of World War II

Wgy

22

TABLE 2 . 1

PARAMETERS OF THE PROGRESS CURVE

 

 

 

 

 

Type of Man-hours per Wright's "Slope"
Pound at Unit
Aircraft Number One (%)
(a)
Fighters 18.5 79
BoMbers 16.0 77
Transports 16. O 77
The Crawford-Strauss Study. An analysis of World War

11 production experience was made by Crawford 6: Strauss in
1947.22 The major purpose of the study has to do with the
acceleration of airframe production; but since progress

curves are important tools in any airframe production pro-

gram, they are given careful attention.

23

The data was derived from the "Source Book" , and was

baSed on the production data of 118 Worl War II models.
Weighted average direct man-hours per pound at specific
plane numbers were determined both for the industry as a
whole and for each type of airplane (fighters, bombers and

transports) . The resulting unit-progress-curve equation for

a\11 models was

-0.32668
x

y = 14.3 (2.8)

23

From (2.8) and (2.3) it is easy to see that:

a = 14.3 and Wright's "slope"=2-O°32668=0.797.

It was probably this study that established the "80 per-cent

rule" as standard in the airframe industry.

Crawford & Strauss found that at least three major
factors were responsible for the dispersion of the individ-
ual progress curves around equation (2.8): (i) type of air-
craft (fighters, bombers or transports); (ii)newness of the
model and of the facility, and (iii) particular circumstances
liproblems that surround production of each individual
model.

As to the relative position of the fighter, bomber and
transport curves, they make the point that the bomber learn-
:ing curve is the lowest because (i) bomber programs during

lflorld War II were given priority over most other programs,
and (ii) the size of bomber aircraft permitted greater access
111 the assembly Operation. The fighter curve is the highest
because: (i) fighters are complex aircraft and (ii) their

des ign changes often.

In connection with the newness of models and facili-
tieS, the 118 models were classified as: (l) Proven models
prOduced in experienced facilities; (2) proven models pro-
duced in new facilities, and (3) new models produced in new

C”r’ experienced facilities. Weighted average curves like the

Strauss c

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24
previous ones were obtained for each class above. On the
basis of visual inspection of such curves Crawford and
Strauss concluded that: The early units of a proven model
produced in an experienced facility require fewer man-hours
per pound than the average for all aircrafts. This advantage
is not maintained, however, after a few hundred units have
been produced. The progress curve for proven models pro—
duced in new facilities follows the average curve for all
aircraft until a few hundred units have been delivered,after
Iflfich the former curve falls slightly below the latter. The
curve for new models produced in either new or old facilities
is consistently higher than the average curve for all air-

craft. These results are summarized in Table 2.2

The third factor - or cluster of factors - which has
been found to affect the level of progress is described as
the special circumstances and problems that surround produg
“tion Of each individual model. Some of these are listed by
the authors as: (i) The length of the production run, (ii)
nflnether or not the model has been engineered for mass produg
tZion, (iii) whether or not proven engineering was available

When production started, (iv) whether or not high production
was started from low production tools, (v) introduction of
design changes, (vi) whether or not old tools were available
when production started, (vii) availability of materials or
component parts, (viii) availability of experienced manpower,
(ix) relative priority attached to a given model, (X) effi‘

cieency of operating controls,(xi) frequency of scheduled

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25
TABLE 2.2

INFLUENCE OF THE NEWNESS OF MODELS AND FACILITIES
ON THE RELATIVE POSITION OF PROGRESS CURVES

 

 

facility

old

model new

curve is lower than
the avg. curve only up
to a few hundred units

curve follows the avg.
Old curve until a few hundred
units; then it falls below

curve is higher curve is higher
new than the avg. curve than the avg. curve
for all planes for all planes.

«ahanges and degree of pressure attached to a program, (xii)
(economical and uneconomical use of outside production,(xiii)
degree to which feeder plants and outlying areas were uti-
ldiZed in order to tap a wider labor market, (xiv) whether or
‘acyt the plant layout was favorable to the production of a

Particular model and (xv) availability of specialized high
24
PrOduction machinery.

The authors conclude that it would be difficult to

theasure the effect of special circumstances or to weigh

dlrect labor progress curves and industry averages for these

I
..!IA

5
.ti'*

”3 ‘
A

..4 o

no;

 

“ «1

VI.‘

 

 

26
factors. However, they believe that the industry average es-
tablished in the study presents a reliable picture of the
relationship of direct man-hours per pound to cumulative prg
duction during World War II, since they include the cumula-
tive effect of all particular circumstances and problems

which surrounded the production program, for each model.

The Stanford-B Model. After World War II a number of
economists and econometricians became interested in learning
curve research. The Air Force had also come to recognize the
importance of the progress curve, and sponsored several
research projects with private organizations to develop
further the application of the theory. Probably the best
known of these were the studies carried out by the Stanford
Research Institute and by the Rand Corporation. Since the

latter were mainly published after 1950 they will be consid-

ered in the next section.

Under contract with the United States Air Force, Air
Imateriel Command, the Stanford Research Institute made a
Study on the Relationships for Determining the Optimum Expan-
W g the Elements 9_f_ a Peace-Time Aircraft Procurement
lixllgggg in order to determine the means of measuring the
Inakimrum rates in aircraft production programs. Since the max
Zinmufllexpansibility rates depend to a large extent on the
1fate of manufacturing time reduction, a decision was made to
atlé'llyze the direct man-hour progress curve. According to the

fiI'lal report the project has resulted in an improved

27
relationship involving direct-labor hours, airframe unit

25
weight and cumulative production.

The Stanford-B formula or the learning formula with a

B-factor can be expressed as follows:

y = a (x + B)n (2.9)

where:

y = direct man-hours required for cumulative

unit number x.

B = equivalent-units of experience available
at the start of a manufacturing program;

1ng10, 4 being a typical value.

n has a similar meaning as the b-exponent
in the Wright model. Usually -l‘<n‘<0,

-0.5 being a typical value.

IEcu'the case B=O, 3 represents man-hours for the first unit.
The curve is asymptotic to a straight line with slope n, and
éippr'oaches such a line over its entire length as b approaches
Ezeanx The effect of the B-factor is to round the initial pop
tZion of the curve thus providing a better approximation in
determining the number of labor hours (and cost) for the

first units produced.

28
Such a curve - or its variants — have satisfactorily
described the general features of many large scale military
aircraft production programs, including the Boeing B-l7,B-47,

B-52, and B-707 programs.

Its is worthy to mention that the reasons for prefer—
ring equation (2.9) rather than equation (2.5) or (2.1) is
not that the former fits the empirical data more closely. In
fact, the Stanford study does not intend to establish this
point. Rather, equation (2.9) is preferred merely because
its parameters are more suitable to predicting airframe pro-

duction performance than the parameters of the conventional

progress curves.

Progress Curves in Great Britain and France

Among British and French manufacturers the progress-
cruve concept was also popular as evidenced by the contribu-

‘tions of E. Mensforth and P. Guibert.

In Great Britain. The first published evidence of
Frrbgress curves existing in Great Britain was indicated by
Eric Mensforth, who was associated with the ministry of air_
craft production in World War 11.27 Mensforth found that
I51:Ufish experience with progress curves was rather similar
tZ<Jthat of American aircraft production. British figures

exhibg the same general trend within limits of 75-85 per-

cent.

29

Mensforth considers the peak volume of production to
be a major factor in the declining of direct-labor hours per
unit. He suggests that when the scale of production is
greater, it permits more specialization and earlier attain-
nmnt of full dexterity. To illustrate this point Mensforth
cites a case of two factories which were producing the ggmg
afircraft with similar equipment and tooling, at rates of 15
and 55 planes per week. The actual man-hour of the latter
were half that of the former. Since both of the factories
were on a piecework basis and the same rate was paid, the

earnings of the one producing 55 units per week were 200 per

cent higher .

In France. The most comprehensive study of the prog

ress curve prior to 1945 was published in France by

P. Guibert?9 The book was translated into English by the
U.S. Air Force but this translation has not received wide-
sruead circulation.3o Guibert considers the gaps of produc-
tion as an important variable influencing unit labor cost.
Healso views the progress curve as approaching é plateau
Either a large number of units have been produced. The begin-
ning of the plateau depends on the rate of production. His
rationale for considering unit man-hour cost as a function
Of the rate of production is that to achieve a given produc-
tion rate, it is necessary to design and build adequate tools,
at tkmnnating time factor?1 Guibert points out that although

the nu"lber of certain tools must increase with the rate of

o I -
. I'Oduction, there are many which can only be con31dered for

.
«.557

a .
.-.-V.

‘ I
.‘O. l!
\

.u- V-

~
n h.

5"“:

c J
Ali
, a

(D

'1
Al'
(I,

larger series

cost dies for

30
(e.g.,the tools of the authomatic lathes,high

presses, jigs for simultaneous assembling and

drilling), and these take a long time to build.

Through empirical studies, Guibert obtained one prog-

ress-curve equation that can be used for any rate of produg

tion:

where:

Guibert

c 31 equations

(a -1)(a - m)(1 - m)(A - 1)

(2.10)
x(o:-l)+A(l-m)-(o.-m)

unit man-hour cost expressed as a ratio of the

man-hour cost of the Ath airplane produced
number of units (planes) produced

the number of units in process when peak produg

tion is attained.

the value of y when x = l

the value of the horizontal asymptote (plateau)

found that m,a,A.may be approximated by empiri

where the only independent variable is the

rate of production §_. Therefore, for a given rate of produc-

tion a. it is

possible to determine m,a,A and to establish y
32

as a function of x through equation (2.10) .

31

Other Contributions

The contributions to progress-curve literature that are
summarized in this chapter were selected for review because
they appeared to have added substance to the development of
the concept. It would be unfair not to include some other
contributions by individuals who seem to have been deeply

interested in augmenting the knowledge about learning curves.

A.B. Berghell. A mathematical treatment of the prog-
ress curve may be found in Berghell's book entitled
Production Engineering i2 the Aircraft Industry,33 which
contains a chapter on "Learning Curves". Aside from Wright's
article, this is perhaps the most popular discussion of prog
ress curves in the forties. Berghell shows the mathematical
relationships existing among the cumulative average curve,
the unit curve, and the cumulative total curve. He also of-
fers an approximation for the equation of the unit curve,

34
derived by the use of empirical equations.

A different problem to those who work with learning
crurves is that of estimating curves for aircraft not yet
{>1fllt. Berghell suggests that if the cumulative average man-
llcnu'data are plotted against cumulative output for, say,
ifcnu'different aircraft, the resulting slopes of these ave;
age curx37es will not be significantly different from one
aucther (i.e. , the _b_ values will be similar), but the le_vgl_
0f the curves (i.e. , the g values) will be different. The

a; ‘ . . .
pread between the curves can be reduced by d1v1ding direct

32
labor hours by airframe weight. Since Berghell assumes that
airframe weight remains the same for each plane, the slopes
do not change. Berghell argues that a heavier airframe will
generally require fewer labor-hours per pound than a lighter
aflrframe at the same cumulative unit number. Therefore,it is
expected that, in terms of direct labor-hours per unit the
curves for the heaviest airframes appear at a higher level,
but in terms of direct labor—hours per pound, the curves for
the heaviest airframes are on a lower level. Berghell then
reads off the direct labor-hours per pound at units 50 and
100 for the four aircraft, plots these eight values against
airframe weight on logarithmic grids, and obtains two linear
curves showing the relationship between direct labor-hours
per pound and airframe weight, one curve for unit number 50
and the other curve for unit number 100. The man-hours per
pound of airframe values for any new airplane are then
obtained at units 50 and 100 by interpolation for the given
airframe'weight. These two cumulative average man-hour per
IDOund points are all that is needed to draw the cumulative

érverage progress curve for the new airplane.

I.M. Laddon. Laddon was Executive Vice-President of
(3cxmolidated Vultee Aircraft Corporation. The article he prg
Sents contains a description of change in the production
iftnumion as the total quantity to be produced by a company
jLE‘inCreased. The author explains the difference in produc-
tiixnlmethods when sixty units are to be produced as opposed

t: .
‘3 several thousand. The company performed much better than

33

36
the customary 80 per-cent curve.
G.W. Carr. One writer who believes the progress curwa
37
should take on the "S" shape is G.W. Carr. He found that,

in several cases, the rate of labor decrease in a given air-
frame varied, yielding different values of b_instead of one
constant value. Carr did not offer a formal relationship
embodying these conclusions, but another modified model,the
"Boeing hump-curve”,is a recognition of his findings.38 The
concavity early in the series was also recognized by the
Stanford Research Institute.39 Carr argues that such concavi
ty results from hiring inexperienced crews at different
points in time during the production of the first several

lots of airframes. The empirical data available to P.Guibert

from the French airframe industry apparently exhibited the
40

same early concavity.
THE FIFTIES

A number of decisive contributions to the theory and
(levelopment of progress curves were published in the 19505.
There was an effort by some researchers to extend the concept
12<>other labor-intensive industries outside the airframe
industry. Also, studies continued to be financed by the

:ALJJ'Force in several research institutes of the country.

34

Extensions of the Concept Outside the Airframe Industry

Werner Hirsch's Studies and Research. Hirsch's stud-

ies were mainly published in Econometrica and the Review of
41 42 _—
Economics and Statistics. In one of his works, Hirsch

 

 

computed a total of 22 empirical progress functions, based
upon historical data from one of the country's largest ma-
chine tool manufacturers. The results of the study may be

summarized as follows:

(i) The hypothesis of linearity (when in logarithmic
coordinates) between the labor per unit and cumulg
tive output was confirmed for all cases. In 17 out
of 22 cases, the correlation coefficient was found
greater than 0.85. In only one instance this coef-
ficient was as small as 0.59.

(ii) The 22 empirical progress functions revealed a
negative slope, i.e., labor hours per unit decremx:
as cumulative production izcreases.

(iii) The average progress ratiz 3 was 19.3 per cent and
the range (16.5 - 24.8).

(iv) Another important conclusion was that different
kinds of operations exhibit different "slopes".
Thus, the 80 per cent learning curve cannot be
universally applied.In fact, Hirsch has shown that
assembly operations are characterized by signifi-

cantly higher progress ratios than are machining

operations. An explanation offered was that in the

 

35
former the labor content is higher than in the
latter.

45
(v) Compounded experience. In a paper ,Armen Alchimn

suggested that older,more experienced manufactunhm;
facilities exhibit a greater rate of decline than
do new facilities.To test this hypothesis Hirsch
used the data of a manufacturer who had produced
more than 15 lots of a certain semi-automatic tur-
ret lathe, in the same plant, when he initiated
production of a greatly improved model. In more
than one respect the new model resembled the previ
ous one, and work on the first appeared to have
constituted valuable experience. In a sense, then,
the improved model was built with the help of much
previously accumulated experience. Thus, the ques-
tion was whether the progress ratio of the new
lathe would be significantly greater than that of
the old one. Progress functions for both lathes
were calculated, and progress ratios (both for as-
sembly and machining work) of the model built with
the compounded experience were found significantly
larger than those of the first model. This conclu-
sion is consistent with Alchian's "experience"

hypothesis.

Hirsch was probably one of the first researchers to reg

ognize the value of the manufacturing progress function

 

36

outside the airframe industry.His contribution represents one

of few published empirical studies in non-airframe industries.

 

46
Stanley E. Bryan's Study. In an insightful article

Professor Bryan related the manufacturing progress phenome-
non to value concepts and collected data on its existence in
a large company's footwear plant. He points out that in
certain types of procurement like the contracting for spe-
cial and non-standard equipment, competition ceases to be a
decisive influence in price determination. In such cases ,
pricing becomes a matter of negotiation. In negotiation,
price is related to cost to a greater extent than to the util
ity of the product. Consequently, the method of estimating
and compiling cost of producing an item becomes crucial both
to the buyer and to the seller. If the manufacturer does not
take into account the progress phenomenon in his estimate he
would quote unrealistically high cost on the labor portion of
the contract. If the buyer is not aware of that phenomenon he
would accept the seller's estimate as being fair.47 According
to Professor Bryan's data, this particular manufacturer of
footwear experienced an approximately 90 per cent "slope"

PrOgress curve.

48
The Schulz and Conway Study. The existence of the

PIOgress phenomenon had been known for some time at IBM-
Endicott, but until 1955 no systematic study had ever been
made to determine whether the decline in direct-labor hours

per unit of output followed any predictable pattern. With

37

this in mind, a group of graduate students from the Industri
a1 Engineering School at Cornell University, under the guid-
ance of Dr. Andrew Schulz, Jr., defined the following objec-
tives for a study of labor hours reduction trends: (i) To
furnish a better basis for product pricing, product replace-
ment, and decision to manufacture. (ii) To provide a better
basis for estimating and planning the amounts of space and
manpower required by a proposed manufacturing program, and
(iii) To help in the planning and budgeting of engineering or

other staff effort for cost reduction activities.

The study carried out to develop the Manufacturing Frog
ress Function consisted in:(i) The collection of a number of
series of labor hours with the corresponding production quap
tities, together with all the supporting information that
could be obtained, such as: engineering changes, manufactur-
ing methods and tooling changes, personnel turnover ratesguui
the like. (ii) The analysis of the information so as to assg
ciate cause and effect, and to isolate the relevant variables,
and (iii) The generalization of the results obtained from the
analysis. The conclusions reached were based on a detailed
study of an accounting machine and verified by sample checks

0n.other IBM machines.

The following factors were found to influence the de-

Cline in labor hours per unit with increased cumulative

(l)

(2)

(3)

(4)

(5)

(6)

(7)

38
The degree of similarity of a machine to a prede-
cessor or to other machines produced had a signif-
icant effect on the rate of progress: the less
similarity, the greater the rate of progress.49

Single-product departments exhibit greater rates

of progress than do multi-product departments.

Product redesign, and tool and methods improvements

result in a greater rate of progress.

Increasing rates of production as a machine program

accelerates result in economies.

Management progress in scheduling and supervision

results in a greater rate of progress.

Increased planning prior production will result in

a lower initial cost and a reduced rate of progress.

Worker learning through repetition, changes in
method, and reduction in scrap and rework result

in a greater rate of progress.

According to the authors of the study, the last item

has either been overemphasized or erroneously indicated as

the main causal factor in the literature prior to 1959. They

indicate that operator learning in the true sense of perfor-

mance of a fixed task is of negligible importance in most

manufacturing progress. Changes in tooling, methods and

Pr0duct design that are usually the result of management and

39
engineering effort rather than operator learning in any

sense, have been found much more significant.

Terminology. An understanding of the following termi

nology will facilitate further discussion:

(i) Manufacturing Progress Function Hours - Those hours

 

over and above the estimated hours which are caused
by the introduction of a new unit into a manufac—

turing system.

(ii) Manufacturing Progress Function Cost - The cost

 

associated with the MPF hours.

(iii) Ultimate Unit pf Production - That point in cumulg

 

 

tive production at which the reduction in manufac-
turing hours per unit from the first unit in the
month to the last unit for the month is between 2%

and 3% and thus can be considered nominal.

(iv) Ultimate Hours - The estimated hours required to

 

produce the unit at the ultimate unit of productnni.

(v) Standard Parts - Component parts used in previous

 

machines.

(vi) Labor Value Index (LVI) - A measure of a component

 

part value determined as follows:

Estimated unit Parts usage No.of machs. produced
x x
hours at ultimate vper machine per month at ultimafla

LVI

u

40

Problem of Aggregation. In order to make the study,
it was necessary to accumulate a great deal of historical
data on manufacturing and assembly hours for the machines and
units under consideration. An analysis of the data showed
that different types of operations exhibited varying progrems
trends. As a result, the operations were broken down as fol-
lows :

(1) Final Assembly Operations: (a) Mechanical Assembly;

(b) Wiring;(c) Inspection, Test and Clear trouble.

(2) Sub-Assembly Operations: (a) Single-product deparp

ments; (b) Multi-product departments.

(3) Manufactured Parts: (a) Standard Parts; (b) New
Parts, further subdivided into: New Parts with

LVI > 140 and New Parts with LVI 5.140.

These breakdowns are referred to as categories. In
analysing the data further for determining the MPF curves,the
data was handled by categories for a machine. For example ,
when studying the trends in cost reduction for parts having
LVIgl40, all parts for a given machine having LVI_<_140
were handled as one figure, rather than as individual
components.It was felt that the study of separate components
WOUld tend to give misleading results, whereas a group of

Parts would be more representative of the category.

The intent was 59 break the labor content down into

 

EéEggories that have reasonably uniform behavior with regard

41

E2.ERE £222.2£ progress. Such classifications, however, are
subject to the criticism that for application of the function
much subjective judgement is required until more knowledge fl;
gained.51

Form of the Model and Parameter Determination. From
the investigation of historical data Schultz and Conway
adopted the Crawford model for the manufacturing progress

curve 2

Y = ax (2.11)

where :

‘13
II

the direct-labor hours required to produaa
the initial unit of production.

b = a constant dependent upon the rate of
progress.

x = cumulative unit of production

yL= lot-average direct-labor hours per unit of

output.

This same curve when plotted on full logarithmic paper he-
comes a straight line. Such line will be designated by the
term progress line. The method of least squares was used so
as to obtain the optimum fit of the progress line to the raw
data. To this end, a code was developed in order to compute
the following information for g_ac_h of thezaforementioned catg
gories: (1) The a parameter; (2) The p parameter; (3) The
Value of y for the thousandth unit;(this permits the plotting

);(4) The

0f the least squares trend line between a and leOO'

42
Wright "slope" percentage; (5) The confidence limits; (6) The

coefficients of correlation of the data.

Plotting the Progress Curve for a New Machine. -The
application of the "slope" percentages thus derived requires
judgement for assembly operations. Since such values were
obtained mainly from the study of a particular machine (call
it the basis machine), all new machines to which a manufactup
ing progress function study is to be applied must be compared

to the basis machine as to complexity and novelty, as well as

 

to any similar predecessors to the machine under consider-
ation. Empirical tables were developed that permit such com-
parisons and the final selection of the Wright "slope" for

a given category of operations. In such tables, the greater
the camplexity of the focused machine as compared to the
basis machine, the greater the rate of progress to be expe-
rienced in future manufacturing.Also, the greater the similap
ity with respect to a predecessor or to other machines pro-
duced, the smaller will be the rate of progress. Two other
factors, namely the amount of tooling completed prior to

initial production, and the rate pf production at the ulti-

 

mate month influence the "slope" determination for some cate_
gories. In the referred tables, the greater the percentage
0f tooling completed prior the inception of production, the
smaller will be the future rate of progress for the category
involved. In addition, the greater the rate of production

scheduled for the ultimate month, the greater the rate of

Progress.

43

The next problem in plotting the progress curve for a
new machine is the determination of the ultimate unit of prg
duction. As a result of the study carried out by the Cornell
Group, the following locations of ultimate units were recom-
mended: (1) Manufactured Parts - 12 months; (2) Final Assem-
bly - 18-24 months. Increased complexity, rate of production
at ultimate and novelty of a machine will dictate the ulti-
mate units at or near the 24 month end of the range. These
relationships are presented in tables similar to those em-
ployed in selecting the wright "slope" for the category.

52
In a paper, Schreiner describes the technique used

at IBM-Endicott to apply the Manufacturing Progress Function
Procedure developed by the Cornell Group to their manufactup
ing activities. In order to plot the progress curve for arm»:
machine it would be sufficient to know the values for the
parameters §_and p. However,this procedure requires the knowl
edge of the direct-labor hours consumed by the production of
the initial unit (i.e., the geparameter). Since knowledge of
the manufacturing progress function cost is required p319;

to the inception of production,this information would not be

available. A different approach is used instead.

When a machine has been designed and tested, Cost Eng;
nearing prepares an estimate on the machine (using,for exam-
ple, predetermined times). These estimates are carried out
for each component, sub-assembly, and final assembly by ope;

ation assuming optimum conditions, i.e., the Methods Engineer

44
considers all tooling complete and operator learning also
complete. The estimates are then separated into the foremen-
tioned seven categories. Next, considering one category at a
time, the direct-labor hours for the category are totalled
This value, yu, is represented in Figure l by the horizontal

53
line at which the progress curve should level off.

The next step is to determine, in units of production,

when the progress curve is expected to level off, that is,to

9)

Category I

  
   

b = Tan ¢

Mfg.Hours/Unit
<

X
1.14 X

 

Cumulative Units of Production

Log-Log Plot

FIGURE 1

Plotting the Progress Line For a Given Category

45
select the ultimate unit of production. This is done by
comparing the machine being studied to predecessor machines
for which the progress rates are known . As mentioned be-
fore, three characteristics are employed in the selection
of ultimates : (1) The relative complexity or difficulty
of the operations ; (2) The novelty or newness of the
machine as compared to earlier operations ; (3) The number
of units to be produced per month at ultimate . These char-
acteristics permit to obtain the entries to the proper
tables of ultimates , already referred . Having selected

the ultimate unit of production x it is now possible

u 3
to locate point (xu , yu) of the progress line (see

Figure 1).

After the mathematical slope of the progress line
through point (xu,yu) is determined, it will be straightfor-
ward to draw the progress curve. Again the focused machine is
compared to previous machines as to complexity of operations
and novelty. The comparisons together with the expected rate
0f production at ultimate yield entries to the proper tables
Of "slopes" already mentioned. Since the Wright "slope" is
given by equation (2.3), it is easy to see that the angle ¢
between the progress line and the horizontal axis can be

calculated by:

 

¢ = tg-1[log(wrigh§o: glope {] (2.12)

46
The knowledge of angle (1) makes it possibleto draw the
progress line through point (xu,yu) and to determine the Q
value for the category (Figure 1). Knowing the g and p param
eters for each of the categories it is possible to draw the

progress curve on arithmetic paper (Figure 2).

Hrs/Unit

4' Mfg.

c2

 

 

 

Cumulative Units of Production

FIGURE 2

Progress Curve in Arithmetic Graph

Airframe Companies Publications

Cost-estimating manuals and the results of progress-

-Curve studies have been prepared by the Glenn L. Martin Com
55

54
panyq by Northrop Aircraft, Inc., North American Aviation
56 57

Inc., and the Boeing Airplane Company.

The Martin. and Northrop studies were designed to trafi1

company personnel in the use of the cumulative average curve.

47

The North American publication was prepared for the instrug
tion of the purchasing personnel in the use of the learning
curve in procurement of subcontracted parts. It is a concise
statement of how a purchasing agent may apply the learning
curve to the many procurement problems which arise in the
course of his duty. The latter three studies are similar to
the Wright presentation, and hence, will not be discussed

further.

The Boeing publication which was authored by W.F.Brown
presents the learning curve as it is used by Boeing, in relg
tively easy to understand form. In speaking about the produg
tion of automobile bodies, the author holds that whereas time
reduction continues to take place even after a large cumula-
tive output has been reached, this reduction will be insig-
nificant and may take a long time to realize.58 This is
contrary to the opinion held by Boeing personnel in the past,
which was:after a certain number of units have been produced,
the progress curve reaches a plateau.59

It should be remembered that most of the publications,
do not contain empirical data to support statements made ,

bUt presumably, the statements are based on company

H 0
experience".

Studies Financed by the Air Force

The RAND Corporation of Stanta Monica, California,under

contract with the Air Force, has prepared several studies of

fi‘l‘. “’1‘
v- .
u.» Is‘

.
'n! r?
'F .

,u- ’v -.

eP v
1. u...

.- an.

-.a I.

a
--5 no»

4
1....
a 4”

~.-_“

' n
"'v- q.

u
'D‘ .1. _

'-
..,’

....“m:

2‘: f' '

-,‘

.“

.
.0
1.3.”
' 1
b_‘

 

 

',
h
J

(I)
[U

48
the progress curve based on the same data that were used in
the Crawford-Strauss and Stanford studies already reviewed.
Of interest to this dissertation are the studies by Alchian

and Asher, to be summarized in the following paragraphs.

60
A.A1chian's Study . This study investigates several

characteristics of the new airframe startups that ocurred
during World War II. In regards to the progress curve,
Alchian poses several questions, the most pertinent of which
are (1) How long does the decline in unit labor hours contip
ue for a given model? (2) Does the progress curve correspond
fundamentally to a linear function on log-log scale? and

(3) Does one progress curve with- given parameters 3 and p
adequately describe the consumption of direct-labor hours per

pound of airframe for all models?

As to the first question, no cessation in the manhour
decline was evident to the author. This conclusion was based
on visual examination of the graphs presented in the Source

Book pf World war II Basic Data: Airframe Industry, Volume I,

 

 

 

already cited.

As to the second question, no attempt is made in
Alchian's study to establish whether or not a suitable altep
native exists to the linear progress curve because of the
inadequate amount of data available for the study. It should
be added that fitting the Crawford model to the data via
least squares regression analysis yielded correlation coeffi

cients exceeding 0.80 in the twenty-two startups analyzed.

49

In answer to the third question, Alchian's study re-
veals that the slope and height of a single progress curve
do not represent the unit man-hour expenditures required for
all aircraft models. Different values of parameters 3 and p
were found for different airframes. Moreover, classifying
the airframes by type (fighter, bomber, and so on) still
resulted in diverse g and p_values within each class. These
findings appear to erase any doubt about the validity of the

"universal 80% — learning curve".

HLAsher's Study . In the second RAND study, Asher at—
tempts to demonstrate that the progress curve in linear terms
does not describe accurately the relationship between unit
manhours and the cumulative output. 1 To show that the prog-
ress curve departs from linearity after a certain cumulative
production number has been reached, Asher examines hourly
data for a number of individual producing departments. It is
concluded that the linear approximation is reasonable for all
departments for an initial quantity of airframes. However,

the different departments exhibit non-similar slopes for'these

 

linear .segments . If an analysis of actual data reveals
that departmental progress curves do, in fact, have signif-
icantly different slopes from each other, then the unit curve
(the sum of the departmental curves) cannot be linear.Instead,
With linear but nonparallel departmental curves, the compos-
ite unit curve must be convex on logarithmic grids and must
aP'Proach as a limit the flattest of the departmental curves.

This mathematical consideration alone is sufficient to

50
demonstrate Asher's point of view. He then, proceeded to sum
the department curves. The aggregate unit curve shows that
it begins to level off at approximately unit 125, and the
author claims that if a linear extrapolation was made between
units 100 to 1,000, the estimating error would be around 25
per-cent. With some reservation about the limited samples
examined in the study, Asher concludes that beyond certain
values of cumulative output, the progress curves examined
develop convexity and thus the conventional linear progress
cannot be considered an accurate description of the zglatiop
ship between unit labor hours and cumulative output.

In examining the continuous-declining characteristics
of the progress curve, Asher found definite discontinuities
in the individual shop data at high cumulative outputs. In
addition, the discontinuities persisted even when the data
from different departments were aggregated into total labor

hour expenditure for an airframe.

Asher also investigated the "slopes" that occur in
different producing departments. Using the Crawford model,he
fOund that an aggregate of sheet metal work, machine shop
Work, and materials processing are characterized by "slope"
Percentages varying from 76% to 87%. Major and final assembly
Work exhibited a faster rate of manhour decrease, yielding

"slopes" between 69% and 75%.

The studies by Alchian and Asher, together with Hirschfis

study in the machine-tool industry.may represent the most

. 1
"PA’

p
Janis

IR... ‘
“N

own...»

3 ~

.Iu.’ ~

"' 5.11“

I

In. '

, .

:~ 1";

‘4 5...,

V I
'A

p
.35 ..
u
.H
‘nt ‘1
\‘

51
objective and rigorous empirical investigation of the prog-

ress curve concept produced in the fifties.
Other contributions

Two other contributions are worth mentioning in this
connection, namely the articles by Koen63 and Andreas.64

Francis T. Koen, of the Missile Systems Division,
Raytheon Company, uses the manufacturing progress function
as the basis for what he calls "Dynamic Evaluation". With it,
he is able to (l) predict production trouble, (2) estimate
manufacturing costs, (3) check estimates and budget perfor-
mance, (4) predict personnel requirements, (5) help determine
make-or-buy decisions, (6) determine relation between

standard cost and quantity values, and (7) establish budgets

and optimum production schedules.

Frank J. Andress believes that "product innovation" is
one of the primary criterions upon which the usefulness of
the manufacturing progress function should be based. This
includes situations where both major and minor design changes
are often incorporated while the product is in production,
Where new products are frequently introduced, and/or where
there are frequent production runs at well-spaced intervals.
Sueh companies would be at the upper end of the manufacturing
Progress function and could, thus, realize the maximum advap

tages of the improvement rate.

52
THE SIXTIES AND SEVENTIES

In the last fifteen years there is evidence ofairenewed
intepest::inl the subject of progress functions, particulap
1y outside the airframe industry. The contributions in this
period are of a somewhat diversified nature. There are
pioneering extensions of the concept to machine-intensive
industries as well as to other labor-intensive industries
not studied before. At least one leading company in the field
of electronic data processing systems continues to develop
and apply the theory of progress functions in its manufac-
turing operations. The progress curve is also demonstrated
to be valid in service activities like overhaul and mainte-
nance. Furthermore, their users seem more careful in iden-

tifying its caveats and possible deviations.
Extension to Machine-Intensive Manufacture

Baloff is best known for extending the use of the
learning curve model to machine-intensive production
65
sYstems as well as to other labor intensive manuface

66
tures different from the airframe industry.

He suggests the use of a modified version of the model
in highly mechanized manufacture (steel, glass-manufacturing,
Paper products and electrical-products)? Some years later
the startup model is extended to three examples of labor
intensive manufacture - automobile assembly, apparel man-

. 68
Ufacture and the production of large musical instruments.

53
Baloff is highly skeptical with respect to past ap-
proaches used in parameter estimation. As indicated earlier,
only a few solutions to the parameter prediction have been

guoposed in the literature.

A well known approach to the estimation of the p param
eter is to assume simply that its value will remain constant
for all startups in an industry, regardless of changes in
product type, processing facility, or company origin. This
assumption, which was apparently quite popular in the air-
frame industry at one time, continues to find support in the
literature.69 The violence to reality that results has been
stressed by Baloff.7O The assumption of a constant p_paramg
ter is inconsistent with the results of empirical examina-
tions of a large number of airframe startups that took place

71
during World War II and in postwar years.

A more refined approach is based on the assumption that
the startups of similar products or processes will experience
identical startup curves. This approach essentially uses the
empirically derived parameters of past startups as best esti
mates of one or both of the parameters in a future startupcxf
a physically similar product or process. Unfortunately, expg
rience suggests that variations in parameter values are not
to be explained this simply. It has been shown by Baloff
that the startups of steel processes that are physically very

Similar exhibit significantly different i and p_parameters,

54
even if the comparison is restricted to a single company in
72
the industry or to a single plant. Interplant comparisons

of the startups of certain classes of airframes have yielded
equivalent findings.73
A third approach that has been mentioned in the lite;
ature represents a refinement of the similarity concept.
Here the focus is on recognizing and evaluating variables or
factors that influence the values of one or both of the parap
eters for a given startup.74 An evaluation of these factors
in relation to experience with past startups could then
puesumably serve as a basis for generating estimates of the
umdel parameters for a future startup. According to Baloff,
there has been no published account of the reliability of
this type of factor approach. Lacking such information, its
Utility in practice must remain an open question.75
Baloff argues that a different approach to the param-
eter estimation problem is suggested by the existence of a
s"trong relationship between the g and 13 parameters among
different startups in the steel and airframe industries and
ill 'the results of a laboratory research on group problemr
sOlving. Though still tentative, this relationship appears
it‘3 ‘hold promise of being developed into an objective means

()1? estimating the p parameter of the model, given a measurg

BEIGEtit of the initial productivity of a startup.

The existence of the relationship was initially reporp

eaci ‘bw'Asher in the airframe industry nearly two decades ago.

55
Asher found a strong direct correlation between the g and p
parameters of the startup model among 12 postwar startups of
fighter-class airframes. The strength of the relationship

observed by Asher was notable, allowing him to fit the fol-

lowing model to the 12 pairs of parameter values:
log b = log m + n log a (2.13)

Following the lead of Asher, Baloff extended this
parameter model approach to the steel industry and to the
glass manufacturing industry.77 The results of a laboratory
experiment on group problem-solving also provided Baloff
evidence of an inverse correlation between the parameters of
the startup model in a learning situation that is similar to
an industrial startup.78 Later he would come to find support
for the parameter model approach in laboratory experiments

79
With group adaptation to a business game and in automobile
80

Startups.

Baloff's empirical contributions may well represent a
tnI‘ning point in the theory and application of the manufac-

tut‘ing progress function.
COutributions by IBM Personnel

After the Schultz and Conway study already reviewed in

an earlier section, IBM personnel became very active in the

--

S‘~:'-1>ject of progress functions. Articles by P.B. Metz, J.G

E<rleip and J.H. Russel will be briefly commented on in the
F

\‘

0 1 lowing paragraphs .

 

SW‘.‘

o nJ.

 

nQn-

View

0;.-
I

dun.

II.
(I)

Gr...

«0;.

 

 

.
‘3‘
\‘.

 

 

56
The Metz Nomograph. In 1962, Philip Metz presented

a nomograph designed to simplify and expedite the application
of the unit progress curve.81 In essence, the nomograph is a
consistent means of estimating the direct-labor hours requing
ment per cumulative unit relative to the known requirement of
a specific cumulative unit. The values of three parameters ,
namely, (xu, yu) and the wright's "slope" percentage are
prerequisite to the application of the progress curve. The
point (xu, yu) represents a known point in cumulative produg
tion. A typical problem is to determine the direct-labor
hours for specific units of pre-defined lots and the average
direct-labor hours for given lots. The analytical solution
consists in calculating segments of areas under the progress
curve through integration between known limits. The nomograph
expedites the computational effort. Moreover, the probability
of a computational error is lessened by the reduced number of
required calculations. According to Metz, the accuracy of the
0btained solutions is at least commensurate with the proce-

dLures normally employed in establishing the required parame-

tetrs.

The Kneip Maintenance Progress Function . ~ In order to
e1Sfect a transition to a maintenance progress function,J,G,Knej_p
Ina'l<:es the following adjustments in concepts: (1) Whereas the
Ink'Et‘n‘ufacturing progress function describes the characteristflx;
(’15 a system through which all of the products must flow, the
11'laintenance progress function describes the system through

which each unit of product flows according to some probability

 

w.
4
.u

*v

57
distribution. That is, a unit may not require service or it
may require a number of service calls; (2) Whereas in the
manufacturing progress function the dependent variable (cost
or labor time) is a production efficiency characteristic, in
the maintenance progress function the dependent variable is
a quality, reliability and serviceabilitv characteristic.82

Having in mind these modifications Kneip adopted the

following model:

v = axb (2.14)
where:

y = the average of the maintenance required.on
a sample of machines from each month's
production of a given production period,
hggrg per unit of production.

a = the value of y for the first unit

x = the cumulative production

b = parameter dependent on the rate of progress.

It! order to evaluate the applicability of the modified model
iJJI describing the improvement in maintenance requirement,two
products, an electro-mechanical machine and an electronic
Ina(Shine were selected.The data on the first product based on

1:}1€3 first twenty-seven months of the product's manufacture

‘

¥

W'EEI‘e taken from the field service histories for the ninety-
day warranty period.The second product data were obtained from

t3V9t) sources: (i) Development models which were mechanically

«a:
.
.w

 

58
operated, simulating an office installation of an equivalent
ninety-day warranty period and (ii) production models in
actual customer installations. These data were derived from
the warranty service reported during the first eight months of

the product's manufacture.

The data were fitted to the logarithmic function by the
method of least squares. 0n the strength of the results of
the regression analysis Kneip concludes that the maintenance
progress function exists, that it relates warranty period
maintenance to cumulative production, and that the relation-

ship is represented by equation (2.14).

Any manufacturing organization which assumes some re-
8ponsibility for its products through a warranty period shouki
find the maintenance progress function of value in predicting
Warranty maintenance requirements on new products and in the
analysis of the maintenance experience on development models

When a maintenance criterion must be met by future production.

83
The Russell Study.The purpose of.J.H.Russell's article

‘153 to explain the deviations from the log-linear model that

result from the addition or subtraction of parallel production

1&11133, A system of parallel lines operating at a fixed rate
84
(>15 learning is used in a computer program to observe the

Ireissults of varying the number of lines employed. The result
jLEB the generation of a specific progress function model for
Eaéicth product profile thus simulated. Therefore,the progress

islltiction can become more useful when the model for the

 

C‘,

q-

.l

59

specific product is predictable.

From the simulation of various product profiles Russell
concludes that the addition and subtraction of parallel lines
of production do create significant deviations in the progrems
function. The author believes that these changes form the
major trend lines of the progress function as applied to a
specific product. In addition, Russell points out that the
effects of major product improvements, operational problems,
inventory policy, lead-time changes, and modifications in
accounting method will also cause deviations to the progress
function. Each should be evaluated separately and its effect
overlaid on the trend lines of the product profile. The study
by Russell represents a pioneering effort towards using simp.

lation methodology in the investigation of adaptation phenom

ena.
0ther Contributions

For their intrinsic interest and relevance to the field
0f manufacturing progress functions, four more contributions

wi-Il.l be concisely reviewed in the subsequent paragraphs.

85
Setting Management Goals‘. James M. White of

S‘tevens Institute of Technology, believes that the manufac-
turing progress function is highly useful in almo‘st‘any situa-
t:ion in which there is some criterion by which to measure the
improvement phenomenon,and which is initially in what Would be

<3<>Ilsidered an "uncontrolled state". Possible cases that would

60

fall within this category are:(l) Reduction of losses due to
waste, (2) reduction of scrap and rejects, (3) decreasing
accident rates, (4) increasing capacity because of poor plan
ning and control of resources, (5) reducing clerical errors,
and other situations. In his article White demonstrates how
to use the progress curve in setting goals for improvement in
waste control. The other cases above mentioned are potential
situations where progress curve theory could be used to
predict the expected rate of improvement thus permitting the

achievement of worthwhile results in management planning.

86
Multi-Product Industries. Paul F. Williams , an in-

dustrial engineer for United Control Corporation, holds that
the manufacturing progress function can be a very beneficial
tool for multi-product industries. He claims there are two
general classes of application of the function. The first is
fOr use on "initial quantities" of production. This would be
uSed for scheduling a new product, or one on which insuffi-
‘Iient records were kept by which to make a manufacturing
jpI-‘Qgress function. The second classification would be for
t'fiallow-on-quantities". This would be used for a product
pI‘esently being produced,or one on which sufficient records
T”ere kept by which to make a progress function . In addition,
t:11£a function is valuable for the following reasons : (1) It
EDITCJVides a systematic, consistent , and objective method
(’13 forecasting production information ; (2) It can be
llssqu to estimate average production costs of both "initial"

El'I'ld "follow-on-quantities";(3) It is a graphic technique and,

bu

61
thus, easy to use; (4) It can be used by management as a yani
stick to measure manufacturing performance, and (5) When it

is properly applied, it can minimize the error of estimatimn.

Petroleum Refining. WLB.Hirschmann,believes that the
progress curve is an underlying natural characteristic of
organized activity, just as the normal curve is an accurate
depiction of normal, random distribution of anything, from
human I.Q.'s to the size of tomatoes.87 He plotted the per-
formance of individual catalytic cracking units at a point
in time against their age at that time. The dependent vari-
able was current capacity expressed as a percentage of
design capacity. The first unit in 1958 was one and one-half
years old and by that time had achieved approximately 116%
0f design capacity. The second unit was four years old and
had achieved about 125% of design capacity. The older units
Show that the performance rapidly improved in the first few
Years, and continued at a slower rate in later years.Another

Sllotting of successive annual points for an individual
cracking unit indicates growth occurring in a step-wise

fashion. However, the pattern of improvement resembles the

ilrrverse of a progress function on arithmetic paper.

If the parameters are changed so that the number of
(1&1378 to process 100,000 barrel is plotted against cumulated
t:}11:oughput on a logarithmic paper, a declining straight line

(:Eitl be drawn through the points as indicated by Figure 3.

 

62

 

‘ A3. of Unit (you-a)

Days Par 100,000 Barrels

 

 

 

 

o v r v v v vvvr V7 r v vfirfr

o 10 100 moo ' ' -- '"
Canaan's 14111101: Bun-01- Run

' ' ' ' "1'. 7

FIGURE 3

Progress Line For a Catalytic Cracking Unit

This Line has a slope of about 9070, as might be expected from

a machine-paced operation which involves comparatively little

dinsect labor.

Hirschmann further states that a manufacturing progress
fliltlction can be determined for any industry. A logarithmic
plot of man-hours per barrel versus cumulated barrels of
Qt‘ude oil refined in the United States since 1860 was made.
He found similar declines for the United States basic steel
industry and for the United States electric power industry.

Hirschmann'ssmdy indicates that the manufacturing progress

63

function is not only applicable to the aircraft industry,but
also to non-aircraft industries.Together with the studies by
Baloff, it represents a pioneering effort towards extending

the progress curve model to highly automated industries where
the adaptation phenomenon might be thought to be either non-

existent or too small to be of value.

Limits of the Progress Curve. An article by W. J.
Abernathy and K. Wayne looks at the progress curve in a new
way.88 It shows the unforeseen consequences of following the
strategy of reducing costs in a product through steady in-
creases in volume: rising fixed costs,a narrowly specialized
work group, and a withered capacity for innovation, to name
just three. To illustrate the changes that accompany a
cost-minimizing strategy, the authors use the case of Ford
Motor Company and its model T. The kindscfifchanges that took
place can be grouped into six categories - product; capital
equipment and process technology; task characteristics and

process structure; scale; material inputs; and labor. Each

category is briefly described as follows.

(1) Product: Standardization increases, models change
less frequently and the product line offers less
diversity. As the implementation of the strategy
continues, the total contribution improves with
acceptance of lower margins accompanying larger

volume.

64
(2) Capital Equipment and Process Technology: Vertical
integration expands and specialization in process
equipment, machine tools,and facilities increase.
The rate of capital investment rises while the

flexibility of these investments declines.

(3) Task Characteristics and Process Structure: The

 

throughput time improves and the division of labor
is extended as the production process is rational-
ized and oriented more toward a line-flow operation.
The amount of direct supervision decreases as the

labor input falls.

(4) Scale: The process is segmented to take advantage

of economies of scale.

(5) Material Inputs: Through either vertical integra-

 

tion or capture of sources of supply, material
inputs come under control. Costs are reduced by
forcing suppliers to develop materials that meet

process needs.

(6) Labor: the heightening rationalization of the
process leads to greater specialization in labor
skills and may ultimately lessen workers' pride in

their jobs and concern for product quality.

The authors point out that the same pattern of change in the
six categories that characterizes Ford history also describes

‘Periods of major reduction in other industries.

65

Implications for Management. According to Abernathy
and Wayne, management needs to recognize that conditions sthp
ulating innovation are different from those favoring effi-
cient, high-volume, established operations. The unfortunate
implication is that product innovation is the enemy of cost
efficiency and vice-versa. The authors point two courses of
action that some major companies have followed. One is to
maintain efforts to continue development of the existing high-
-volume product lines. This requires setting the industry
pace in periodically inaugurating major product changes whiha
stressing cost reduction via the learning curve between model
changes. This course of action, exemplified by IBM, amountstn
maintaining comparatively less efficient overall operations.
The second course of action is to take a ,decentralized ap-
proach in which separate organizations or plants in the cor-
porate framework adopt different strategies within the same
line of business. One organization in the company will pursue
profits with a traditional product to the limit of the learp

ing curve while othersvfill.develop new products and processes.

SUMMARY

The present chapter contains a review of the histori-
cal development of the manufacturing progress function and a
summary of the more important contributions to the progress

curve literature that are relevant“ to this dissertation.

66
Although the progress curve was discovered in 1922, it

was largely unknown until World War II.

T.P. Wright is given credit for originating the formu-
lation of the progress curve theory in 1936. His statement,
that cumulative average man-hours per unit decline by a
constant percentage every time the output is doubled, re-

mains the most popular formulation in existence.

In the forties, a number of modifications of the origi
nal model were proposed. However, empirical verification of
the modified models has been fragmentary and they have found
little acceptance. In spite of this, Crawford contributedtim
"unit" learning curve and noticed that different "slopes"
might exist for different airframes. Ie is probably the
first author to perceive the link between rate of progress ha

a job and its degree of complexity and novelty.

Several relevant contributions to the development of
progress curves were published in the 19503. There is an
effort by some researchers like Hirsch, Bryan, Schultz and
Conway to extend the concept to labor—intensive industries
other than the airframe industry. In addition, studies con-
tinued to be financed by the United States Air Force in.some
research institutes of the country. The studies by Alchian
and Asher, in the airframe industry,together with Hirsch's
Study in the machine-tool industry,and Schultz and Conway's
research in manufacturing of electronic and electro-mechani-

cal products may well represent the most objective and

67
rigorous empirical investigation of the progress curve con-

cept produced in the fifties.

The last fifteen years have seen some pioneering extep
sions of the progress curve concept to machine-intensive
industries as well as to diverse labor-intensive industries.
The studies carried out by Baloff constitute the most comprg
hensive investigation of the progress function in the men-

tioned period.

Review of the Main Hypotheses

It is worthwhile to recapitulate the principal hypoth-
eses raised by the forementioned authors and generally

accepted:

(1) Linearity , when in logarithmic coordinates , be-

 

tween the 1abor per unit and cumulative output was

generally confirmed except in Asher's study.

(2) There is no one single progress curve that can be

universallyapplied to all types of operations

 

involved in the manufacturing of a given product.
Also, there is not a curve that .can represent_
the adaptation phenomenon for all products in a

given firm or industry.

(3) Assembly operations experiment significantly higher

 

rates of progress than machining operations.

 

68
(4) As to the newness or novelty hypothesis the findings
are contradictory. According to Crawford the rate of

progress is an increasing function of the complexity

 

of the job and of the lack of experience of the
worker. Schulz and Conway agree with Crawford when
they state that the degree of similarity of a
machine to a predecessor has a significant effect
on the rate of progress; the lggg similarity, the
greater the rate of progress. They also observed
that the greater complexity, the less the rate of
progress. Nevertheless, Alchian and Hirsch found

in diverse settings that the greater the similarity
of a product to a predecessor, the greater the rate

of progress experienced.

(5) There existszistrong correlation between the paramg
ters of the startup model among startups that oc-
curred in a given production facility (Baloff)
and among startups that occurred in different

facilities (Asher).

(6) Plateaux predictability continues to be a contro-

Versial issue.

CHAPTER III

MANUFACTURING PROGRESS FUNCTIONS: A MATHEMATICAL EXPOSITION

The interested reader of the literature on manufactug
ing progress functions will have certainly noticed that the
field lacks uniformity as to mathematical notation and more
precise definition of the variables involved. A number of
theoretical results are stated and used without formal math
ematical demonstration. Assumptions are often implicit in

the derivation of important results.

In spite of being discrete, the manufacturing progress
function may be advantageously treated as a continuous func-
tion under certain conditions. The consequent simplification
achieved in the final formulas and calculations using the
progress function is worthwhile. Nevertheless,these features
have been largely neglected in the literature. Such deficiep
cies and others that will be pointed out in the course of
this chapter have led to a state of confusion in the design,

interpretation and evaluation of related empirical research.

In the present chapter the mathematical treatment of
Progress functions will be rewritten. The systematic exposi-

tion here proposed is probably novel in the literature.

69

70

FOUR TYPES OF PROGRESS FUNCTIONS

There are at least four types of progress functions,al
though they are mathematically related to one another. All
four can be expressed as power functions. They are similar
in form but have different meaning. The following notations,

terminology and definitions will be used in this dissertatflxi
Type 1: The Unit Progress Function

The functional relationship specified for the unit

progress function is:

y = ax'b (3.1)
where
x = the cumulative unit of production
y = the direct-labor hours required to produce
that xth unit in cumulative production
3 and b = parameters; a>0, Ogbgl
Interpretation of Parameters a and b.‘ Parameter i

represents the direct labor hours required to produce the
initial unit of production: for x = l, y = a(l)'b = a. Param
eter 2.13 dependent upon the rate of progress. Geometrically,
it is the slope of the progress line, as already mentioned
in Chapter 2 (p.12). Another interpretation is suggested by
the concept of elasticity of I function. The elasticity of

X at the point g is given by:

(3.2)

and

Replacing dz and E in (3.2) by their equivalents already ob-

dx y
tained, it follows that

Ex - (3.3)

Thus, parameter b_may be called the progress e1asticity,i.e.,
the ratio of an infinitely small relative change in the
direct labor requirement associated with a correspondingly
small relative change in cumulative output. For example, in
the "eighty per cent curve" parameter b is 0.32, i.e., a l
per cent increase in cumulative output is associated with a
0.32 per cent decline in direct labor requirement.Recalling
that the rate of progress has been defined as the complement
to one of the "slope", in the case of the"eighty per cent

curve" the rate of progress is

(l — 0.80) = 0.20

Mei

but

1 n
ma >1
($33.).

‘4”,
J.“

u

72
Therefore, the rate of progress corresponding to a progress

elasticity or parameter b’of 0.32 is about 0.20.

Domain and Range of y. - Usually (but not always) §_is
a positive integer. In some instances, however, 5 may repre-
sent a continuous variable.2 For x = 0, the function is not
defined. Thus, in general, the domain X of the unit progress
function y is a subset of the positive reals. Usually, the

domain X of y is a subset of the set of positive integers.

3
The range set Y of y is a set of positive real numbers.

The function y is the set of ordered pairs (x,y) where
y = y(x) = ax’b. These specifications may be more compactly

expressed by using set theory notation. Thus, generally

X = Dom y = {x : xeR+}
but usually
X = Dom y = {x : xeN}
where
R+= the set of positive reals
N = the set of positive integers: 1,2,3,...
Also
Y(X)= Ran y= {y:y=ax'b, xeR+(or EN), 552+, beI}C'R+
and
y = {(x, y(x)) : xeX}
Where I = the unit interval of real numbers.

73

Type 2: The Cumulative Average Progress Function

The functional relationship specified for the cumula-

tive average progress function is the following:

-b

y = ax (3.4)
where
y = the cumulative average direct labor
hours per unit
x = the cumulative number of units produced
3 and _b = parameters; a>0, 0_<_b_<_l
Meaning of a, b and y, Parameters g and b have the

same meaning as in the case of the unit progress function.As
to the meaning of y, assume a table is available containing
a series of values of 5 and the corresponding direct labor

hours input per unit, y, for each g:

1 X2. . .Xi. . .Xn

y[y1 y2...yi...yn

xlx

 

The cumulative average direct labor hours per unit up to and

including unit xi is defined as follows:

y. 3 i- y. , i = l,2,...,n (3.5)
1 xij

1 J

"raw-

Equation (3.4) means that the hyperbolic rule of correspon-
dence is now between the set of xis and the set of yis where

the yis are calculated according to (3.5).

. ‘4'
,‘—\

(L)

74

Domain and Range of y > Using set theory notation:
X = Dom y = {x : xeR+} (in general)
or
X = Dom y = {x :xeN} (more usually)
Also

: y = ax ,xeR+(or 5N), aeR+,beI}C R+

7(X) = Ran y = {y

y = {(x, y(x)) : xeX}
where the ys have the meaning given by (3.5)

Type 3: The Cumulative Total Progress Function

The cumulative total progress function is expressed by:

yT = axB (3.6)
where
yT = cumulative total direct labor hours
x =the cumulative number of units produced
_a_ and B = parameters; a>0, 05B_<_l
Meaning of a, B, and yT. Again, parameter E repre-

sents the direct labor hours required to produce the initial
unit of production. Parameter B may be seen as the elasticity
0f yT at the point g, As to the meaning of yT, assuming as
before that a table of values of g and y is available, the
cumulative total direct labor hours expended in producing

the first xi units is given by:

C.»

‘A

MH-
'~<

i = l,2,...,n (3.7)

Equation (3.6) signifies that the rule of association is now

between the set of xis and the set of yT 3 calculated
i
according to (3.7).

Domain and Range of yT. Using set theory notation:
X = Dom YT = {x : xeR+} (in general)
or
X = Dom yT = {x : xeN} (more often)
Also

- = . = B
YT(X) Ran yT {yT.yT ax ,xeR+(or eN),a€R+,BeI} C R+

and

YT = {(X, YT(X)) : xeX}
where the yTs have the meaning given by (3.7).

Type 4: The Lot Average Progress Function

The lot average progress function is expressed by:

“ ’b (3 8)
= ax .
yL
where
y: = lot average direct labor hours per unit
x =the cumulative number of units produced
a and b = parameters; a>0 , Ogbgl

76
Meaning of a, b, and y:. Parameter g represents the
direct labor hours required to produce the initial unit of
production. Parameter b is the elasticity of y: at the point

E- The meaning of y: may be approached as follows. Let

xk = the counting from the first unit produced up

to and including the last unit of lot k,

k = l,2,...,m.
Similarly,
xk_1 = the number of units from the first unit
produced up to and including the last unit
of lot (k-l).
Therefore

xk - xk-l = number of units in lot k.

The lot average direct labor hours per unit considering lot};

is defined as follows:

yL = , k = l,2,...,n1 (3.9)
k xk-xk-l

 

In equation (3.8), the rule of correspondence is now between

the set of xks and the set of y 3 calculated according to
(3.9).

 

:a

310'

ave

I «r

77

Domain and Range of y:. Again, employing set theory
notation:
X = Dom y: = {x : xeR+} (in general)
or
X = Dom y: = {x : xeN} (more frequently)
Also

- ‘ _ _ _ _ . = -b
YL(X)—Ran —{y .y ax ,xeR+(or eNlaeR+,beI} C R+

‘4

= {(x,yi(x)) : xeX}
where the yi's have the meaning given by (3.9).

Fitting the Progress Function to Empirical Data . The
crucial problem has been the conformity of the forementioned
functional relationships to empirical reality. It is worth
mentioning that several other forms have been considered in
the literature of progress functions.4 However, the power
function continues to be universally employed by the research
ers in the field. The preference for the power function may
‘be justified on the following grounds: (1) Adequate fitting
to the available data as produced by the firms involved, and
(2) simplicity of utilization, particularly when in the form
10f double-logarithmic graph. In the following section it will

be shown how the four types of progress functions previously

described are mathematically related to one another.

78
FUNDAMENTAL PROBLEMS

Statement of Fundamental Problems

Almost always progress function users have faced the

following four basic problems:

Problem I. Given y = y (x) , determine:

YT = yT(x) . y = y(x), and yL = yL(x)

Problem II. Given y = y(x), determine:
YT = YT(X). y = y(X). and yL = yL(X)

Problem III. Given yT = yT(x), determine:

F = §(X). y y(X). and 5 = §L(X)

L

Problem IV. Given y: yL(x), determine:

yT = yT(X). F = §(X), and Y = y(X).

Whenever the expression "gi_vgp the function. . . is used
in this chapter, it signifies that the focused function can
be statistically fitted to the empirical data and moreover,
that the chosen function is the one that yields the best

fitting of the four types of progress functions already

79

defined. From such point on the other three progress func-
tions may be determined through mathematical analysis, as

follows.
Solution of Problem I

(a) Given the cumulative average progress function
y = y(x), the cumulative total progress function

may be determined by using definition equation

(3.5):

yT(x) = §(x) - x (3.10)

In case the discrete function y = y(x) is approxi-

ax'b, it follows from (3.10) that:

mated by y

l-b
ax , x = 1,2,... (3.11)

yT (X)

(b) The unit progress function will be determined by
subtracting the cumulative total hours consumed
in the production of the first (x-l) units from the
cumulative total hours consumed in the production
of the first 5 units. The difference is exactly am:

the hours consumed in producing unit 5 alone:
= - -l .12
y(X) YT(X) yT(x ) (3 )
or according to (3.11) and (3.12):

y(x) = a [él-b - (x-l)1-€] ,x=l,2,... (3.13)

80
(c) The lot average progress function is easily deter-
‘mined by adopting the same notation of the previous

section. Therefore, according to equations (3.9)

and (3.11):

 

 

‘4

or l-b l-b
a(xk - xk_1)
yL = , k=l,2,...,m,x =0 (3.UD

k xk-xk-l O

 

Using equation (3.14) and given the number of units in

each lot k, k=l,2,...,m, it is possible to calculate a table

Lk

termin I = _ .
e YL YL(X)

of values of

as a function of xk and x and thus to d9

k-l

Using the Continuity Assumption to Approximate Solu-
tions . As it was previously noted, progress functions are
often discrete. Nevertheless it is viable to approximate so-
lutions by assuming continuity of the progress function in
the interval of interest. Take, for example equation (3.13).

For values of g'not very close to the first units of produg

tion:

NIH

y(x) a fox y(x) dx (3.15)

81
Equation (3.15) follows from the definition of the av-
erage or mean value of function y = y(x) over the interval
from zero to x. It is assumed that y(x) is continuous over
the interval of interest.5 Since y = y(x) is being approxi-
mated by y = ax-b, it follows from (3.15) that

X _ l’b
f y(x) dx 5 y(x)-x = ax (3.16)
0

Derivation of (3.16) with respect to 5 yields:

b

a (1-b)x- (3-17)

Ill

y(X)

Equation (3.16) is the same as (3.11). Recall that
equation (3.11) was derived by assuming that the discrete

function y = y(x) could be approximated by y = ax'b.

Calculation of the unit progress function through
(3.17) is evidently faster and simpler than through the exact
equation (3.13) even if machine computation is considered.
Since §'= y(x) is given, it suffices to multiply the ye by

(l-b) in order to obtain the ys.
It is worth mentioning that the function:

f(1og x) = log a + log(1-b) - b log x

is asymptote of

F (log x) = log {a [él-b - (x—1)1-€]}

82

This result is used very often in the literature but no
formal proof is offered except the recourse to geometrical
intuition. In Appendix B this author states it as a theorem
and contributes a formal mathematical demonstration. The ra-
tionale for approximating equation (3.13) by equation (3.17)
(for values of 5 not close to the first units) stems from

this theorem.

Solution of Problem II

(a) Given the unit progress function y = y(x), the cumg
lative total direct labor hours up to and including
unit g is given by definition (3.7):

yT(x) 3 z y(x) x = 1,2,... (3.18)

x
In case the discrete function y = y(x) is approxi-

mated by y = ax-b, then:

yT(x)= Z ax-b= a x-b, x= 1,2,... (3.19)

)3
x x

(b) The cumulative average progress function is deter-
mined through definition (3.5) and equation (3.18):

y (X) £34")

T = -—-———— (3.20)
X

 

370:) =

Therefore, in case y=y(x) is approximated by y=ax-b,

the cumulative average function is given by:

83

-b

y(x) =3 x , x = 1,2,... (3.21)

X
x

(c) In order to determine the lot average progress

function, given the unit progress function, let:

xk the counting from the first unit produced

up to and including the last unit of lot 5,

k=l,2,...,m, xo=0.

xk-l = the counting from the first unit produced
up to and including the last unit of lot

(k-l).

In case the discrete function y=y(x) is approxi-

mated by y=ax-b, equation (3.19) yields:

and

The total direct labor hours expended in lot k.is
obtained by subtracting the last equation from the

first one:

yT = x“b - 23 x ) (3.22)
k

"k xk-l _b
a(2
l

84

(xk_l+j) , k=l,2,...,m,x0=0 (3.23)

The lot average direct labor hours for lot k_wi11

be given by:

_ -b
a .
y = -——————- Z (xk +3) (3.24)
Lk xk xk—l J=1 1
Given the number of units in each lot §,k=l,2,...mr

it is possible to calculate a table of values of

yLk as a function of xk and xk-l through equation

(3.24) and thus to determine yi= yi(x).

Approximating the Summation Formulas by Integrals.
The formulas involving summations may be approximated by

improper integrals. Take, for example, formula (3.19):

x _ x _ -b
x b= a lim f x b = 1%E-x1 (3.25)

Y (X) s a f
T 0 x1+0 x1

Using definition (3.5) and the approximation in (3.25%
the cumulative average progress function given by equation
(3.21) may be expressed as:
y (X)

T a -b

—— x (3.26)

_ A
y(X) — l-b

 

~
-
-

85

Also, it follows from equation (3.25) that:

 

 

l-b
Tk(X )~ _I_— xk
and
~ a x1-b
yT(xk-l) - 1 xk_1
Thus, l-b
_ __ YT“? ' yT(-"1<1 _a("1<
YL - ’
k xk xk- 1

Solution of Problem III

(l-b)(xk-xk_1)

(3.27)

This problem is trivial once the solution to problem I

is presented.

(a) Given the cumulative total progress function,

yT = yT(x), the cumulative average progress func-

tion is merely:

YT(X)

lll>

 

F (X)
X

In case
YT

yT(x) is approximated by y

(3.28)

‘ ax the

T -' 9

cumulative average progress function will be of the

form:

y(x) = ax _ , x=l,2,...

(3.29)

(b) The unit progress function may be obtained as in

Problem I:

86
Y(X) = YT(X) - YT(x-1)

Given that yT= yT(x) is of the form yT = axB ,

y(x) = a £43 - (x-1)B:] ,x=l,2,... (3.30)

(c) The lot average progress function is determined by
the same method already explained in the case of

Problem I. The resulting expression is:

B B
_ 8(Xk - Xk_1)
Y = , k=l,2,...,m, x = O (3.31)

Lk xk xk-l

 

Solution of Problem IV

(a) Given the lot average progress function §i=yi(x),
the total direct labor hours consumed by the manu-

facturing of lot k is given by:

ka= yLk(xk - xk_1) , k=l,2,...,m, x =0 (3.32)

The cumulative total direct labor hours since the
inception of production up to and including unit

xk (the last of lot k) is, therefore:

k

= 2 — .- , k=l,2,..., , =0 3.33
YT(xk) j=1 ij(xJ xj_1) m x0 ( )

87

In case §L= yi(x) is being approximated by yi=ax'b,

the cumulative total direct labor hours from unit

number one up to and including unit x is,according

k
to (3.33):
k -b
yT(xk)— ajil xj (xj-Xj-l)’ k=l,2,...,m, xo=0 (3.34)

(b) The cumulative average direct labor hours per unit

up to xk will be, by definition:

y(X)
— A
y(xk)=-I—-13— (3.35)
xk
If y: = yi(x) is being approximated by §L= ax-b,
then, according to (3.34) and (3.35):
_()=-§-1§x-b(-x )k=12 mx=0 (336)
yxk x = o x. ’- , ’ ”"’ ’ O '

k j 1 J J J 1

(c) Given the lot average progress function yi=yi(x),
the unit progress function cannot be exactly deter-
mined. However, an assumption can be made, i.e.,
that the lot average direct labor hours considering
lot k'is approximately equal to the lot average
direct labor hours considering lot 5 deleted by its

last unit xk. Symbolically,

88

YT YT - y(xk)

 

_ A g
yL = -—-:%3——— ' E 1 _ (3.37)
k xk xk-l Xk xk-l
If (3.37) holds, then:
y(xk) a yLk (3.38)

i.e., the unit progress function can be approxi-
mated by the lot average progress function. An al-
ternative way of checking the validity of (3.38),

given the same assumption (3.37) is as follows.

The cumulative total direct labor hours up to

and including unit xk is, according to (3.33):

k
y (x ) = E y (x - x. ) =
T k j=1 Lj J J l
k-l _ _
= 2 ,- , + x -x 3.39
j=1 YLj(xJ XJ-1) yLk( k k_1) ( )

In view of (3.37), the cumulative total direct
labor hours up to and including unit (xk - 1) may
be expressed as follows:

k-l

YT(xk- 1) s jil §Lj<xj- xj_1) + §ik(xk- 1 - xk_l) (3IED

89
Subtracting (3.40) from (3.39) yields the same re-
sult as in (3.38):

Y (xk) a yL
k
In case yt = yL(x) is approximated by yi= ax'b,then:
-b
y (xk> z axk (3.41)
Note.- : Problem IV as well as item (c) of Problems I,

II, and III are not formally treated in the literature. The
unit progress function and the lot average progress function
are taken as the same model. The approximation and the assunp
tion behind it are not explained. The confusion may have its
origin in the fact that the unit progress function is rarely
encountered in a real world situation. Labor expenditure is
typically recorded on a monthly, as opposed to a per-unit
basis. As a result, direct labor hours data are normally cal
culated and reported in relation to standard accounting time
periods, yielding average direct labor hours per unit figures
for the "lot" of product produced during the accounting
period. Also, cumulative output statistics (x) indicate the
total output of the product (from inception of manufacture)
that is achieved at thg gag of the accounting period. Thus,
What exists in practice is the lot average function §L=yi(x)

as previously defined, not the unit function y = y (x).

90
Nevertheless, in the literature the lot average function is

designated by the term "unit function"...

PARAMETER CALCULATION

Application of the progress function to practical prob

lems involves two requirements:

(1) It must be known which type of progress function

 

best fits the empirical data. In practice the cumg
lative average progress function and the lot aver-
age progress function are very popular .. In some
special situations where data is available in a

per unit basis the unit progress function may also

be considered.

(2) Parameters g and b_must be known or somehow calcu-

lated so that the progress function can be applied.

In this section it is assumed that the cumulative ave;
age progress function (type 2) is the best fitted to the
available empirical data. Once given the type of progress
function there exist two general classes of problems involy

ing parameter calculation:

(A) Given a point in the progress curve and the slope
of the progress line, determine the progress func-

tion.

91

(B) Given two points in the progress curve, determine

the progress function.

Class (A) involves solely parameter E determination
since parameter b (the slope of the progress line) is given.

In class (B) both parameters must be calculated.

Each class contains four problems. Their statements
and respective solutions will be the subject of this section.
Statement of Problem Al

Given a point in the cumulative average progress curve,
(xi, y(xi)) and the slope of the progress line (parameter b),

determine the cumulative average progress function y = y(x).

Solution of Problem A1

Since, by assumption, y = ax'b, one must have:

-b
ax.
1

F (xi)

3" (xi) x: (3.42)

O O 8

Parameter b’is given and parameter 3 can be calculated by

(3.42). Thus, problem Al is solved, i.e.,

I = ax'b = §(xi) x‘i’ x'b (3.43)

92

Statement of Problem A2

Given a point in the cumulative total progress curve,
(xi, yT(xi)), and parameter b, determine the cumulative ave;

age progress function y = y(x).

Solution of Problem A2

From equation (3.11):

l-b
yT (xi) " axi

- y (X.)
_ T i
a — ——I:b—- (3.44)
x.
1

Since b'is known and E can be calculated by (3.44),
problem A2 is solved. The cumulative average progress func-
tion will be given by:

y (x.)
_ -b T i -b
= = -————— 3.45
y ax 1-b x ( )
i

X

Statement of Problem A3

Given a point in the unit progress curve, (xi, y(xi))
and parameter b, determine the cumulative average progress

function y = y(x).

93

Solution of Problem A3

According to equation (3.13):

_ 1-b
Y(xi)=aEc:b-(xi-l)]

 

.'. a = (3.46)

Again, since b is known and 3 can be calculated through
(3.46), problem A3 is solved. The cumulative average progress

function will be given by:

 

.V = ax = x (3.47)

Statement of Problem A4

Given a point in the lot average progress curve,
(Xk, §L(xk)) and parameter b, determine the cumulative aver-

age progress function y = y(x).

Solution of Problem A4

One possible solution is to use approximation (3.36),

FL (Xk) E y(xk)

94
In this case, problem A4 is reduced to problem A3 already

solved.

If the number of units in lot k is known, say, nk, than
equation (3.14) may be used to determine parameter 3. Since

xk_1 = xk ‘ nk’ it follows from (3.14):

a = 1-b l-b (3'48)
xk Wham)

 

Thus, the cumulative average progress function will be ex-

pressed by:

? nk
y = ax"b = l-b Lk 1-b x'b (3.49)
- (xk - nk)

 

Statement of Problem Bl

Given two points in the cumulative average progress

curve, (xi, y(xi)) and (xj, y(x,)), determine the cumulative

average progress function y y(x).

Solution of Problem B1

Since, by assumption, y = ax b, one must have:

T (Xi) = ax;b (3.50)

and

95
-b

. (3.51)
J

— x. = ax
y ( J)
Dividing (3.50) by (3.51), taking logarithms and solving for
2, yields:
log [mp/571x33]

b = (3.52)
log (xj/xi)

 

After calculating b, parameter p may be calculated from (3.5»

or from (3.51). Therefore, using (3.50):
— b
a = y(X.) X. (3'53)
1 1

Once parameters 2 and g are calculated, problem B1 is
solved. The cumulative average progress function will be

given by:

where b and 3 are calculated through equations (3.52) and

(3.53), respectively.

Statement of Problem B2

Given two points in the cumulative total progress curve,
(xi, yT(xi)) and (xj, yT(xj)), determine the cumulative ave;

age progress function y = y(x).

96

Solution of Problem BZ

According to equation (3.11):

l-b

y (X,) = aX, (3.54)
T 1 1
and
l-b
x. = a 3.55
YT < J> xj < >

Dividing (3.54) by (3.55), applying logarithms and solving
for 2, yields:
log [:yT<xj_>/yT<xj )1

b = 1 - log (Xi/xj) (3.56)

 

Parameter g may be calculated through equation (3.54) or
(3.55). Thus, from (3.54):
Y (X.)
_ T 1
X.

1

The cumulative average progress function will be ex-

pressed by

-b
= ax

‘4'

where b and g are given by equations (3.56) and (3.57),respe§

tively.

97

Statement of Problem B3

Given two points in the unit progress curve, (xi,y(xi))
and (xj, y(xj), determine the cumulative average progress

function y = y(x).

Solution of Problem B3

(a) An approximate solution may be developed by using
equation (3.17). Since (xi, y(xi)) and (xj,y(xj))
are points in the unit progress curve, one must

have:

b

I2

y(xi) a (1 - b) x; (3.58)

and
b

I2

a (l-b) x:
J

y(xj) (3.59)

for xi and xj not very close to the first units of the series.

Dividing (3.58) by (3.59), taking logarithms and solving for

 

b, yields:
log Bap/yap]
3 (3.60)
108 (xj/xi)
Also, from (3.58):
y(x ) xb
a a -—-lL——;£ (3.61)
(1 -1»

The cumulative average progress function will be given by

— -b
y = ax

98

where b and g are calculated through (3.60) and (3.61),respeg

tively.

(b) A more exact solution stems from the application

of formula (3.13). Thus,

y(x.) = a [%}-b - (X. - l)1-€] (3.62)
1 1 l
and
y(x.) = a [:x]:-b - (x. - DMD] (3.63)
J J J

Dividing (3.62) by (3.63) yields:

 

 

 

l-b l-b
Y(xi) = xi - (xi - 1)
l-b l-b (3.64)
y(x.) _ _ 1
J xj (xj )
Now, let:
xi = A, (xi - l) = C, xj= D, (xj - l) = u,
y(xi)
l - b = B, and = F
y(x.)
J
Then equation (3.64) reduces to
AB CB
F = —B—_—B (3.65)
D - E

where B is the only unknown. Equation (3.65) may be solved by

Newton-Raphson method as follows.

99

In order to apply the method, equation (3.65) is written in
the form f(B) = 0, i.e.,

B B
AB — CB - F(D — E ) = 0 (3.66)

An assumption is made that f(B) is analytic for values
6
of the variable B near the root sought. Also, suppose that
7
Bn is a known approximate value of the root. Then, the

Taylor's expansion about Bn may be formed as follows:

2
man + h) = f(Bn) + hf'(Bn) + %T f"(Bn) + ...=0 (3.67)

If all terms of second and higher degree in h are neglected,
then:

= -f<Bn)/f' (Bn) (3.68)
and

Bn+1 = Bn - f(Bn)/f'(Bn) (3.69)

Applying iteration formula (3.69), yields:

B B B B

 

f(Bn) = A n - c n - F(D n - E n) (3.70)
B Bn Bn Bn
f'(Bn) = A n 1nA - c 111C - 17(1) lnD - E 11111) (3.71)
and
B B B B
B =B- An‘cn'FG’n‘En) (3.72)
n+1 n B B B B

A n1nA - c n1nc - F(D nlnD - E nlnE)

100

After some iterations it is possible to obtain an ade-

8
quate approximation 3* of B. Thus, parameter b may be calcu-

lated with the required approximation from:
b a l - B (3.73)
Parameter g may be calculated from (3.62) or (3.63). Thus:

( -)
a = * y x1 * (3.74)
x? - (xi - 1)B

 

The cumulative average progress function will be given by:

_ y(x.) *_
y = ax b = 1 -— xB l (3.75)

 

.‘lc
where B is calculated through iteration formula (3.72).

Statement of Problem B4

Given two points in the lot average progress curve,
(xk, yi(xk)) and (x1, yi(x1)), determine the cumulative ave;

age progress function §'= y(x).

One possible solution is to use approximation (3.36)

Then:

37 §Y()
Lk xk

101

and

S; 3 Y(X1)
Ll

In this case, problem B4 is reduced to problem B3 al—

ready solved.

If the number of units in lot k and lot i are known,say,

nk and n1, a more exact solution may be developed by using

equation (3.14), as follows:

l-b 1-b:]
y _ a xk ' (xk " nk)

 

 

L
k “k
and
[ 1-b 145]
§ = a x1 (x1 - n1)
L1 111

Dividing (3.76) by (3.77) yields:

 

 

5’- :Clt-b-(xk-n)l-_ n
_ILE:L 1‘ —. 1
y "l-b l-b‘ n
L1 x1 - (xl - n1) k

 

 

Let:

xk = A 9 ()ck - 11k): C: X1=D, (XI-Ill): E:

(3.76)

(3.77)

(3.78)

102

yLk/YLl
(l-b)=B,and -—n-—/n——=F (3.79)
l k

Then, equation (3.78) becomes:
B B
F ___. _.._____B B (3.80)

where B is the only unknown. Equation (3.80) may be solved
for B by Newton-Raphson method, as in Problem B3. The itera-
tion formula is again (3.72). The coefficients A, C, D, E,
and F are now given by (3.79). After a number of iterations
an adequate approximation B* of B is reached. Thus, parame-

ter b_may be calculated with the required approximation from:

Parameter g may be calculated from (3.76) or from (3.77).

Therefore:

Y
_ Lk nk

x: -<xk-nk>B

 

The cumulative average progress function will be given by:

f = ax‘b = k k x3 ' 1 (3.82)

 

103
*
where B is calculated through iteration formula (3.72) with

A, C, D, E, and F given by (3.79).

SUMMARY

The field of manufacturing progress functions lacks
notation uniformity, precise definition of the variables and
functional relationships involved, and formal proofs of impq;

tant results.

The fact that the progress function may be advanta-
geously treated as a continuous function so as to simplify
final formulas and speed up solutions has been largely ne-
glected in the literature. Instead, formidable formulas that
are given to computers to digest are preferred in the name

of exactness...

In this chapteréuloriginal mathematical exposition of
the progress function is offered. Initially, four types of
progress function are identified. Functional relationships
for the four types are clearly and compactly defined with
recourse to set theory notation. Parameters g and b are care-

fully explained and suggestively interpreted.

In a second section, four fundamental problems which
I
users might have faced consciously or unconsciously, are fo;

mally stated and solved, at times by more than one method.

Finally, parameter calculation problems are classified

and solved by exact or approximate formulas.

CHAPTER IV
QUADRATURE AND SUMMATION OF PROGRESS FUNCTIONS

This chapter represents a continuation of the mathema;
ical exposition on progress functions initiated with Chapter
III. Two related topics of practical relevance are now ap-
proached: the integration of progress functions and the de-
batable problem of their aggregation.Original approximations

are proposed for both problems.
QUADRATURE OF PROGRESS FUNCTIONS
Cumulative Total Hours Calculation

Let the manufacturing progress function be expressed
as a unit progress function:

y = ax-b (4.1)

Also, let (xu, yu) be the point in the progress curve where
a plateau begins (Figure 4). This point has already been de-
fined in Chapter II (page 39 ). For the moment it is assumed

that it can be practically determined.

104

105

Mfg.Hrs/Unit

 

‘4
c:

XI
111
l

 

 

Cumulative Units of Production

FIGURE 4

Progress Curve Ultimate Point (xu,yu)

and Manufacturing Progress Function Hours.

Recall that with equation (4.1) holding for the unit

curve, the exact expression for the cumulative total hours

is given by:

y = a 2 x (4.2)
x

Approximation Methods . (a) An approximation method
that eliminates calculation of every unit value in equation
(4.2) was suggested in a study prepared by Boeing Airplane
Company, already mentioned in Chapter II (page 20 , equation
2.7). Essentially, the integral is taken of equation (4.1)

with respect to ; between the limits 0.5 and (xu + 0.5) so as

106

to improve the approximation. Thus:

+
y ~ 7X“ 0.sax_b dx = i (x +0 5)1'b 0 51'b (4 3)
T - 0.5 l-b u . . ..

(b) However, a simpler approximation was also suggested in

Chapter III. From equation (3.25) it follows that

 

a 1-b
YT(xu) - 1- xu (4.4)
for xu not very close to the first units in the series.
But, according to equation (4.1):
_ b
a — yuxu (4.5)
Therefore, from (4.4) and (4.5):
X Y
s u u
x : [.6
1T (Xu) l-b (+ )

where yT(xu) does not depend on parameter 3.

The same reasoning might be applied to equation (4.3).

Hence, from (4.3) and (4.5), it follows that:

b
x y _
yT(x ) 5 u u [Exu+0.5)l-b - 0.51 g] (4.7)

u l-b

 

107

(c) A third approach to be proposed herein is suggested by
l
Euler-MacLaurin summation formula:

X

n 1 1 82 h . .
:Of(xi)= hfx: f(x)dx+§ [f(x0)+ f(xrfl +—2'_—[:f (xg- f (x0)] +

3

 

 

B4 h
+ 4: E“ (xn) - fm(x0):] + ...+
20-3
B h ~ _
+ 216.2 [f(Zp—3) (b) _ f(21> 3) (3)]
(2p-2)!
20
B h~
+ nu —2}’———— (4.8)
(29)!

Formula (4.8) is useful for finding the approximate sum of
any number of consecutive values of a function when these
values are given for equidistant values of x, provided the

integral L§n(x)dx can be easily evaluated. In this formula
‘0

h is the distance between the equidistant values of ;, so
that n11= xn - xO . The last term is a remainder where u
designates an average value of f(2p)(x) between x0 and xn .The
B's are Bernoulli numbers. Recall that these are the numbers
Bl’BZ""’Bn’ defined by the expansion of the following

generating function:

 

x =]_+—I—!-+-—2T-—+...+T+... (4.9)

108

convergent provided that [x] < 2 . Also, recall from the

related theory that:

and B

The B's with even indices have alternate signs and the

following values:

_ _5_ = _ 691 = Z ___ 3617
B10 66’ BIZ 2730’ 314 6’ B16 510’

= 43867 B = _ 174611 B = 854513
13 798 ’ 20 330 ’ 22 138 ’

 

B

Replacing the B's in (4.8) for their values, yields:

 

n 1 X“ 1 h , ,
iEof(xi)= h f f(x)dx + 7 f(x0)+ f(xn) + I2 f (xn)- f (x0)
3 5
h h
‘ 776 E... (Kn)- f'" “‘0’} 30240 Evan)— 970(0)]

7

h vii vii
- —-———-— f x - f x - R (4.10
1209600 I: ( n) ( 0)] )

109

The Inherent Error in Euler-MacLaurin Formula . In
(4.10) the terms on the right side, beginning with
(h/12)[f'(xn) - f'(xOZ], form an asymptotic series. In comput
ing with such a series it is important to know what term to
stop with in order to get the most accurate result. One get
the most accurate result by stopping with the term just be-
fore the smallest,since according to C.V.L. Charlier,in stopping
with any term in Euler-MacLaurin' formula the error committed
is less than twice the first neglected term. It can also be
shown that the first two terms of Euler-MacLaurin formula

4
will give a more accurate result than Simpson's Rule.

By taking the first three terms in formula (4.10), the

 

third formula for approximating equation (4.2) can be derived

as follows:

=a :u x-bc'aijéuf(x)dx4-ljf(l)+ f(x Z]+-l;[f'(x )- f'(ffl}
YT 1 = l E’ ‘ u 12 n

(4.11)
b x 'b 1 1 b
Since, f(x) = x" ,7 u x' dx = —-(x ' - 1), f(1)= 1,
1 l-b u
f(x ) = x'b, f'(x) = - bx-(b+l), f'(x ) = - b x-(b+l), and
u U u u
f'(l) = - b, it follows from (4.11) that:
.. _L 1-b_ l -b _b_ _ -(b+1)
YT(xn) = a [l-b (Xu 1) + 2(1+xu ) + 12 (l x )

.13)

110
or from (4.5):

y (x ) 5y xb [—_1:— xi'b-l)+%— (1+x11b)+j% (l-x;(b+l)% (4.14)

Manufacturing Progress Function Hours Calculation

In Chapter II the term "MPF hours" was defined as theme
hours over and above the estimated hours which are caused by
the introduction of a new unit into a manufacturing system.
The MPF hours or the direct labor hours associated with the
'manufacturing progress function can be measured by the shaded
area in Fig.4. Knowledge of the MPF hours may be of extreme
importance to management in deciding about the implementathn1
onf a new product or a major change in existing products.The
practical development of such matters will be delayed until
a later chapter. In the following paragraphs four methods

for calculating the MPF hours are proposed and assessed.

Exact Method. An obvious approach is suggested by
equation (4.2). Noting that the shaded area in Figure 4 is
the difference between the total area under the progress

curve and the rectangular area x yu, and also using equation

u
(4.1), the MPF hours consumed by units number 1 up to and

including unit xu can be exactly determined as follows:

MPF Hours

II
0)

 

= a (z: x'b - xl'b) (4.15)
1 u
Also, according to (4.5):
‘L‘! _ b xu _b
MPF ...-ours - yuxu 2i x - xuyu
1. x“
-b

= xuyu ( l-b Z x — l) (4.16)

xu 1

Note that formula (4.16) does not require the knowledge
of parameter E. Formulas (4.15) and (4.16) are better suited

for machine computation. Tables can be programmed and devel-

oped on a digital computer.

Approximate Methods. Equations (4.3), (4.4) and
(4.13) suggest three possible approximations to the problem
of calculating the MPF hours. Again, subtraction of the rec--~
tangular area xuyu from the total area under the progress

curve,and equation (4.1) yield the following results:

(a) Using equation (4.3)

l-b l-b
MPF Hours 3 I%b [Exu+0.5) ~ 0.5 :] - xuyu

= a{'f}jg [(1H1+0.5)1-b - 0.514)] - xi-b} (4.17)

Also, according to (4.5):

l-b -b
(xu+ 0.5) - 0.51 J
- 1

l-b
(l-b) xu

(4.18)

 

MPF Hours 5 x y [:
Uu

112
(b) Using equation (4.4)

 

 

‘ ~ _;;_ l-b _ ab l-b
MPF Hours l-b xu - xuyu I:— xu (4.19)
Also, from (4.5):
A
MPF Hours a 14:1 xl'b - xuyu
b
= 1:71; am. (410)

Equations (4.19) and (4.20) are much simpler than
(4.17) and (4.18) and are well suited for manual calcula-

tion.

(c) Using equation (4.13)

 

. [T1— (.13-b- 1>+;<1+x; b) +

llz

MPF Hours -b

b -
I2 (1 - xu(b+l)% ' Xuyu

 

l l-b l -b
= a [%:g (xu — l) + E (1+xu ) +
j; -(b+l) l-b
12 (l - xu ) - xu 7] (4.21)
or from (4.5):
l _
MPF Hours 5 xuyuy{ {Eb[%:g'(x: b-1)+ i‘bef%9 +
x x
u U

 

b 1
I2 (1 - b+l{] - l } (4.22)
u

113

Equations (4.21) and (4.22) derived from Euler-
-MacLaurin summation formula are the most accurate of the
three proposed approximations. They can be easily handled by

a pocket calculator.

Accuracy of the Proposed Formulas

In order to demonstrate the accuracy of formulas (4.3),
(4.4), and (4.13) the cumulative total hours values were
machine computed for xu = 1, 2, 3, 4, 5, 10, 20, 30, 40, 50,
100, 200, 300, 400, 500, and. 999 using the exact formula
(4.2) and also calculated with a HP-25 Scientific Programma-
ble Pocket Calculator through the approximation formulas. In
all cases parameter §_was taken as 100 hours. Table 4.1 was
developed for b = 0.152003 (90% progress curve). The per-
cent deviations from the exact values are also shown in the
referred table. Such results were plotted on semi-log

graph (Figure 5).

THE AGGREGATION PROBLEM

Consider the sum of two progress functions given by:

y1 = alx-b1

and. -b2
y2 = 82x

The sum is

- -b1 -b2 (4.23)
y1 + y2 = alx + azx

EXACT AND APPROXIMATE CUMULATIVE TOTAL

114

TABLE 4.1

 

 

 

 

HOURS; a = 100, b = 0.152003 (90% curve)
Formula Formula % Formula % Formula %
xu

(4.2) (4.3) dev. (4.4) dev. (4.13) dev.

1 100.000000 100.800810 0.801 117.920906 17.9 100.000000 0
2 190-000000 190.968570 0.510 212.260916 11.7 190.036653 0.0193
3 270.620600 275.658038 0o378 299.366015 9.01 270.661171 0.0108
0 355.620600 356.695583 0.302 382.076873 7.00 355.662119 0.0117
5 033.919300 035.017196 0.253 061.668002 6.13 033.961129 0.0096
10 799.007900 800.592303 0.103 831.003177 3.95 799,090071 0.0053
20 1060.776000 1061.903251 0.080 1095.805815 2.00 1060,818365 0.0029
30 2072.689300 2073.863996 0.057 2109.599805 1.78 2072.731906 0.0021
00 2650.271300 2655.009680 0.000 2692.050601 1.00 2650.310187 0.0016
50 3210.195500 32150376092 0.037 3253.322750 1.22 3210.238600 0.0013
100 5810.101800 5815.287300 0.020 5855.981338 0.720 5810.106067 0.0008
200 10096.006000 10097.59535 0.011 10500.76709 0.023 10096.05236 0.0000
300 10820-002800 10821.59509 0.008 10866.10092 0.308 10820.05157 0.0003
000 18926.783000 18927.97795 0.006 18973.38198 0.206 18926.83017 0.0003
500 22878.510500 22879.70727 0.005 22925.78159 0-207 22878-56330 0-0002
999 01182.189600 01183.39816 0.003 01231.01311 0.120 01182.25397 0.0002

 

115

 

 

 

 

 

 

 

 

 

 

100

 

~~3——- _ --2»——~—~-—3-—-—- 4- 5 s 1 7,, ~ .3 - -~9 - ~10

-7.- q. .._.T.._

x (cumulative units)

‘ FIGURES - EXACT AND APPROXIMATE CUMULATIVE wTOTAL _ .1,
HQURS vs. CUMULATIVE PRODUCTION

a = 100, b= 0.152003 (90% CURVE)“

 

 

116

A log-log plot of (y1 + yz) versus ; is a convex curve
whose shape depends upon parameters (a1, a2, b1, b2). The

plot will be linear if and only if bl = b2 = b

-b
yl + y2 = (a1 + 82) X (4~24)

Strictly speaking, if the model is assumed to hold for
two separate portions of a task it cannot also be assumed to
hold for their sum unless the separate curves have equal
slopes - which will not in general be the case. Several
authors contend that this fact precludes the use of the
linear model for operations, departments, sections and total
of the same project.5

Notwithstanding, it is the purpose of this section to
show that it is theoretically correct and even desirable in
practical work to approximate the sum of m_progress functions

of different parameters by a progress function of the same

functional form as the addend functions.

Statement of the Problem

Given m progress functions

determine a function of the form

such that
Y

Ill
":43
<1

117

Solution of the Aggregation Problem

Expressing a Progress Function as a Power Series.

The result known as Taylor's series may be posed as follows:

y(x)=y(c)+ 1%(f:l(x—c)+ iii—(Ig- (x-c)2+. . .+ 1531;92- (x-c)n+. . .
n

(4.25)

Taylor's series may be used for the expansion of any
given function y(x) in a power series in (x-c), provided the
expansion exists . For a function to admit of a Taylor's
series expansion in powers of (x-c), it is necessary that the
function be finite and possess finite derivatives of all
orders at x = c. The series in the right member of (4.25)

will have a sum equal to y(x) whenever the series converges.

-b
Let y(x) = ax . Therefore, expanding y(x) into

Taylor's series yields:

y(x) = ax-b
y'(x) = -abx-(b+l)
y"(x) = ab(b+l) x-(b+2)

y'"(x) = -ab(b+l)(b+2) x‘(b+3)

-(b+n)

y(n)(x) = (-1)n ab(b+l)(b+2)...(b+n-l)x , n=l,2,3,...

or equivalently

n a(b-l)!b(b+l)(b+2)...(b+n-l) X-(b+n)
(b-l)!

 

y“) (x) = <-1>

118

n a(b+n-l)! x-(b+n)

 

=(-1> (H)! , “0,1,2”.
Also
Y(c) = ac-b
y'(¢) = -abc’(b+1)
y"(c) = ab(b+l) c‘(b+2)
Y'"(C) = -ab(b+1)(b+2) c'(b+3)

 

n a(b+n-l)! c-(b+n)

3"“) (c) = <-1>
(b-l)!

. n = 0,1,2....

By substituting these values in equation (4.25), it

follows that:

 

—b -b abc
c .—

-(b+l) -(b+2)
y=ax =a (x—c)+ab(b+l)C

1! 2’

 

(x-C) -

 

n a(b+n-l)! C-(b+n) (x-c

n
I1!(b-l)! ) 4' '°'

+ (-1)

(_l)n a(b+n-1)! C-(b+n)

0 n!(b_1)! (x-c)n (4.26)

 

ll
"v18

i

Existence of the Expansion. In order to prove that

the expansion in (4.26) exists, it suffices to prove that

lim Rn = 0

n+oo
where Rn is the remainder term of the expansion. It is given

by:

(n)
Rn = Y [E :19 (x'cij (x-c)n, 0<0<1 (4.27)

119

Since y(n)(x) = (-1)n ab(b+l)...(b+n-1)x-(b+n),n=l,2,

it follows that:

 

y(n)[§+6(x-CX]=(-l)n ab(b+l)...(b+n-l)[§+e(x-CX]-(b+n)
Hence, from (4.27):
n

R =(_1)n ab(b+l)(b+2)...(b+:;l) . (XQ?) (4.28)
n [E + 6 (x - CZ] n

But |y(n) (x)! decreases with x,and 0 < 0 < 1. Thus:

NOE: + e(x-c)jl < y‘n)<Ix-cl) (4.29)
for all c - IEO. < x < c + Igg-

Also, let N be a fixed positive integer greater than

2|x-cl. And for n>N, put n = N + k. Then,

Since

Effie '—"-?1- 63—8 <N—‘2> (rm?) .

it follows that

N
|(X'C2n| < (X'C) . _1_
n! N! 2k

120

And for n2>N,

 

N N
[Rn] <Y(n)(Ix-cl) B-Cl 2 . 1

N ! 2“

When n+w, x, c and N remain fixed but ;%~*O. Hence
2

R -+O
n

<x<c+_c_

c
1+6 l-G

 

for all c -

Therefore , Taylor's expansion, given by (4.26) is

valid for all g in the interval above.

Interval of Convergence. The interval of convergence
for (4.26) may be practically determined by d'Alembert critg
rion (ratio test) applied to the corresponding series of

absolute values. Let

 

 

= <-1> §§iili§i' c"b+“’<x-c>n
Then
= _ n+1 a (b+n)! _(b+n+1) - n+1
‘1 +1 ( 1) (n+l)l(b-l)! c (x C)

And after the necessary simplifications

l un+ll_ lb+nc x-c x-c

 

 

_'n+bc

 

121

X-C <1 for convergence (4.31)

 

 

u
' n+1 = l.

 

 

It follows from (4.31) that the interval of convergence is
0 < x < 2c (4.32)

At the end points of the interval of convergence, the
simple ratio test can never be effective. Convergence or

divergence will have to be tested by other criteria.

At the end point x = 2c the series has:

n a(b+n-l)! _
un = (‘1) nl(b-l)1 C (b+n) C“

n a(n+l)(n+2)...(n+b—l)
cb(b-l)!

 

= (-1) (4.33)

Since the series has alternate signs and the nth term
in (4.33) numerically decreasestx>zero as 2 becomes infinite

the series is convergent at x = 2c. To see this:

 

lim a(n+l)(n+2)...(n+b-l)== .a lim nb-l = O

n+w Cb(b-1)! c (b-l)! n+m

(recall that Ogbgl in a progress function)

At the end point x = O the progress function is not
defined. Thus, the Taylor's expansion given by (4.26) is
valid for all 5 such that

O<x32c (4.34)

122
Deriving the Aggregation Formula.‘ Using (4.26) to

expand the progress functions

 

 

 

_bj
, = a x ‘ , '=l,2, ,m
YJ ] J
about x=c:
- b-+l
-b -bj anJc ( J )
= J: _ _
yJ aJx ajc 11 (X C) +
.. .+
albj(bj+1)c (b3 2) 2
+ ‘2! (x - c) -
_ ,+
n aj (bj+n-1)!c 03.] n) n
+ (-l) n!(bj-1)! (x-c) +n.. (4.35)

valid for O<<X¢32c , j=l,2,...,m, n = O,l,2,...

From the theory on power series, assuming that the
series are convergent in some interval, then they can be
added or subtracted term by term for each value of §_common
to their intervals of convergence. Since the yj's are conver
gent in the same interval, their sum is also a convergent

series in the same interval. Thus:

- ,+
(bJ 1

m m 'b. m ajbjc
2 y.= 2 a;c 3- 2 (x-c) +
= j=1 1!

 

 

-(b-+n)
. .+ - J
aj(bJ n l)!c

m
E (-1) n, (bj‘l) :](x-c)n +. .. (4.36)

 

123

valid for O < x 5 2c

Recall that the initial statement of the problem is to

find a progress function of the form

y = Ax—B (4.37)
such that
m
y 5 2 y' (4.38)
i=1 1

Expansion of (4.37) into a Taylor's series about x=c

 

 

 

yields:
-(B+1) -(B+2)
y: -B=Ac-B-ABC 1! (x-C)+3AB(B+£;f (x-c) _
-(B+n)
_ n A(B+n-l)!c _ n
+ ( l) nl(B-1)! (x c) +... (4.39)

valid for O<<x:;2c , n = O,l,2,...

124

From (4.38) one must have:

-(B+1)

-(B+2)

 

 

 

 

AC-B _ ABC (X-C) +AB<B+1>C (X-C)2 _
l! 2!
-(B+n) m -b.
_ r1A(B+n-l)!c _ ‘n = J _
+( l) n!(B-1)! (x c) +... jil ajc
b b (b +1) -(b1+2)
m a. . -(b.+1 111 a, . . C -.
[:22 _a_—11c J ](x-c)+£2 J J J2, ](x-c)2-...
j=1 .=1 '

+ [:f (-1)n
j=l

- (b.+n)

aj(b.+n-l)!c

 

J J n
n!(bi-1)! (X-C) +a..

(4.40)

If the expansions in (4.40) are truncated after the

second term and the method of undetermined coefficients is

employed, it follows that:

 

-B m ‘b.
Ac = 22 ac 1 (4.41)
i=1 J
and
_ m. -(b.+l)
ABc (3+1) = z a.b.c J (4.42)
'=1 J J
J
Solving (4.41) and (4.42) for A and B yields:
m -bj
Z a.b.c
_ j=1 J J
B — 'm -b (4.43)
2 a.c J
and
B
A = c D (4.44)

125

where D is the denominator in (4.43)

-b.
Thus, given y1 = ajx J , aj>0, 03b131, it is possible
to find y = Ax'B, (A>O , O<B<l) such that
m
y 5 X y
j=1 J

the parameters of y being given by equations (4 43) and

(4.44).

. . . *
Pr00031tion. Let b* = min b. and b

J
j = l,2,...,m. Then

*
b*<B<b

Proof. Since

then according to (4.43), (4.45) and (4.46)

*
b*:<B:<b

’

(4.45)

(4.46)

(4.47)

Consequently, since 0:;bj131 it follows from (4.47) that

O<:B<<l

126

Accuracy of the Proposed Aggregation Method

The following sample calculation and graph are aimed
at demonstrating the accuracy of the proposed aggregation
method which is based upon Taylor's series expansion. Let

yl= 150 x‘0-322, y2= 350 {0°152 and y3=500 x’0'515

be three hypothetical progress functions. The calculations
and graph plotting are carried out according to the suc-

ceeding steps:

(1) Each individual function yl(x), y2(x), and y3(x) is
computed for x = 1,2,3,4,5,10,20,30,40,50,100,200,300,
400,500 and 1000 (Table 4.2, columns (2),(3), and (4)).

(2) The aggregate function is then calculated for each given
5 by summing up the corresponding values y1(x), y2(x),and
y3(x) for the three addend functions.Call it the exact ag

gregate func‘tionCI‘able 4.2, column (5)).

(3) Parameters B and A of the approximate aggregate function

 

are computed through formulas (4.43) and (4.44). From

(4.33) one must have

0<x52c -- c

|V
Nix

Also, from the given data

x e [1, 1000]

EXACT AND APPROXIMATE AGGREGATE

127

TABLE 4 . 2

PROGRESS FUNCTION-INTERVAL [:1, 1000:]

 

 

 

 

(1) (2) (3) (4) (5) (6) (7)

x ,1... :2- ,3- g v "4
ISO/X.322 350/x' 500/x'515 1 .J GEM/x.213 dev

1 150.000 350.000 500.000 1000 000 664 000 33.6

2 119.994 315.001 349.896 784 891 572.860 27.0

3 105.307 296.173 283.957 685 437 525 462 23.3

4 95.9904 283.501 244.855 624.346 494 230 20.8

5 89.3353 274.047 218.273 581.655 471 289 19.0

10 71.4647 246.643 152.746 470 854 406 601 13.6

20 57.1689 221.979 106.891 386 039 350 791 9.13

30 50.1716 208.711 86.7466 345 629 321 767 6.90

40 45.7328 199.781 74.8013 320.315 302 642 5.52

50 42.5621 193.119 66.6808 302.362 288 594 4.55

100 34.0480 173.807 46.6627 254 518 248 982 2.18

200 27.2370 156.427 32.6542 216 318 214.807 0.70

300 23.9033 147.077 26.5004 197.481 197 034 0.23

400 21.7885 140.784 22.8512 185 424 185 323 0.054

500 20.2779 136.090 20.3704 176 738 176.721 0.010

1000 16.2215 122.481 14.2551 152 958 152 464 0.32

 

128

Therefore, by taking the mid-point of the interval:

150 x 0.322 350 x 0.152 500 x 0.515

 

 

 

 

 

0.322 + 0.152 + 0.515
B = 500 500 500 :=37.705864
176.737834
150 + 350 + 500
50013"2 5000'I52 5000'SIS
= 0.213343
and

B

A = c - D = 5000'213343 x 176.737834 = 664.064309

Thus, the aggregate function may be approximated by:

a 664 x'0°213 (4.48)

V

(4) The approximate aggregate function in (4.48) is calcu-
lated for the same values of 5 as already mentioned in

step 1 (Table 4.2 , column 6 ).

(5) Per cent deviations of the approximate values obtained
in step (4) with respect to the corresponding exact val-

ues from step (2) are evaluated (Table 4.2, column (7)).

(6) The exact and the approximate aggregate functions are

plotted on arithmetic graph paper (Figure 6).

>129

 

 

Moog .Q €555 .. mzonoza BEBE 3% ”5.384% 2.. H95 o HERE

Amid: gwumgov x

98 SH o3

on S

 

2056sz HOSE

ZOE—[Eh
fig
ngomg

 

9:

cow

8:
com

08

o8

o8

HE:
has:

 

 

 

130
The results in Table 4.2 and Figure 6 show that the
approximate aggregate function in (4.48) is quite effective
in representing the exact aggregate function after the 40th

or 50th unit up to the 1000th unit in the series.

If the focus is on the early units of the series it
suffices to change the value of g and recalculate the para-
meters A and B for a new approximate aggregate function in
that interval. For example, assuming that the interval of

interest is now

x e [1, 20]

then

150 x 0.322 350 x 0.152 500 x 0.515

 

 

 

 

 

0 322 + 0.152 + 0.515
3:: 10 10 10 _ 139.165507
' 470.853278
150 + 350 + 500
100‘322 100'I52 100'SIS
= 0.295560
and
A = C3 - D = 100'295560 x 470.853278 = 929.920021

The aggregate function may be represented by

-0.296

y E 930 x (4.49)

131
A second table is calculated by following steps 1-5
previously mentioned (Table 4.3). The results are also
plotted on arithmetic graph paper (Figure 7). It is manifest
that the approximate aggregate function in (4.49) fairly well

represents the exact aggregate function in the interval

[1, 20] and even beyond the 20th unit.

TABLE 4 . 3

EXACT AND APPROXIMATE AGGREGATE
PROGRESS FUNCTION-INTERVAL E1, 20]

 

 

 

 

3 y 5 Z
x 1 Yj 930 x-0.296 dev.
1 1000.000 930.000 7.00
2 784.891 757.492 3.49
3 685.437 671.823 1.99
4 624.346 616.983 1.18
5 581.655 577.548 0.71
10 470.854 470.417 0.093
20 386.039 383.158 0.75
30 345.629 339.825 1.68
40 320.315 312.085 2.57
50 302.362 292.138 3.38
100 254.518 237.948 6.51

 

_.132_ .,

 

flow . .Q A<>§ .. mZOHHUsz mmmmuomm §§< Magnum Qz< HOSE N. mmDUHm

Amuwg o>wumH25ov x
8 2 m .. m N H

1“

 

 

8 a
can
con

1 6523
3658.4. 8..
Egg
com
/ so
2:.
zoEozE mmmmuomm Ba 2.
8m
T 82

 

3.25
A .61:

 

 

 

133
Accuracy in the Quadrature of

Approximate Aggregate Functions

The main interest of approximate aggregation of prog-
ress functions may reside in estimating the total direct
labor hours and the MPF hours to be consumed by a prospec-
tive manufacturing program where one or several new products
are involved. Since this is one of the potential applicatflnns
of the manufacturing progress function to be treated in
chapter VII no further comments are necessary at this point.
However, to show how accurate the integratiOn of the
approximate aggregate function isthe following sample checking
is carried out. The same data of the previous subsection is

assumed.

(1) For each given progress function y1(x), y2(x) and y3(x)
the g§§§§ total hours expended in the manufacturing of
unit 1 up to and including unit 999 are machine computed
through formula (4.2). Thus:

999
yT = 150 z x‘0°322= 150 x 158.562733

1 1

23784.40995

999 -0.152
YT = 350 z x = 350 x 411.821896

2 1

144137.6636

999 -0 Slr
y = 500 2 x ' ’= 500 x 57.3720489

T3 1

28686.02445

134
(2) The total hours expended in the three products (or actig
ities) represented by progress functions y1(x), y2(x)
and y3(x) are computed
3
2 YT. = 196608.0981
1 1
(3) Now the approximate aggregate function in (4.48) is used
in order to calculate the same total hours consumed by
y1(x), y2(x) and y3(x). Since the focused interval is
sufficiently large, formula (4.4) may be used:

a A 1-B _ 664 0.787 _
y - 1:3 X — 67737 X 999 - 193575.9442

T
(4) The per cent deviation of the approximate result obtained
in step 3 with respect to the exact result from step 2
is computed:

% dev. = 193576 - 196608 X 100% = _ 1.54 %

196608

 

It may be concluded from step (4) that even using the
less accurate formula (4.4) the value of the integral is

approximated with reasonable accuracy.

Again, considering the interval [1, 20] and going
through the same previously mentioned steps, the following

results are obtained:

3 20 -0 22 20 _ 20 _
exact total hours= ZyT = 150 Z x '3 +350 2 x O'152+500 Z x 0'5”:
1 i 1 1 1

= 10388.7271

135

From formula (4.4) and equation (4.49):

0.704
a 930 x 20 = 10885.17477
0.704

 

approximate total hours yT

10885—10389
10389

 

% dev.

x 100% = 4.78%

()1: from formula (4.3) and equation (4.49)

= —2§9— 20 50'704 0 50'704 — 10265 1192
YT - 0.704 ( ' - ' ) _ '
% dev. = 10265'10389 x 100% = - 1.19%

 

10389

CHAPTER V

A MODEL OF THE PROGRESS PHENOMENON IN
MANUFACTURING INDUSTRIES

The objectives of the study included the derivation of
£1 symbolic-analytic model of the manufacturing progress phe-

nomenon. Such a model will be presented in this chapter.

This purpose cannot be accomplished without a means of
understanding how system adaptation can exist in a manufac-
turing concern. It is worthwhile to expend some effort in
questioning the causes of, or reasons for, the systematic
gains in productivity embodied in the progress curve,however

tentative the resulting explanation might be.

Although system adaptation can exist in a wide variety
of production systems, methodological problems can impose
Serious restrictions on a study of the phenomenon. These
problems, and their influence on the study, are also dis-

cussed in this chapter.
A CONCEPT OF MANUFACTURING PROGRESS

The manufacturing progress function is essentially an
e .
mp 1rical concept. There is no rational or deductive proof
1: -
hat, can be advanced to support it or the assumptions upon

Vol-1 -
10h it is based. While these assumptions may be intuitively

136

A

137
appealing, they must be empirically validated. The three
assumptions upon which the MPF is based are: (1) The amount
of time required to complete a given task or unit of produc-
tion will decrease each successive time the task is under-
taken; (2) The rate at which this reduction in unit time will
occur will be a decreasing one, and (3) This reduction in

‘unit time will follow a specific predictable pattern.

It has been demonstrated in the literature that these
Tassumptions are valid for many classes of products and manu-
:facturing processes. Moreover, they may serve the purpose of

Iaroviding a preliminary characterization of the manufacturing

garogress phenomenon.

Iféictors in Manufacturing Progress

There are several factors which may influence or con-
1:::ribute to manufacturing progress: (1) The production worker,
(2) Management, and (3) Staff or supporting personnel. Their
C=<Dtntribution will vary over time and in relative importance

ifzrtimlone industrial environment to another.

The Production Worker. It seems reasonable to expect

that the production worker will improve his performance as
:Elea })ecomes more acquainted with a new task. This should be
e"7354(51ent in a reduction in the time to complete a unit of prg
duction as well as a reduction in the number of rejects. It
i;53 éafilso suggested that the learning of the worker Will have

a: .1:
“ Eiiiher immediate and significant effect in terms of

138
manufacturing progress but the change in rate of production

he contributes will not extend over a long period of time,

particularly in routine and repetitive tasks.

Management and/or Supervision. An efficient supervi

sion should be able to motivate the worker towards a reduc-
tion in the time necessary to complete a unit of production.
Similarly, management may be able to impress upon the staff
personnel the need for further savings through their efforts.
In a situation where more than one product is being manufac-
tured simultaneously in a plant the attention dispensed by
Inanagement to one or more of the products will influence the
Inagnitude of manufacturing progress for that product or
jgrroducts. On the other hand, the relative attention or lack
t:11ereof given one product over time may be reflected in a
j§>arogress curve exhibiting plateaux of no improvement followed

‘E33r'sharp drops reflecting sudden management emphasis.

Staff and/or Supporting Personnel. It is suggested

that a substantial portion of the manufacturing progress will
t><3 the result of the efforts of the supporting personnel who
in many organizations are listed under the headings of the
“’éaitfiious staff functions. It is also suggested that the im-
33:=7<>‘Jements contributed by the staff functions will be more
<3"13’ £1 continuing, long-ranging and long-lasting nature and
that they will be the significant factor after the initial
1‘€3“Ellfining has been accomplished by the production workers.

53 .
QLIQ of the typical contributions from the various staff

 

139

offices are discussed in the following paragraphs.

Tooling. The type of tooling used, the degree of
completion or development prior to production and the changes

made during production will influence the magnitude of manu-

facturing progress.

Methods Engineering. The extent to which work methods
in detail are designed prior to production and then the
(emphasis on improving these methods during production throug

1work simplification and similar programs will affect total

rnanufacturing improvements.

Production Planning and Control. This function may
iFéicilitate progress through improved planning, routing,
saczheduling, dispatching, and follow-up. This should result
IiJrl an increased utilization of machines, tools, and labor ,
EiIan ultimately in a reduction in the direct labor hours nec-

essary to manufacture a unit of production.

Materials Management. When a materials management
<-‘-<>‘151<2ept of organization is used this function may include the
15>]:"C>ciuction planning and control, the purchasing of raw mate-

ITCi-éills and component parts, the control and the storage of

13“relaroduction, in-process and finished inventory and the
.nu£3“t:<2rials handling function. If it is performed effectively,

In. - .
a":erial shortages and the disruption of production should

140

be reduced or eliminated.

Product Engineering. This function is responsible for
product design, specifications, testing, and the like. The
extent to which manufacturability was considered prior to
production and the degree of change required subsequently
during production will influence total manufacturing improvg

ment.

Quality Control. The extent to which a quality assug
ance program can reduce rework and repair operations, and

scrap losses will affect the total manufacturing progress.

The degree to which each and all of the above listed
:Eactors may influence manufacturing progress will depend.upan
tihe amount of available or possible improvement remaining
ilumediately after the first unit of production has been.built
IIt: seems reasonable to accept that a company that has spent
In11ch.time and money getting the entire organization ready
to start production of the first unit has a smaller potential
fc>Ii‘manufacturing improvement after production has started
tZIIEIn.a company that has started unprepared and plans to
I'etrlove the inadequacies during manufacture. There is simply
more room for progress for the second company than for the
f.ij-tll‘sm. Let indices 1 and 2 designate the prepared and the
unprepared company, respectively. Thus, in terms of the power

1311 . .
1r1<2tion model, their progress functions may be written as

if
Q:Llows:

141

-b1 -b2
Y1 — alx , and 72 = 32X

a <<a , and b <<b
1 1

[\3

provided that the greater potential for improvement of the
second company be fully realized. The fact that there exists
more room for improvement does not necessarily imply that
the available improvement will materialize. By a similar
reasoning, company 1 will probably produce the first unit in
less time than company 2 because it is better prepared to do
so. However, it may also happen that the inequality a1‘< a2
V0111 not come true in spite of the greater/preproduction
Giffort of company 1. Management action may be the crucial
(jesterminant of actual progress in the manufacturing organizg
‘ttion , once there exists a potential for improvement. The
degree to which such a potential is exploited is related to
1:11£3 drive and resourcefulness of management and its skill in
Stimulating supervisors and technical people to be creative

Euf‘ldworkers to be productive.
I‘IBJ'IDcheses About the Progress Phenomenon - A Reexamination

It is believed that in theoretical work a minimum num-

I)
Gag)?” of hypotheses is desirable in explaining any kind of

142

phenomenon. It is our contention that a reduction can be
achieved in the number of available hypotheses about the ma-
nufacturing progress phenomenon and that the explanatory

power of the remaining hypotheses can be greatly enhanced.

A Rationale for the Novelty Hypothesis. According to
Crawford the rate of progress is a function of the experience
of the worker. The greater the lack of experience of the
worker, the greater the rate of improvement. Schultz and
Conway agree with Crawford when they state that the degree
of similarity of a machine to a predecessor has a significant
effect on the rate of progress; the less similarity, the
greater the rate of progress (see Chapter II,pp 16, pp 38 ,
and pp 68). An organization which has had experience doing
a similar type of work probably will not exhibit much improvg
rnent when the "new" job begins because there is not much roan
lieft for improvement in the "new" operation. By the same

“txaken, an organization which is unfamiliar with this type of
E>lroduct and with this type of work will have a greater potep
tIiQal for manufacturing progress. This is a possible ratio-
Iléalle for the novelty hypothesis already mentioned in Chapter

II-

Preproduction Preparation, a Kind of Experience. The
nChlelty hypothesis can be strengthened if preparation prior
t:(:, Iproduction can be considered a kind of experience - -
particularly if the fabrication of prototypes is part of the

I) .
j':“‘3:r)‘roductn.on effort.

143
The Assembly and Machining Progress Hypothesis. Some
researchers have found that assembly operations exhibit
significantly higher rates of progress than machining opera-
tions (see Chapter II, pp 34 , and pp 50 ). The usual
explanation is that in assembly work there is a relatively
large scope for learning; in machine work the ability to
reduce labor hours is greatly restricted by the fact that the
machines cannot "learn" to run any faster.1 However, this hy-
pothesis can be explained by the novelty hypothesis.Consider,
for example Hirsch's study of the progress phenomenon in ma-
chine-tool manufacturing. This study represents one of the
few instances where the Assembly and Machining Progress hy—
pothesis was explicitly tested. According to Hirsch, the
possible reasons why the machining improvement rate is con-
siderably less than the assembly improvement rate are: (l)
Pfiany times the same part is used in both old and new models,
(Jr even in different machines that, after many lots have been
Ibiroduced, the actual improvement rate for each additional lot
Ibeacomes almost negligible; (2) The accomplished machine
oPerator can do little to improve his efficiency, since most
C’15 his basic motions are the same, regardless of the part
being produced, and (3) The third reason is due to the great
jit1<2onsistency in the ratio of new to old parts used in the
assembly of the different machines. Thus, machines comprised
()rf? a.large quantity of new parts will have a high rate of
iTttrE>rovement, while machines with few new parts will have a

l . O O O O 0
:)‘K7 improvement rate. Hirsch's explanation is con31stent w1th

144

the findings of Schultz and Conway.

The Parameter Correlation Hypothesis. As mentioned hl

Chapter II a strong correlation between the parameters of fire

progress curve was found by H. Asher and N. Baloff. In

Asher's study the correlation was found among startups that
occurred in different facilities. In Baloff's study the cor-
relation was among startups that occurred in the same facil—
ity. However, the findings of both studies are consistent,
i e., low values of parameter Q are associated with small
p-parameter values, and high values of g with large, b_values.
A tentative explanation is that management reacts to the
reported level of initial productivity in ways that could
tend to accelerate or decelerate the progress phenomenon.
Startups that show a relatively "poor" beginning frequently
liave more technical resources allocated to them, and are gen-
errally "nurtured" to a greater extent, than startups that
begin relatively well. Furthermore, judging from the comments
(>1? many production personnel, it appears that the motivation
1F<>r progress within the production organization will be
jtrrVersely correlated with the initial productivity. The moti
‘V’éitzion for productivity improvement is apparently stronger
if?(>1? "poor" startups than for those that show a relatively
large initial productivity. Both the "internal" motivation
(szf the production organization and "external" management
I:‘h‘tr'isfissure seem to be influenced in this way.4

145
MANUFACTURING PROGRESS - - A MODEL

The conceptual framework presented in the previous

section offers a basis for a model of the progress phenome-

non in manufacturing industries.

It seems desirable that such a model preserve the
simplicity of wright's formulation while reflecting the
generally accepted linearity hypothesis. Recall that this
hypothesis states that linearity exists - - when in loga-

rithmic coordinates - - between the labor per unit input and

cumulative output.

The model should also provide a means of predicting
‘the exact course of a startup, not just its functional form.
ffhe usefulness of the model could be enhanced appreciably if
some method of predicting the _a_ and 't_) parameters of the
model for a given startup could be developed. The parameter
(:cyrrelation hypothesis and the novelty hypothesis provide

some basis to develop such a method.

In explaining these hypotheses a common element emergai
t311€3 potential for improvement. Implicit in the explanation
€33i4\7en.in.the last section was the assumption that the poten-
tial for improvement can be somehow estimated a priori and
that management will strive to reach the estimated productiy

i‘t2T57 'target. A model of the progress phenomenon should recog-

I1 ‘ .
3L“==<e the potential for improvement and include a method for

(1
e termining its magnitude.

146

The Traditional Model: Some Qualifications.

The power function y = ax.b offers a very efficient
description of the way in which the progress phenomenon
occurs in a manufacturing concern. The model has successful-
ly explained a large number of startups in diverse industrial
settings; This is well documented in the literature (Chapter
II). Nevertheless, the use of the device as a predictor re-
quires some clear definition of the estimating values of the

parameters of the function, namely, the g and b parameters.

Much of its use in the airframe industry implies the
'use of the manhour expenditure on the first airframe produuai
as a basis for extrapolation and use of a uniform 80% charag
‘teristic to define the Wright "slope". However, it seems
cioubtful, that the labor expenditure on the first unit
tJroduced can be determined with any accuracy. Even in the ah;
:Edrame industry parameter 3 represents a theoretical manhour
(arcpenditure on the first airframe produced. This is so be-
CZEIuse.the actual manhours figure is rarely known. Labor expeg
diture is typically recorded on a monthly as opposed to a peg
“11111t basis. As a result, the general practice in empirical
i1'1\7estigations has been to determine the y value (the average
value) and the corresponding value of x, for a "lot" of
Se\feral airframe units. The size of a lot will correspond to
the number of units produced during the month-long accountitg
I>ej:‘iod. Therefore, for the first empirical observation, x1

W‘
:L‘:LC1 represent the number of units produced during the first

F1"

147

month of production, and yl will be the average number of

manhours per unit expended on these x1 units.

Use of this lot convention, whose adoption is dictated
by industrial accounting procedures, has several consequencmm
the value of the g parameter is a theoretical one, except in
the rare instance that only one airframe unit is produced
during the first month of production. It can be argued that,
even if the first accounting period yielded a cost for the
first unit of an airframe, this cost figure would be suspect.
It is a rare cost system that could accurately determine the
cost of a single unit - - let alone the first unit - - of an
airframe, given the nature of the production process and the

industry.

Moreover, it seems unlikely that first piece labor
liours can be determined soon enough to enable their use as a
pxredictor. To be useful for production decisions related to
(Brigineering effort, production planning, manpower planning,
<31? design changes, estimates far in advance of production ana

necessary.

Estimating the Ultimate Point (xu, yu)

Given the difficulties in estimating the manhours con-
S"J-l'l'led by the first unit produced, the progress function may
gs't24i—ll be determined if some other point in future production
31-53 estimated. The so-called ultimate point (xu, yu) was

61
lready defined in Chapter II (p.39 ). Consider now some

148

aspects of its determination in practical work.

Conventional estimating procedures, say the Methods-
Time Measurement system (M.T.M.), are usually intended to
give an estimate of the time that will be required after the

6
operation has "settled down". This estimate corresponds to

the ordinate yu of the ultimate point. However, it is neces-
sary to associate the conventional estimate yu with a specif

ic point in cumulative production, x The most satisfactory

u'
way of doing this is to try not to disturb the existing
estimating procedure but to examine its past performance to
determine how long it took after production began for the

actual time to decrease to the vicinity of the estimate.

Presumably, (xu, yu) should be the point where the
Inlateau begins. In practice it may not happen exactly this
vvay. This fact does not impede the determination of the prog
ress function. It suffices that the estimated labor hours per
tJIlit at ultimate (yu) be reached at point xu in cumulative
EDITOduction. Therefore (xu, yu) continues to be a point on the
EDITOgress curve and can be used instead of the first point.In
13jilnns that have experience with production standards the

E>Zliateau can be anticipated with reasonable accuracy.

Once determination of point (x

u’ yu) is made, the dif-

Ifzi~<1ulties of obtaining first item labor hour are avoided. In
51':1<iition, the estimate yu is a practical means of defining
he potential for improvement. It represents a productivity

t:
éaljréget that management will try to reach. The potential for

149

improvement is given by (a - yu), i.e., the difference
between the first unit labor hours and the ultimate unit

labor hours.

Estimating Parameter b

The parameter correlation hypothesis provides a means
of determining the b-parameter. As previously mentioned,this
hypothesis was confirmed in at least two empirical studies.
The findings of our research - - to be presented in Chapter
6 - - also disclose the possibility of formalizing this
relationship in terms of a parameter model in which the two
parameters could be functionally related in the case of
startups that occurred in the same plant. Such results indi-

cate the usefulness of the following empirical equation:

b=m+nlna (5.1)
a
where a — yu

and m, n are constants in the model.

A Symbolic-Analytic Model of the

Manufacturing Progress Phenomenon

It is suggested by this author that the manufacturing
progress phenomenon can be described and its course predicted

by the following empirical equations:

150

g = ax (5.2)
b=m+nlna (5.3)
where
= 1L
9 yu
__ a
“'1'."
and

(xu, yu) is the estimated ultimate point.

If equations (5.2) and (5.3) prove to be empirically
valid, prediction of parameters 3 and b_for a new startup can

be developed as follows.

Equation (5.2) yields for x = x -

u'
1 = ax5b
u
_ b
a — x
u
By substitution in equation (5.3):
b
b = m + n 1n (x )
u
or solving for b
b = m (5.4)

 

1 - n 1n x

151

 

 

Finally,
m
_ b _ (l-n 1n xu)

a — xu — xu (5.5)

and
m
(_ )
a = y x l n ln xu (5.6)
u u

Thus, parameters g and p can be predicted by using formulas
7
(5.4) and (5.6).

GENERAL METHODOLOGY OF THE EMPIRICAL RESEARCH

An examination of the progress phenomenon in several
manufacturing concerns was a central feature of the study
objectives. The intention is to test the model offered in
the last section by using real data. A first methodological
decision was the selection of an adequate sample. As will be
apparent the empirical nature of the study did place very
definite practical limitations on the sample. The selection
of the study sample was largely shaped by two conditions - -
industrial cooperation and availability of reliable and

usable data.
Criteria in Sample Selection

Since empirical studies in non-airframe manufacturing

firms are rare, one of the criteria used in selecting the

152
sample was the desire to extend the model to a large number

of concerns outside the airframe industry.

A rapid rate of implementation of new products or'nmjor
changes in existing products was another criterion. The
reasons for this requirement are: the availability of suffi-
cient historical data for meaningful analysis and the future
usefulness of the research findings in predicting manufac-

turing startups.

Another aim was to gather data from facilities located

in diverse countries for the purpose of comparison.

It appeared desirable to analyse manhour data experi-
enced in production situations which have not used a progress

curve as a control device.

Finally it was intended to obtain aggregate data for
whole industries, for the purpose of possible macroeconomic

applications.

The Sample

Nine manufacturing firms representing five different
industries were ultimately selected for inclusion in the
study sample. These firms will be designated by letters A,B,

C,D,E,F,G,H, and I.

Firms A,B,C, and D are members of the electronic data

proCessing systems industry.

153

Firms A,B, and C are respectivehrthe Brazilian,CanadiaL
and Japanese representatives of the same multinational enteg
prise in the field of business machines manufacturing.In Eran
A twelve different examples of manufacturing startups were
analyzed. One startup was studied in each of the firms B and
C. The processes considered cover the manufacturing of both
conventional and advanced products in the field. Startups in
these three firms occurred in a period ranging from 1960

until mid-1973.

Firm D is a leading manufacturer of business machines
in the United States. The net sales for 1958 amount to over
half a billion dollars. Data was obtained for two different
programs. The first set of data cover man-hours expended in
the manufacture of an electronic data processing system. The
second set represents small computer components and one
control unit. A total of fourteen startups were available for
study. The data obtained was that for assembly operations

only and coversaaperiod ranging from 1953 until 1960.

Firm E is an American firm in the electrongcs manufac-
turing field, which started operations in 1946. Sales
ran: approximately ten million dollars annually in 1960.
Small electronic motors is the speciality of the company.The
firm's output for which data was obtained consists entirely
of specialized components used in electronic data processing
systems and automated control systems. In firm E fifteen

startups were analyzed. The manufacture of the focused

154
computer components started in 1949 and ended in 1951. Only

assembly time data was obtained from firm E.

Firm F is one of the oldest printing press and bindery
machine manufacturers in the United States].-0 It is considered
a leader in its field. Sales in 1959 were over twenty-five
million dollars.At firm F data was obtained for all products
manufactured from 1950 until 1960 yielding a total of twenty-

five startups. Data obtained from this manufacturer repre—

sents both assembly and machining time.

Firm G is one of the oldest Brazilian shipyards. Its
shipbuilding capacity was 10,000 tons of steel in 1973. The
sales in the same year were over ten million dollars.At this
firm data was obtained for only one type of ship. However ,
the nineteen startups made available cover all manufacturing
operations involved in its production. Startups occurred in.a

period ranging from August 1971 until March 1973.

Firm H is the leading Brazilian manufacturer of oil
drilling tools and equipment. At this firm data was made
available for only three of the products manufactured. Data
represents assembly time. The three startups occurred from

July 1971 until May 1973.

Firm I is a Brazilian manufacturer of light airplanes.
Data was obtained for parts, major assemblies and complete
airframes, yielding a total of sixty-nine startups that

ocCurred in two different plants during 1972.

155

Whole Industries. Aggregate data was obtained for the
Brazilian Mechanical and Electrical Industry in a period that
is considered the "take-off" period of Brazilian industrial-
ization in these fields (1960 - 1964). This industry will
be represented by letter J. Data is subdivided into nine
groups as follows: (J1) Castings, (J2) Forgings, (J3) Mechap
ical Machinery, (J4) Electrical Machinery, (J5) Industrial
Equipment, (J6) Locomotives, Wagons and Railway Equipment,
(J7) Shipbuilding, (J8) Roadbuilding Tractors and Equipment,
and (J9) Farm Machinery. A more detailed description of the

products included in each aggregate is found in the following

11
Chapter.
Summarizing the above, nine manufacturers - - repre-
senting five different industries - - are included in the

sample. A hundred and fifty-nine separate cases of product
and process startups that occurred in four different
countries and nine distinct plants have been analyzed . In
addition aggregate data was obtained for whole industries in

one country, yielding nine more startups.

Measuring the Progress Phenomenon

After selecting the sample, a second methodological

question had to be considered, that is, the way of measuring

the startups.

p 156

Each of the startups was analyzed in relation to two
variables, an index of productivity (dependent variable) and
a measure of cumulative output (independent variable), as
indicatedin.equation (5.2).The progress of the different
startups was examined periodically in relation to the normal
production statistics developed by the participating firms.
As to the dependent variable,direct-labor hours data are
usually calculated and reported in relation to standard
accounting time periods, yielding average direct-labor hours
per unit figures for the lot of product produced during the
accounting period. As to the independent variable, cumulathna
output statistics indicate the total output of the product
(from inception of manufacture) that is achieved at the end

of the accounting period.

Direct comparisons of productivity between different
products or processes are not possible with a direct—labor
per unit measure. For example, process 1 may have a theoreti
cal ultimate of yu1,whereas process 2's theoretical ultimate

is y It is meaningless to compare the productivity of the

u2
two processes in absolute terms. In addition, the direct-laun:
hours per unit measure creates a disclosure problem; the pa;
ticipating firms do not wish to have the actual labor hours

per unit of their operations and products revealed in abso-

lute terms.

These inconveniences are largely avoidable. As already

mentioned, it is standard practice to develop predetermined

157
engineered standards of performance (yu). Thus it is possflfle
to define an index of productivity ($1) that relates the
actual direct-labor hours per unit cogsumed to the produc—

tion standard and take it as the independent variable. This

is the meaning of y in equation (5.2).

A final general methodological question refers to the
measurement period used in the study. Production records are
summarized on a monthly basis in all individual firms. The
physical output of the various products and operations, as
well as the productivity indices, are calculated at the end
of each calendar month. In most cases, monthly observations
give a good description of the course of a startup. In a few
examples, bi-weekly or weekly observations might have provid
ed a better representation of the progress phenomenon. The
main effect was a reduction in sample size, which has some

statistical implications.

In the next chapter the findings in the firms and indus
tries will be described and analyzed. There, specific meth-

odology will be introduced whenever needed.

CHAPTER VI

EMPIRICAL FINDINGS

In this chapter the manufacturing progress model pres-
ented in chapter V will be tested with real data from sever-
al industries. The analysis of the startups encountered in
each of the industries studied has been divided into six
separate sections, in this order: (1) Manufacturing of Elec-
tronic Data Processing Systems and Components (Firms A,B,C,D
and E); (2) Manufacturing of Bindery Machines and Printing
Presses (Firm F); (3) Shipbuilding (Firm G); (4) Manufactur-
ing of Oil Drilling Tools and Equipment (Firm H); (5) Produg
tion of Airframes (Firm I) and (6) Mechanical and Electrical

Industry (designated by letter J).

STARTUP ANALYSIS IN THE MANUFACTURING OF ELECTRONIC
DATA PROCESSING SYSTEMS AND COMPONENTS

Characteristics of the Products and

Manufacturing Processes Studied

Wherever possible, some of the characteristics of pro-
ducts and processes studied will be revealed. For obvious

reasons, however, a complete disclosure is not possible.

Firms A,B and C. The Products studied at firms A, B

and C comprise’ card punch machines of two types and the

158

159

central processing unit (CPU) of a third generation computer.
The card punches will be designated by letters X (a modern
model) and Y (an older model). Technically, the focused prod
ucts and components are electromechanical or electronic. The
card punch is one of the units in the so-called peripheral
equipment of almost all electronic data processing systems.
Before any data can be read and properly processed, it must
be recorded in: proper form. Data recording is performed by
the card punch. Bearing a striking resemblance to an electric
typewriter, the card punch's keyboard contains alphabetic ,
numeric, and special characters. Recording is accomplished
by stroking the keys one key at a time, as the card moves
from right to left. Modern card punches print or interpret

directly over the columns as they punch ' holes in the card.

The central processing unit studied is a two-level
memory system, made up of large-size processor (main) storage
which functions as a backing storage for smaller, monolithic
buffer storage. The processor is capable of requesting eight
bytes from the buffer every 80 nanoseconds. Memory capacity

is up to 3,100,000 characters.

The manufacturing of these products is broadly divided
into the following processes: (1) parts machining, (2) subag
sembly, (3) final assembly and (4) testing. Because of the
various features and combinations of features available to
customers, the major portion of the machines i1; built to

cUstomer order. It is not a mass production organization in

160

comparison with an appliance or an automobile manufacturer.
The machine shop delivers parts on a short production run
basis. The subassembly departments produce units in lots,and
the final assembly departments operate on a continuous prodq;
tion basis. Engineering changes resulting from customer
requests for new features, cost reductions, or standardiza-
tion of components or units are introduced throughout pro-

duction of a machine.

Table 6.1 summarizes the startups analyzed in firms A,

B and C of the same multinational company.

Firms D and E. The products of firm D comprise
several second generation computer components. Although these
parts would at a later date be assembled by the buyer into a
unified data processing system, each one of the computer
components was especially designed and different from the
other units. The company made these to order for one of the
larger electronics manufacturers in the United States. It
should be stressed that the data obtained was for ‘assem-
bly operations only. Close to 70 per cent of the parts are
procured from outside vendors, and these are charged to

material cost.

The products of firm E for which data was obtained
consist entirely of second generation computer components
and data storage units. As with firm D, only assembly data
was obtained from firm E,because 1) Almost 80 per cent of

the parts going into the assembly are procured from outside

161

TABLE 6.1

STARTUPS ANALYSED IN FIRMS A, B AND C

 

 

Firm Country Startup Code General Description

 

 

.A Brazil A1,A2,A3, A4 Startups A1 through A4 comprise
and AS the final assembly operations
of the CPU of a third generation
computer.
Startup A5 corresponds to the

total final assembly of that

unit.

A6 Total manufacturing,card punch X.

A7 Parts machining, card punch X.

A8 Subassembly, card punch X.

A9 Final Assembly, card punch X.

A10 Testing, card punch X.

All Final Assembly and Testing, card
punch X.

A12 Assembly of major units of card
punch Y.

?B Canada B1 Final Assembly and Testing, card

punch X.

C Japan C1 Final Assembly, card punch Y.

 

162
vendors, and 2) manhour data from parts produced by the com-
pany are misleading - - company E has a policy of manufactur
ing certain parts in lot sizes greater than assembly lots.
The parts in excess of assembly needs are stored for future
assembly needs or issued as spare parts orders are received.

Table 6.2 summarizes the startups studied in firms D and E.

TABLE 6.2

STARTUPS ANALYZED IN FIRMS D AND E

 

 

 

 

Firm Country Startup Code General Description
TD United States Dl Control Unit no. 2

D2 High Speed Printer no. 1
D3 High Speed Printer no. 2
D4 Card Reader Unit no. 1
D5 Tape Perforation Unit no.1
D6 Magnetic Tape Unit no. 1
D7 Control Unit no. 1A
D8 Computer no. 1A
D9 Computer no. 2A
D10 Computer no. 3A
D11 Computer no. 4A
D12 Computer no. 5A
D13 Computer no. 6A

D14 Computer no. 7A

163
TABLE 6.2 (cont'd)

 

 

Firm Country Startup Code General Description

E Umited States E1 Computer no. 2B
E2 Computer no. 3B
E3 Computer no. 4B
E4 Computer no. 5B
E5 Computer no. 6B
E6 Data Storage Unit no. 1B
E7 Data Storage Unit no. 2B
E8 Data Storage Unit no. 3B
E9 Computer no. 7B
E10 Computer no. 8B
Ell Control Unit no. 1B
E12 Computer no. 1B

 

Startup Measurement and Data

In firms A, B and C data is usually reported in rela-

tion to standard accounting time periods (month),yie1ding

average direct labor hours per unit figures for the lot of

product delivered during the accounting period. However, in

the case of startups A1 through AS, the direct labor hours

were available on a per unit basis. To obtain a common basis

of analysis cumulative average man-hours per unit (y) were

computed for all startups analyzed in this dissertation.

164
The dependent variable in equation (5.2) is a produc-

tivity index given by:

=1
9! Yu

This index relates the actual performance of the opera
tion (y) to predetermined, engineered standards of perfor-
mance (yu). Therefore, for each startup, the cumulative
average man—hours per unit figures were divided by the re-

spective engineered estimate yu.

Cumulative output statistics indicate the total output
of the product (from inception of manufacture) that is
achieved at the end of the accounting period. In the case of
startups Al through AS, the completion of each unit is not

necessarily achieved at the end of the accounting period.

Startup data in this modified form pertaining to firms

A,B and C is shown in Tables 6.3 through 6.7 (Appendix F ).

In firms D and E data is usually reported in lot-aver-
age form. In the case of startups D1 through D6 data was
available on a per unit basis. Cumulative average man-hours
per unit figures (y) were computed for all startups. Since
engineered estimates (yu) were not available, the dependent
variable y was calculated by dividing the cumulative average
figures (y) by the y value of the last observation in each

startup.

165

The job order cost system used by Company D is similar
to job order cost systems found in other situations.Brief1y,
when an order is placed on a computer unit, a ledger is set
up for the particular order. The particular computer will
have a unit number assigned to it and the various operations
on the unit are assigned job numbers. The worker's time
tickets are then posted to the particular job number and
then to the ledger. The company charges the time of the in-

spection force to direct labor.

Firm E has a modified job order cost system of accumu—
lating time data for fixed interval units in the total quan-
tity produced. Thus, the first two lots consist of five unifis
each. Thereafter time data is accumulated for lots of ten ,

and occasionally for a lot of twenty.

Data in this modified form for firms D and E is shown

in Tables 6.8 through 6.10 (Appendix F ).

Startup Analysis

The descriptive efficiency of the manufacturing pro-
gress model, given by equation (5.2) is supported by the
results of regression analysis of the startups obtained from
firms A,B,C,D and E. The detailed results are displayed in
Trable 6.11, startups A1 through E12 (Appendix F ). In each
(:ase, the results were obtained by regressing the data
1relating to the startup phase against the log-transformed

:form of the proposed model using the least-squares criterion.

166
Clearly, by taking logarithms of both sides of equation

(5.2), it follows that
log y = log a - blogx (6.1)

All regressions were machine-computed (CDC 6500)using

a FORTRAN program written by this author (Appendix C).

Table 6.11 shows the coefficients of correlation (r)
and determination (r2), the t-ratios, the a parameter values
and the b parameter values obtained in each regression, as
well as the number of observations (N) upon which the regreg
sions are based, the Wright "slope" and the startup codes

previously assigned.

The closeness of fit can be judged by the correlation
coefficients (r) or the coefficients of determination (r2).
The coefficient of determination, being the square of the
correlation coefficient, is more sensitive to any deviations
from the regression line. Thus, it is a more revealing mea-
sure of the closeness of fit. Also, it indicates the propor-

tion of the total variance of the dependent variable that

is explained by the regression line.

The t-ratios given in the table measure the relative
sizes of the regression coefficients (b) and the standard
(deviation of the regression coefficient. The t-ratio is
<3btained by dividing the calculated regression coefficient b

13y its own standard error.

167
Examination of the t-ratios and coefficients of deter-
mination (Table 6.11) pertaining to startups A1 through E12
will demonstrate the general efficiency of the fit provided

2 values vary from 0.590 to

by the regression model. The r
1.000, with a median value of 0.960. Hence, in one-half of
the cases,the startup model explains 96% or more of the

total variance in the dependent variable. In only three
out of forty cases does the regression fail to explain at

least 8 Z of the variance. Also, in only six out of forty

cases,the coefficients of determination were less than 0.910.

The t-ratios are also generally impressive, ranging
from 2.08 to 61.2. (In the case of startup D7 the t-ratio

was extremely large.) The median value is 10.1.

If one is willing to make the necessary assumptions of
normality and common variance, the null hypotheses p = O and
B = 0 can be rejected at the 0.99 level of significance in
all cases except the following: D5, where the level is 0.975;
A7, where the level is 0.95; A8 and D11, where the level is
0.90, and D8 where the level is 0.75.

Parameter Analysis

The parameter model given by equation (5.3) is sup-
jported by the results of regression analysis of the startup
fparameters obtained from firms A,D and E. Data from firms B

.and C is insufficient for a parameter analysis.

168

For firm A, the degree of correlation in the avail-
able data can be judged from Table 6.12, where the ¢1and p
values of the different process startups have been presented
in order of decreasing a value for each product analyzed.
Startups A5, A6 and All are not included because they repre-
sent aggregation of individual processes already taken into
account in the table. Startup A12 is not included because it
refers to an older model of card punch. This startup occurred

some ten years before the startups that are considered in the

 

 

 

 

table.
TABLE 6.12
FIRM A PARAMETERS E AND E
Startup General Description a b
Code
A3 Final assembly operations 2.707 0.311
A4 of the CPU of a 2 492 0.231
A1 third generation 1.898 0.173
_y _____ cww9£_____________lpzflgfl
A10 Testing, card punch X 4.876 0.205
A9 Final Assembly, cd punch X 3.253 0.132
A7 Parts Machining, cd punch X 1.721 0.0875
A8 Subassembly, cd punch X 1.516 0.0509

 

169
The strength of the relationships can be assessed from

the regression results shown in Tables 6.13 and 6.14.

TABLE 6.13

PARAMETER MODEL REGRESSION RESULTS
(FIRM A, CPU OF A 3rd. GENERATION COMPUTER)

 

 

Number of
m n r2 t-ratio
Observations

 

 

4 -0.0l77 0.305 0.889 4.01

 

TABLE 6.14

PARAMETER MODEL REGRESSION RESULTS
(FIRM A, CARD PUNCH X)

 

 

Number of

m n r t-ratio
Observations

 

 

4 0.0090 0.118 0.954 6.46

 

Both the coefficients of determination (r2) and the
t-ratios are sufficiently large to conclude that the parame-
ter model given by equation (5.3) provides a good descrip-

tion of the data available from firm A.

170
If the assumptions of normality and common variance
are made, the null hypotheses p = 0 and B = 0 can be
rejected at the 0.975 level of significance in the case of
card punch X (Table 6.14) and near the 0.95 level in the came

of the central processing unit (Table 6.13).

It can be concluded that the parameter model expressed
by equation (5.3) was found empirically valid among processes
of the same product, for card punch X and for the central
processing unit of a third generation computer. The assem-
bling of the central processing unit is a final "product"

for firm A.

The available data was insufficient to check the empig

ical validity of the model among products of firm A.

As to firms D and E, their data may be used as an indi
rect means of supporting the credibility of the correlation
between the jgand 2 parameters. Since the engineered esti-
mates (yu) for each startup were not available, a difficulty
arose as to the way of calculating the indices of produc-
tivity (y) as defined in (5.2). As previously mentioned, the
indices of productivity in the case of firms D and E starfiqu;
(Tables 6.8, 6.9 and 6.10) were calculated by dividing the
cumulative average figures (y) by the y value of the last
observation in each startup. Intuitively, the last observa-
tion in each startup is closer to a possible steady-state
phase (estimated by yu) than any other previous observation

in that startup, provided that the progress function is a

171

strictly decreasing one. Thus, the crand 2 parameters per-
taining to startups D1 through E12 (Table 6.11, Appendix )
were obtained by regressing the data in this modified form
against the log-transformed form of the proposed model

(equation 6.1), using the least-squares criterion.

The degree of correlation in the available data can now
be judged from Table 6.15 (Firm D) and Table 6.16 (Firm E),
where the czand E values of the different product startups

have been presented in order of decreasing czvalue.

The strength of the relationships can be assessed from

the regression results shown in Tables 6.17 and 6.18.

Although the regression results are not quite as favor
able as those obtained in the case of firm A, they still sub
stantiate the possibility'that the parameter model would have
been empirically supported if the true estimates yu were
used in order to calculate the productivity indices y in the
case of firms D and E.This possibility is strengthned if the
data is adjusted according to the following rationale. Intui
tively, if a generic startup X exhibits a smaller percentual
decline in the labor hours per unit from the (x - l)th obseg
vation to the xth observation (the xth observation being the
last one available) than another startup Y, then startup X
is probably closer to a steady-state phase than startup Y.
From a practical point of view , the beginning of a plateau
might be defined as that point in cumulative production at

which the reduction in labor hours per unit from the (x-l)th

172
TABLE 6.15

FIRM D PARAMETERS _g AND 13

 

 

 

 

Startup
General Description a b
Code .
* D12 Computer no. 5A 7.002 0.406
' D10 Computer no. 3A 6.945 0.409
D13 Computer no. 6A 6.683 0.409
* D7 Control Unit no. 1A 5.252 0.333
D11 Computer no. 4A 3.521 0.220
* D2 High Speed Printer no. 1 3.230 0.297
* D6 Magnetic Tape Unit no. 1 2.882 0.205
* D1 Control Unit no. 2 2.649 0.368
* D4 Card Reader Unit no. 1 2.483 0.275
D9 Computer no. 2A 1.625 0.0929
* D3 High Speed Printer no. 2 1.341 0.0936
D14 Computer no. 7A 1.333 0.0533
D5 Tape Perforation Unit no. 1 1.230 0.0543
D8 Computer no. 1A 1.156 0.0226

 

173
TABLE 6.16

FIRM E PARAMETERS

3 AND E

 

 

 

 

Startup
General Description a b

Code

E9 Computer no. 7B 9.602 0.365

El Computer no. 2B 8.889 0.377
* Ell Control Unit no. 1B 5.948 0.211
* E6 Data Storage Unit no.1B 4.497 0.146

E7 Data Storage Unit no.2B 4.359 0.276
* E12 Computer no. 1B 4.349 0.238

E4 Computer no. 5B 4.034 0.235
* E5 Computer no. 6B 3.911 0.175
* E3 Computer no. 48 3.548 0.174
* E10 Computer no. 8B 3.076 0.179

E8 Data Storage Unit no.3B 3.017 0.214
* E2 Computer no. 3B 2.471 0.142

 

174
TABLE 6.17

PARAMETER MODEL REGRESSION RESULTS

 

 

 

 

 

(FIRM D)
Number of 2
m n r t-ratio
Observations
14 0.0242 0.204 0.870 8.97
TABLE 6.18

PARAMETER MODEL REGRESSION RESULTS
(FIRM E)

 

 

Number of 2
m n r t-ratio
Observations

 

 

12 0.0312 0.161 0.687 4.69

 

175

observation to the xth observation (the xth observation.behug
the last one) is some negligible figure. Recall that this
procedure was adopted in a previously mentioned research by

Schultz and Conway (Chapter II, p.39 ).

The startups with an asterisk in Tables 6.15 and 6.16
have the largest percentual decline in the labor hours per
unit from the (x - l)th to the xth observation (the xth
observation being the last one). From the available data ,
startups D12 , D1 ,1D4 and D3 have a percentual decline
greater than 3% in the labor hours per unit from the (x-l)fl1
observation to the last observation x. If they are excluded
from the regression computation, then the results obtained

from the remaining startups in Table 6.15 are as shown in
Table 6.19.
TABLE 6.19

PARAMETER MODEL REGRESSION RESULTS
(FIRM D, STARTUPS D12, le, D4 and D3 EXCLUDED)

 

 

Number of 2
m n r t-ratio
Observations

 

 

10 -0.0039 0.211 0.973 16.98

 

176
Now, if all startups with more than TLZ of percentual
decline are excluded from the computation (i.e., all start-
ups with an asterisk in Table 6.15), then the regression
results obtained from the remaining startups in Table 6.15

are as shown in Table 6.20.

TABLE 6.20

PARAMETER MODEL REGRESSION RESULTS

(FIRM D, STARRED STARTUPS IN TABLE 6.15 EXCLUDED)

 

 

 

 

Number of 2
m n r t-ratio
Observations
7 -0.0070 0.211 0.988 20.29

 

The increasingly improved regression results from
Tables 6.17, 6.19 and 6.20 seem to support the conjecture
that the parameter model would have been empirically vali-
dated among products of firm D if the true estimates yu were
available for calculating the productivity indices y in the
case of firm D startups. The results from Table 6.20 show
that if one is willing to make the assumptions of normality
and common variance, the null hypotheses p = O and B = 0 can

be rejected at the 0.995 level of significance.

177

Similarly,in the case of firm E, Tables 6.21 and 6.22
exhibit the regression results when the startups with more
than 2% and 1% of percentual decline,respective1y,are excluded
from the regression computation. Again, the increasingly im-
proved regression results from Tables 6.18,6.21 and 6.22.8mmn
to support the conjecture that the parameter model wouldluwe
been empirically validated among products of firm E if the
true estimates yu were available for calculating the productiy
ity indices y in the case of firm E startups.The results Emma

Table 6.22 show that if one is willing to make the assumptions
TABLE 6.21

PARAMETER MODEL REGRESSION RESULTS
(FIRM E, STARTUPS E11 AND E10 EXCLUDED)

;

 

 

 

Number of m n r2 t-ratio
Observations
10 -0.0204 0.171 0.760 5.03
TABLE 6.22

PARAMETER MODEL REGRESSION RESULTS
(FIRM E, STARRED STARTUPS IN TABLE 6.15 EXCLUDED)

 

 

 

 

Number Of 2 _ .
Observations m n r t ratio
5 0.0530 0.143 0.967 9.35

 

pof normality and common variance,the null hypothesesga=0 and

B=0 can be rejected at the 0.995 level of significance.

178
STARTUP ANALYSIS IN THE MANUFACTURING OF
BINDERY MACHINES AND PRINTING PRESSES

Characteristics of the Products and

Manufacturing Processes Studied

The products studied at firm F comprise bindery ma-
chines and printing presses of several types. Technically ,
the focused products are electromechanical. The means and
methods of manufacture are labor-intensive and feature a
large component of assembly work. The manufacture of these
products is broadly divided into machining and assembly work.
The company initiates the production process only after
orders are received. Normally, action on the first few
purchase orders is delayed until enough additional orders
are received to make up what is considered a sufficient
number of units for the lot. Over 95% of the parts used in
assembly are manufactured by the company. It may be added
that a large percentage of the parts are bought in the form
of unfinished castings and then machined to specification in

the plant.

Table 6.23 summarizes the startups analyzed in firm F.

Startup Measurement and Data

In firm F,data is reported in lot-average form. Cumulg
tive average man-hours per unit figures (y) were computed

for all startups. Since engineered estimates (yu) were not

179
TABLE 6.23

STARTUPS ANALYZED IN FIRM F (U.S.)

 

 

 

 

Startup Code Designation
Fl through F16 Binding machines (no.1 through no.16)
F17 through F28 Printing presses (no.1 through no.12)

 

available, the dependent variable g was calculated by divid-
ing the cumulative average figures (y) by the y value of the
last observation in each startup. Cumulative output statis-
tics (x) indicate the total output of the product (from incep
tion of manufacture) that is achieved at the last unit of

each lot.

The job order cost system used by Company F is similar
to the cost system found in Company E or in Company D. For
this reason no further description is necessary.Data obtained
from this manufacturer represents both assembly and ma-

chining time.

Data in this modified form for firm F may be found in

Table 6.24 (Appendix F ).

180

Startup Analysis

The descriptive efficiency of the manufacturing pro-
gress model given by equation (5.2) is also supported by the
results of regression analysis of the startups obtained from
F. Examination of the t-ratios and coefficients of determing
tion (Table 6.11) pertaining to startups Fl through F28
reveahsthat the r2 values vary from 0.657 to 0.999, with a
median value of 0.985. Hence, in one-half of the cases, the
startup model explains 98.5% or more of the total variance
in the .dependent variable. In only two out of twenty-eight
cases does the regression fail to explain at least 88.9% of

the variance.

The t-ratios are also generally large, ranging from

3.05 to 81.05, with a median value of 16.3.

If the assumptions of normality and common variance
are made, the null hypotheses p = 0 and B = 0 can be.rejected
at the 0.99 level of significance in all cases except the
following: F3 and F13, where the level is 0.975; F2, where

the level is near 0.95 and F15, where the level is 0.90.

Parameter Analysis

As in the case of firms D and E, for which the engi-
neered estimates (yu) were not available,the data from firm F
may still be used as an indirect means of supporting the

'credibility of the correlation between the g_and.b_parameters.

181
The procedure adopted for firms D and E is played
again . The initial degree of correlation in the available
data from firm F can be judged from Table 6.25, where the g
and b_values of the different product startups have been

arranged in order of decreasing 3 value.

.TABLE 6.25

FIRM F PARAMETERS _g AND 9

 

 

 

 

 

Startup -

Designation a b
Code
F16 Binding machine no.16 4.94 0.246
F24 Printing press no. 8 4.74 0.228
F20 Printing press no. 4 4.47 0.235
F13 Binding machine no.13 3.85 0.271
F1 Binding machine no. 1 3.57 0.238
F17 Printing press no. 1 3.42 0.237
F15 Binding machine no.15 3.22 0.219
F22 Printing press no. 6 3.17 0.201
F28 Printing press no.12 3.16 0.227
F23 Printing press no. 7 2.80 0.133
F19 Printing press no. 3 2.71 0.186
F18 Printing press no. 2 2.70 0.176
F14 Binding machine no.14 2.59 0.172
F25 Printing press no. 9 2.41 0.120
F21 Printing press no. 5 2.02 0.123

182

TABLE 6.25 (cont'd)

 

 

 

 

Startup

Designation a b
Code
F5 Binding machine no. 5 2.01 0.147
F6 Binding machine no. 6 1.99 0.150
F4 Binding machine no. 4 1.97 0.184
F27 Printing press no.11 1.95 0.131
F9 Binding machine no. 9 1.93 0.136
F3 Binding machine no. 3 1.83 0.110
F12 Binding machine no.12 1,80 0,148
F10 Binding machine no.10 1.80 0.128
F7 Binding machine no. 7 1.77 0.123
F26 Printing press no.10 1.57 0.0731
F2 Binding machine no. 2 1.49 0.0604
F11 Binding machine no.1l 1.48 0.0758
F8 Binding machine no. 8 1.36 0.0551

 

The strength of the relationship can be assessed from

the regression results shown in Table 6.26.

Tables 6.27 and 6.28 exhibit the regression results
when the startups with more than 5% and 2% of ultimate per-
centual decline, respectively, are excluded from the regres-

sion computation.

183
TABLE 6.26

PARAMETER MODEL REGRESSION RESULTS

 

 

 

 

 

 

 

 

 

 

 

 

(FIRM F)
Number of
m n r2 t-ratio
Observations
28 0.0292 0.150 0.835 11.5
TABLE 6.27
PARAMETER MODEL REGRESSION RESULTS
(FIRM F, STARTUP F4 EXCLUDED)
Number of
m n r2 t-ratio
Observations
27 0.0244 0.153 0.865 12.6
TABLE 6.28
PARAMETER MODEL REGRESSION RESULTS
(FIRM F,STARTUPS F4 AND F5 EXCLUDED)
Number of 2 _
Observations m n r t ratio

 

 

26 0.0229 0.154 0.867 12.5

 

184

The improved regression results from Tables 6.27 and
6.28 seem to indicate that the parameter model would have
been better supported among products of firm F if the true
estimates yu were available for calculating the productivity
indices y in the case of firm F startups. Although the coef-
ficient of determination (r2) is not quite as favorable as
those obtained in the case of firms D and E, it demonstrates
that little of the variance in parameter b was left unex-
plained by the regression (See Table 6.28). The t-ratio also
remains reliably large. The results from Table 6.28 show that
if normality and common variance are assumed, then the null
hypotheses p = 0 and B = 0 can be rejected at the 0.995 level

of significance.

Since the products studied at Firm F comprise bindery
machines and printing presses it seemed worthwhile to
check if the parameter model would be supported among
products of the same kind. The initial degree of correlation
can be assessed from Tables 6.29 (binding machines) and 6.30
(printing presses) where the a and b parameter values have

been arranged in order of decreasing a value (Appendix F ).

The strength of the relationships can be judged from
the regression results exhibited in Tables 6.31 (binding ma-

chines) and 6.32 (printing presses).

Tables 6.33 and 6.34 exhibit the regression results for
(binding machines when the startups with more than 2% and 1%

of ultimate percentual decline, respectively, are excluded

185
TABLE 6.31

PARAMETER MODEL REGRESSION RESULTS
(FIRM F, BINDING MACHINES)

 

 

Number of

 

 

 

 

 

 

 

 

 

 

 

 

m n r2 t-ratio
Observations
16 0.0270 0.162 0.884 10.3
TABLE 6.32
PARAMETER MODEL REGRESSION RESULTS
(FIRM F, PRINTING PRESSES)
Number of
m n r2 t-ratio
Observations
12 0.0160 0.153 0.809 6.50
TABLE 6.33
PARAMETER MODEL REGRESSION RESULTS
(FIRM F, BINDING MACHINES, F4 AND F5 EXCLUDED)
Number of 2 t- t’
Observations m n r ra 1°
14 0.0209 0.165 0.922 11.9

 

186
TABLE 6.34

PARAMETER MODEL REGRESSION RESULTS
(FIRM F, BINDING MACHINES; F4,F5,F3 AND F2 EXCLUDED)

 

 

Number of
m n r t-ratio
Observations

 

 

12 0.0304 0.158 0.924 11.0

 

from the regression computation.

Similarly, Tables 6.35 and 6.36 show the regression
results for printing presses when startups with ultimate peg
centual decline greater than 0.5% and 0.3%, respectively,are
excluded from the regression computation. Again, the
results seem to indicate that the parameter model would have
been even better supported among products of the ggmg kind
at firm F, had the true estimates yu been available for calcg
lating the productivity indices y in the case of the focused

startups.

The regression results from Tables 6.34 (binding ma-
chines) and 6.36 (printing presses) show that if normality
and common variance are assumed, then the null hypotheses
p = 0 and B = 0 can be rejected at the 0.995 level of signif

Iicance in both cases.

187
TABLE 6.35

PARAMETER MODEL REGRESSION RESULTS
(FIRM F, PRINTING PRESSES , F28 EXCLUDED)

 

 

 

 

Number of 2
m n r t-ratio
Observations
11 0.0170 0.148 0.834 6.73
TABLE 6.36

PARAMETER MODEL REGRESSION RESULTS
(FIRM F, PRINTING PRESSES; F28,F21,F17 AND F22 EXCLUDED)

 

 

Number of 2
m n r t-ratio
Observations

 

 

8 0.0204 0.139 0.862 6.13

 

188
STARTUP ANALYSIS IN SHIPBUILDING

Characteristics of The Product and

Manufacturing Processes Studied

The product studied at firm G is a ferry-boat of 1,250
dead-weight tons (DWT) of which five units were built. The
means and methods of manufacture are labor—intensive and

feature a large component of assembly work.

The production of these ships is divided into three
main groups of operations called: (1) Services, (2) Hull and

(3) Equipment.

The Services group includes the following operatflxm:
machining, transportation and carpentry , and accounts for
approximately 11 per cent of the total manhours expended in

the production of the ship.

The Hull group comprises the following operations:

lofting, laying out, manual cutting, automatic cutting,bend-

 

ing, subassembly preparation, hull blocks subassembly, manual

 

welding, automatic welding, chamfering, carbon chamfering,
planing, and scaffolding. This group uses approximately
55 per cent of the total manhours expended in the production
of the ship. The three underlined operations use 75 per
cent of the labor hours expended in the group, or 41 per cent

of the total manhours expended in the construction of the

'ship.

189
The Equipment group comprehends the following opera-
tions: piping (fabrication), piping (assembly), parts fabri-
cation, deck equipment subassembly, machinery subassembly,
electrical assembly, machinery assembly, carpentry, painting,
and rigging. This group uses approximately 34 per cent

of the total manhours expended in the production of the Ship.

Table 6.37 summarizes the startups analyzed in Firm G.

Startup Measurement and Data

In firm G the data was available on a per unit (ship)
basis. The reports issued by the Production Planning and
Control Department of that firm exhibit the direct labor
hours per unit expended in each startup from G1 through G19.
Cumulative average manhours per unit (y) were computed for
all startups analyzed. Then, for each startup the cumulative
average manhours per unit figues were divided by the respec-
tive engineered estimate yu yielding the productivity index

figures (y) of equation (5.2).

Cumulative output statistics indicate the total output
of the product (ships) from inception of manufacture. The
completion of each ship is not necessarily achieved at the

end of the accounting period.

Startup data in this modified form is shown in Table

6.38 (Appendix F ).

190
TABLE 6.37

STARTUPS ANALYZED IN
FIRM G (BRAZIL)

 

 

 

 

Startup
General Description
Code
G1 Production of Complete Ferry-Boats
G2 Services Group of Operations
G3 Hull Group of Operations
G4 Equipment Group of Operations
G5 Carpentry (fabrication)
G6 Manual Cutting (Hull)
G7 Hull Blocks Subassembly
G8 Manual Welding
G9 Piping (fabrication)
GlO Piping (assembly)
G11 Parts Fabrication
G12 Deck Equipment Subassembly
G13 Machinery Subassembly
G14 Electrical Assembly
G15 Machinery Assembly
G16 Carpentry (assembly)
G17 Painting
G18 Rigging
G19 Manual Cutting (Equipment)

 

191

Startup Analysis

The descriptive efficiency of the manufacturing prog-
ress model given by equation (5.2) is generally supported by
the results of the regression analysis of the startups ob-

tained from firm G.

Examination of the coefficients of determination (Table
6.11) pertaining to startups G1 through G19 reveals that the
r2 values vary from 0.054 to 0.999, with a median value of
0.924. Hence, in one-half of the cases, the startup model
explains 92.4% or more of the total variance in the .depen-
dent variable. In six out of nineteen cases the regression
fails to explain at least 81.7% of the variance. However, in
three of these six cases the available data cannot be consid
ered reliable. According to the Chief-Engineer of the Produg
tion Planning and Control Department, the data pertaining to
startups G13, 615 and G8 should not be considered reliable

since the timekeepers were new in the job and were being

trained for that purpose.

Unfortunately it was not possible to have the data
corrected within the period of time that the authoraflJocated
for his research in the focused shipyard. Therefore,it seems
advisable to exclude startups G13, G15 and G8 from the
present analysis. In doing so, the new median value for r2
will be 0.929. Now, in only three out of sixteen cases does

pthe regression fail to explain at least 81.7% of the total

variance in the dependent variable. The t-ratios are

192

generally large, ranging from 2.39 to 53.08, with a median

value of 6.27.

If the assumptions of normality and common variance are
made, the null hypotheses p = 0 and B = 0 can be rejected at
the 0.99 level of significance in all cases except the follql
ing: G1 and G4, where the level is 0.975; G12 and G2, where
the levels are 0.95 or near 0.95, respectively, and G7 and

G10, where the level is 0.90.

Parameter Analysis

The parameter model given by equation (5.3) is sup-
ported by the results of regression analysis of the startup
parameters obtained from firm G. The degree of correlation
in the available data can be judged from Table 6.39, where
the cland p values of the different process startups have
been presented in order of decreasing a.va1ue for the only
product analyzed. Startups G1, G2, G3 and G4 are not included
because they represent aggregation of individual processes
already taken into account in the table. Startups G13, G15

and G were excluded for reasons previously explained.

The strength of the relationship can be judged from

the regression results shown in Table 6.40.

Both the coefficient of determination r2 and the
t-ratio are sufficiently large to conclude that the parame—

’ ter model given by equation (5.3) provides a good description

193
TABLE 6.39

FIRM G PARAMETERS CIAND b

 

 

 

 

Startup
General Description a b

Code

G16 Carpentry (assembly) 2.109 0.2663
G14 Electrical Assembly 1.964 0.2517
G10 Piping (assembly) 1.955 0.1397
09 Piping (fabrication) 1.448 0.1131
Gll Parts Fabrication 1.361 0.1007
617 Painting 1.356 0.1058
G18 Rigging 1.293 0.0900
G6 Manual Cutting (Hull) 1.275 0.0878
G5 Carpentry (fabrication) 1.273 0.0727
619 Manual Cutting (Equipment) 1.150 0.0509
G7 Hull Blocks Assembly 1.101 0.0303
G12 Deck Equipment Subassembly 1.022 0.0363

 

194

TABLE 6.40

PARAMETER MODEL REGRESSION RESULTS
(FIRM A, FERRY-BOAT OF 1250 DWT)

 

 

 

 

Number of 2
Observations m n r t-ratio
12 0.0110 0.298 0.867 8.07

 

of the data from firm G. If the assumptions of normality and
common variance are made, the null hypotheses p = 0 and B==0

can be rejected at the 0.995 level of significance.

It can be concluded that the parameter model.expressed
by equation (5.3) is empirically supported among processes
of the same product, in the case of the particular type of
ferry-boat analysed in firm G. Since this was the only
product studied in that firm the empirical validity of the

parameter model among its products cannot be established.

STARTUP ANALYSIS IN THE MANUFACTURING
OF OIL DRILLING AND PRODUCTION EQUIPMENT

Characteristics of The Products and

Manufacturing Processes Studied

The products studied at firm.H may be generally de-

scribed as tools and equipment for oil drilling mmiproduction.

195
The manufacturing of these products is broadly divided into:
(1) Machining operations and (2) Assembly operations. The
bulk of the machining work corresponds to the cutting of
internal threads. The assembly operations are mainly per-
formed by welding. Table 6.41 summarizes the startups ana-

lyzed in firm H.

TABLE 6.41

STARTUPS ANALYZED IN
FIRM H (BRAZIL)

 

 

 

 

 

 

Startup
Designation
Code
H1 Packers
H2 Cement Retainers
H3 Shoes

 

Startup Measurement and Data

In firm.H the data was available in lot-average form.
Cumulative average manhours per unit figures (§) were com-
puted for all startups.Then, for each startup the cumulative
average manhours per unit figures were divided by the respeg
tive engineered estimate yu yielding the productivity index

. figures (g) of equation (5.2).

196
Cumulative output statistics indicate the total output
of the product (from inception of manufacture) that is

achieved at the end of the accounting period (month).

Startup data in this modified form is shown in Table

6.42 (Appendix F ).

Startup Analysis

The descriptive efficiency of the manufacturing prog-
ress model given by equation (5.2) is generally supported by
the results of the regression analysis of the startups avail
able from firm H (Table 6.11). In all cases the startup model
explains 84.5% or more of the total variance in the indepen-
dent variable. If the assumptions of normality and common
variance are made, the null hypotheses p = 0 and B = 0 can
be rejected at the 0.99 level of significance in two cases

(H3 and H2), and at the 0.95 level in one case (H1).

Parameter Analysis

The small number of observations available is not suf-
ficient for a reliable regression analysis of the startup
parameters obtained from firm H.Notwithstanding,the existing

data favors the parameter model given by equation (5.3).

The degree of correlation in the available data can be
judged from Table 6.43, where the a and b_parameter values

‘ of the different product startups have been arranged in

197

order of decreasing a value.

TABLE 6.43

FIPI'I H PARAMETERS a AND b

 

 

 

 

Startup
General Description a b
Code
H3 Shoes , 3.132 0.2564
H2 Cement Retainers 2.520 0.1524
H1 Packers 1.450 0.0702

 

The strength of the relationship can be assessed from
the regression results shown in Table 6.44.
TABLE 6.44

PARAMETER MODEL REGRESSION RESULTS
(FIRM H, OIL DRILLING AND PRODUCTION EQUIPMENT)

 

r
r

Number of 2
m n r t-ratio

Observations

 

 

3 -0.0220 0.2236 0.905 3.08

 

198
If the assumptions of normality and common variance
are made, the null hypotheses p = 0 and B = 0 can be rejected

at the 0.80 level of significance.
STARTUP ANALYSIS IN THE AIRFRAME INDUSTRY

Characteristics of the Products and

Manufacturing Processes Studied

The products studied at firm I comprise light airplanes

of four types, to be designated by the letters W,X,Y and Z.

Airplane W is a two-seat plane fabricated with steel
tubes, wood and fabric, and propelled by a 90 HP engine. It

is intended for primary training purposes.

Airplane)(iszu1executive four-seat plane, entirely fab-

ricated with aluminum alloys and propelled by a 180 HP enghua

Airplane Y is the military version of airplane X.
Finally, airplane Z is a two-or-three-seat 290 HP-engine
plane, intended for intermediate trainingzuuiacrobatics, and

equipped for navigation instruction.

The means and methods of manufacture in the airframe
industry are labor-intensive and feature a large component
of assembly work. Airframe production can be roughly classi-
fied into eleven classes of operations, as follows: (1) Ma-
chining Operations, (2) Sheet metal work, (3) Processing: i3

cluding painting, heat treating, etc., (4) Primary Assembly,

199
(5) Wing subassembly, (6) Fuselage subassembly, (7) Miscellg

neous subassembly (e.g., control surfacesauuielectrical
tubing), (8) Major wing assembly, (9) Major fuselage assembly,
(10) Final Assembly, and (11) Ramp work and miscellaneous

operations.

This classification of operations gives a good indica-
tion of the sequential nature of the production flow. The
producing departments are interrelated; the output of one
serves as the input of others. The assembly operations are
almost totally labor—paced. The machining and fabrication
Operations are partially machine-paced, but owing to the
discontinuous job-lot nature of much of this work, labor-

pacing is still very important.

Table 6.45 summarizes the startups analyzed in firm I.

Startup Measurement and Data

In firm I the data was available both in lot-average
and in cumulative-average form (y). For each startup, the
cumulative average manhours per unit figures were divided by
the respective engineered estimate yu, yielding the produc-

tivity index figures (g) of equation (5.2).

Cumulative output statistics indicate the total output
of the product (airplanes) from inception of manufacture
Startup data in this modified form is shown in Table 6.46

(Startup I1 through I4).

200
TABLE 6.45

STARTUPS ANALYZED
IN FIRM I (BRAZIL)

 

 

 

 

 

 

 

 

Startup
General Description
Code
11 Total Manufacturing, airplane W
12 Total Manufacturing, airplane X
I3 Total Manufacturing, airplane Y
I4 Major Assemblies , airplane Z
15 through 169 Spare parts, airplanes W,X,Y and Z
TABLE 6.46
AIRFRAME INDUSTRY (BRAZIL)
FIRM I
Startup
Code x = cumulative production 0 = productivity index
11 x... 26 52 83 116 146 164
y...1.30 1.29 1.18 1.11 1.08 1.12
I2 xd.. 5 15 25 35 45 55 60 65 75 80
y...l.45 1.43 1.40 1.35 1.29 1.21 1.18 1.16 1.13 1.11
I3 x... 5 10 15 20 25 30 34 39 40
y...l.54 1.46 1.39 1.33 1.28 1.20 1.20 1.17 1.17
I4 x... 2 5 9 14 20 28 36 44 52

 

...2.64 2.27 2.05 1.87 1.69 1.53 1.42 1.35 1.29

k:

201
In the case of startups 15 through 169, parameters g and b
as well as the engineered estimates (yu) for each spare part
were already available in firm 1. However, the startup data
was not made available by that firm. The modified parameter
czwas calculated by dividing parameter g'by the correspond-
ing engineered estimate yu. Thus, Table 6.47 (Appendix F )
exhibits the parameters a and b for each spare part. The

data is arranged in order of decreasing a value.

Startup Analysis

The descriptive efficiency of the manufacturing program;
model given by equation (5.2) is supported by the results of
the regression analysis of the startups available from firm
I (Table 6.11). In all cases the startup model explains 78.5%
or more of the total variance in the dependent variable. If
the assumptions of normality and common variance are made,the
null hypotheses p = 0 and B = 0 can be rejected at the 0.995

level of significance in all cases.

Parameter Analysis

The parameter model given by equation (5.3) is sup-
ported by the results of regression analysis of the startup
parameters obtained from firm I. The degree of correlation
in the available sata can be judged from Tables 6.48

(Products) and 6.47 (Spare parts, Appendix F ).

202
TABLE 6.48

FIRM I PARAMETERS a AND E

 

 

 

 

(PRODUCTS)
Startup
General Description a b
Code
14 Major Assemblies, airplane Z 3.240 0.226
I3 Manufacturing, airplane Y 1.994 0.142
11 Manufacturing, airplane W 1.877 0.106
I2 Manufacturing, airplane X 1.833 0.103

 

The strength of the relationships can be judged from the
regression results exhibited in Tables 6.49 (products) and

6.50 (spare parts).

TABLE 6.49

PARAMETER MODEL REGRESSION RESULTS
(FIRM I PRODUCTS, STARTUPS I1 THROUGH I4)

 

 

Number of 2
m n r t-ratio
Observations ”

 

 

4 -0.0179 0.209 0.965 7.43

 

203
TABLE 6.50

PARAMETER MODEL REGRESSION RESULTS)
(FIRM I, SPARE PARTS, STARTUPS IS THROUGH I69)

 

 

 

 

Number of 2
m n r t-ratio
Observations
-4 3
65 1.424x10 0.217 1.000 5.22x10

 

Both the coefficients of determination r2 and the
t-ratios are sufficiently large to conclude that the param-
eter model given by equation (5.3) provides a good descrip-
tion of the data from firm I. If the assumptions of normalflnr
and common variance are made, the null hypotheses p = 0 and
B = 0 can be rejected at the 0.975 level of significance,in
the case of products (Table 6.49) and at the 0.9995 level ,

in the case of spare parts (Table 6.50).

It can be concluded that the parameter model given by
equation (5.3) is empirically supported among products of

firm I and also among spare parts of products of firm I.

204
STARTUP ANALYSIS IN THE MECHANICAL
AND ELECTRICAL INDUSTRY

An industry is an aggregate of components. It can be
argued that if progress in components is widespread, it
should be reflected in aggregate performance. W.B. Hirschmann
has shown that a logarithmic plot of manhours per barrel
versus cumulative barrels of crude oil refined in the United
States since 1860 results in a fairly regular type ofdeclineas
such a reasoning would lead us to expect. Hirschmann has
shown that other industries (e.g. electric power and basic
steel industries) also exhibit similar declines.1 It seems

reasonable to infer that improvement curve patterns can exist

in other areas as well.

In this section the data gathered from the Brazilian
mechanical and electrical industry during its "take-off"
period (1960—1964) is analyzed and discussed. The data is
found to support the manufacturing progress model given by

equations (5.2) and (5.3).
The Product Groups Studied

As mentioned in Chapter V, the product groups analyzed
in the Brazilian mechanical and electrical industry are the
following: (J1) Castings, (J2) Forgings, (J3) Mechanical
Machinery, (J4) Electrical Machinery, (J5) Industrial Equip-
ment, (J6) Locomotives, Wagons and Railway Equipment, (J7)

Shipbuilding, (J8) Roadbuilding Tractors and Equipment, and

205
(J9) Farm Machinery.

2
According to the source document the above classifi-

cation was used as a basis for the determination of the
input-output matrix of the mechanical and electrical indus-
trial sector of the Brazilian economy. The products consid-
ered in each group have similar technological processing. A
list of the most representative products in each group is

given below.

(J1) Castings. Cast iron parts processed by manual molding
and machine molding. Cast steel parts produced by manual

molding and machine molding.

(J2) Forgings. Forged carbon and alloy steel parts produced

by drop forging or press forging processes.

(J3) Mechanical Machinery. Machine—tools, cranes, centrifu

gal pumps and stonebreakers.

(J4) Electrical Machinery. Generators and alternators,

transformers, DC and AC motors.

(J5) Industrial Equipment. Liquid gas tanks, coolers, and

steam boilers.

(J6) Locomotives, Wagons and Railway Equipment. Vans, ore
wagons, passenger-cars, electrical locomotives, diesel-

hydraulic and diesel-electric locomotives.

206
(J7) Shipbuilding (powerplants excluded). Passenger-liners,
cargo ships, freezers, tank-ships, tugs, motor-boats,

barges, flatboats and ferry-boats.

(J8) Roadbuilding Tractors and Equipment. Bulldozers,
caterpillar tractors, levellers, dump trucks, scrapers,

asphalters, crushers and loaders.

(J9) Farm Machinery. Farm tractors (micro, light, medium

and heavy), ploughs, sowing machines and harvesters.

Startup Measurement and Data

For each startup in industry J, the data was available
in manhours per metric ton, for each year of the period
(1960-1964). Also available was the number of metric tons
produced per year for each group of products considered. Cu-
mulative average manhours per metric ton figures (y) were com
puted for all startups. Then, for each startup, the cumula-
tive average manhours per unit figures were divided by the
y value of the last observation in each startup (1964 figure)

yielding the productivity index figures (y) of equation (5.2).

Cumulative output statistics indicate the aggregate
output (in metric tons) of all products in each focused group
from inception of manufacture (1960). Startup data in this

modified form is shown in Table 6.51 (Appendix F ).

207

Startup Analysis

The descriptive efficiency of the manufacturing prog-
ress model given by equation (5.2) is supported by the
results of the regression analysis of the startups from indug
try J. Examination of the coefficients of determination
(Table 6.11) pertaining to startups J1 through J9 reveals

that the r2

values vary from 0.882 to 0.992, with a median
value of 0.979. Hence, in one-half of the cases, the startup
model explains 97.9% or more of the total variance in the
dependent- variable. In only two out of nine cases does the

regression fail to explain at least 90.4% of the variance.

The t-ratios are also generally large, ranging from

4.74 to 19.38, with a median value of 11.82.

If the assumptions of normality and common variance are
made, the null hypotheses p = 0 and B = 0 can be rejected at
the 0.995 level of significance in all cases except the fol-

lowing : J7, J6 and J9, where the level is 0.975.

Parameter Analysis

Since the last observation in each startup corresponds
to the beginning of a plateau (1964), the data from industry
J can be used as a direct means of supporting the credibility
of the correlation between cland 2 parameters. The degree of
correlation in the available data can be judged from Table

6.52, where the a and p values of the different product gram;

I [‘1 lull

208

startups have been arranged in order of decreasing a value.

TABLE 6.52

INDUSTRY J PARAMETERS a AND b

 

 

 

 

Startup
General Description a b
Code
J8 Roadbuilding Tractors and Equip. 5.960 0.1598
J4 Electrical Machinery 3.708 0.0969
J7 Shipbuilding 3.293 0.0883
J9 Farm Machinery 3.219 0.0891
J5 Industrial Equipment 2.672 0.0732
J1 Castings 2.423 0.0655
J6 Locomotives,Wagons and Rwy.Equip. 2.058 0.0594
J3 Mechanical Machinery 1.968 0.0498
J2 Forgings 1.831 0.0457

 

The strength of the relationship can be judged from

the regression results exhibited in Table 6.53.

Both the coefficient of determination r2 and the
t-ratio are sufficiently large to conclude that the parameter
model given by equation (5.3) provides a good description of
the data from industry J. If the assumptions of normality

and common variance are made, the null hypotheses p=0 and B=0

209
TABLE 6.53

PARAMETER MODEL REGRESSION RESULTS
(INDUSTRY J, STARTUPS J1 THROUGH J9)

 

 

 

 

Number of 2
m n r t-ratio
Observations
9 -0.0133 0.0909 0.965 9.07

 

can be rejected at the 0.995 level of significance.

It can be concluded that the parameter model given by
equation (5.3) is empirically supported among groups of prod
ucts of similar technology within the mechanical and electri

cal industrial sector of the Brazilian economy in the period

1960-1964.

SUMMARY

In the foregoing chapter the manufacturing progress
model presented in chapter V was tested with real data from
nine manufacturers representing five different industries.
A hundred and fifty-nine separate cases of product and
process startups that occurred in four different countries
and nine distinct plants have been analyzed. In addition ,
aggregate data was obtained for whole industries in one

country, yielding nine more startups.

210
The major points that have emerged from the discussion

can be recapitulated as follows.

The descriptive efficiency of the manufacturing prog-
ress model given by equation (5.2) is generally supported by
the results of regression analysis of the startups obtained
from firms A,B,C,D,E,F,G,H,I, and industry J. Considering all

2 vary from

startups, the coefficients of determination r
0.590 to 1.000, with a median value of 0.9685. Hence, in one~
half of the cases, the startup model explains 96.85% or more
of the total variance in the dependent variable.ln only amI

per cent of the cases does the regression fail to explain at

least 81.7% of the total variance in the dependent variable.

The t-ratios are also generally impressive, ranging
from 2.08 to 99.99, with a median value of 10.89. If the
assumptions of normality and common variance are made, the
null hypotheses p = 0 and B = 0 can be rejected at the 0.95

level of significance in 95% of the startups analyzed.

The descriptive efficiency of the parameter model
given by equation (5.3) is generally supported by the results
of regression analysis of the startup parameters from firms
A,D,E,F,G,H,I and industry J. Data available from firms B
and C was insufficient for a parameter regression analysis.
The parameter model is empirically supported among processes
of the same product (firms A and G), among products within
the same plant (firms D,E,F,H and I),and among groups of

products of similar technology within the same industrial

211

sector of a foreign economy (industry J).

The findings of this research together with similar
results obtained by Asher in the airframe industry and by
Baloff in the steel, glass manufacturing,and automobile
industries3 constitute adequate evidence to suggest that the
parameter model can be developed into an effective means of
predicting the parameter b of a new startup. However, addi-

tional investigation and validation is a mandatory require-

ment for successful industrial application of the model.

One aspect of the model that requires attention is the
stability of the coefficients m and g, In Appendix D this

subject is explored by using the available data.

CHAPTER VII

IMPLICATIONS OF THE FINDINGS

The present study is concluded with a discussion of due
industrial implications of the findings reported in Chapter
VI. The importance of recognizing and predicting the manufag
turing progress phenomenon is now related to several decision-
making functions that are encountered in an industrial concern
as well as in economic planning at the national level.An.ove£
all design of a computerized Manufacturing Progress Function

(MPF) System is also suggested.
IMPLICATIONS AT FIRM LEVEL

In Chapter II the term "Manufacturing Progress Functnx1
Hours" (MPF Hrs) was defined as those direct-labor hours over
and above the estimated hours which are caused by the intro-
duction of a new unit into a manufacturing system. The term
"Manufacturing Progress Function Cost" (MPF Cost) was defined
as the cost associated with the MPF hours. In Chapter IV some

methods for calculating the MPF hours were proposed and assessed.

Knowledge of the MPF hours and cost may be of extreme
importance to management,first,in deciding whether or not to
put a proposed product or‘a major change inan existing product
into production,and secondly, in pricing a product to recoup

these introductory costs.

212

213

The Manufacturing Progress Function is applicable not
only 1 to complete machines,but tosmaller units or features as
well. Even manufacturers who produce new models each year
seldom produce units which are completely new. The MPF can
then be applied to major changes in existing products to
determine their effect on the cost. Many of the proposed
changes are of a cost reduction nature. But not every pro-
posed change can result in lower costs. It may be advisable
to submit all major changes to an MPF study. If the MPF Cost
exceeds a given percentage of the anticipated savings for a
predetermined period (say, one or two years), the change
might well be returned to Product Engineering for reanalysis.
Also, if more than one choice exists for Engineering Changes,
the Manufacturing Progress Function provides additional cost

information for the proper selection.

Once it has been decided to produce a new machine other
questions must be answered. How many assembly personnel will
be required each month while production is building up? How
much floor space will be required and when? The Manufacturing
Progress Function provides answers to these questions. In
addition, the knowledge of the MPF Cost - - prior to and.dur-
ingproduction - - makes possible the channeling of special engi
neering and staff efforts to those programs or activities of
a program which show high MPF Costs. By doing this, it is
possible to effect a more rapid rate of progress and reduce

the MPF Cost through engineering changes or methods changes.

214
In the following subsections the practical use of the
Manufacturing Progress Function at firm level is illustrated

with a sample problem.

Problem Statement

The management of firm X wishes to know the MPF Hours
and Cost associated with the implementation of a certain
product P so as to decide whether or not to put product P
into production and to price it in order to recoup the intrg

ductory MPF Cost.

The manufacturing of product P comprehends the follow-
ing operations: (1) Machining, (2) Subassembly, (3) Mechani-
cal Assembly, (4) Electrical Assembly and (5) Testing.

The following data is also known:

TABLE 7.1

MANUFACTURING PROGRESS FUNCTION AND COST DATA

 

 

 

 

 

 

 

 

Y T r .

u u (monetary units
Operation (hrs/unit) (months) per hour )
Machining 30 12 10
Subassembly 10 13 9
Mechanical Assy. 6 20 6
Electrical Assy. 20 18 10

Testing 4 24. 12

 

215

TABLE 7.2

PRODUCTION SCHEDULE
(PRODUCT P)

 

 

Month Year 1 Cum. Year 2 Cum.
JAN 10 10 40 280
FEB 10 20 40 320
MAR 10 30 40 360
APR 10 40 40 400
MAY 10 50 40 440
JUN 10 60 70 510
JUL 20 80 80 590
AUG 20 100 80 670
SEP 20 120 80 750
OCT 40 160 80 830
NOV 40 200 80 910
DEC 40 240 80 990

 

In Table 7.1, Tu is the estimated period of time (in
months) elapsed from the inception of production until the
ultimate unit of production xu is reached. Recall that xu can
be practically defined as that point in cumulative productflxi
at which the reduction in manufacturing hours per unit from
the first unit in the month to the last unit for the month
is between 2% and 3% (or some other negligible figure) and

thus can be considered nominal. The yu values correspond to

216
the estimated direct-labor hours required to produce a unit
at the ultimate unit of production. The E values are the
labor and burden hourly rates for each type of manufacturing
operation. The production schedule of product P appears in

Table 7.2

From past research it is anticipated that parameters b
and a.of the operation startups for product P will be related

as follows:
b = 0.0410 + 0.132 1n a (7.1)

It is also known that the best fit to the empirical data was

achieved with the unit progress function.

Management also wishes to know how many assembly persqg
nelznuihow much floor space will be required each month While
production is building up, considering a time horizon of 12

months.

Solution

(3) Determination of The Ultimate Unit of Production

For Each Operation.

Once Tu is estimated for each category of operation
from historical data, the value of xu is obtained from the
production schedule by searching for the unit in cumulative
production that corresponds to the estimated Tu' For example,

for the machining operation, Tu = 12 months (see Table 7.1).

217

From the production schedule (Table 7.2) the unit in
cumulative production that corresponds to Tu = 12 months is
the 240th unit. Thus, xu = 240. The other xu values for each
operation can be determined in a similar way. The results are
exhibited in Table 7.3, together with the corresponding yu

values, for better visualization.

TABLE 7.3

ULTIMATE HOURS AND UNITS
(PRODUCT P)

 

 

 

 

Operation xu yu
Machining 240 30
Subassembly 280 10
Mechanical Assy. 670 6
Electrical Assy. 510 20
Testing 990 4

 

(b) Calculation of Parameter b Values.

Prom empirical equation (7.1) it follows that m=0.0410
and n=0.132. Since the best fit to the empirical data was
achieved with the unit progress function, parameter p values can
be calculated through formula (5.4), as follows:

0.0410

. . _ m _ =
MaChlnlng b ‘ 1 - n 1n xu ‘ 1 — 0.132 1n 240 0°148

 

 

218
0.0410

 

 

 

 

Subassembly b = 1 _ 0.132 1n 280 = 0.160
Mech. Assy. b = 1 _ 0?lg:lIn 670 = 0.291
Elect.Assy. b = 1 _ 071g21$n 510 = 0.232
Testing b = 1 _ 0?1331In 990 = 0.458

(c) Calculation of Parameter g Values

The parameter 3 value of each manufacturing operation

can be calculated through formula (5.6) as exhibited in

 

 

 

 

Table 7.4 .
TABLE 7.4

PARAMETER g VALUES CALCULATED

ACCORDING TO FORMULA (5.6)
Operation . xu yu b a
Machining 240 30 0.148 67.51
Subassembly 280 10 0.160 24.63
Mech.Assy. 670 6 0.291 39.86
Elect. Assy. 510 20 0.232 84.95

Testing 990 4 0.458 94.20

 

219

For example, in the case of the machining operation,

formula (5.6) yields:

0.148

a = y x: = 30 x 240 = 67.51

(d) MPF Hours and Cost Calculation

Exact Method

 

The MPF Hours and Cost associated with the imple-
mentation of product P may be calculated as exhibited in
Table 7.5 . In column (1) and for each operation, the exact
total hours expended in the manufacturing of unit 1 up to
and including unit xu were machine-computed1 through formula
(4.2), repeated below for easy reference:

X

51
y = a x
T 1

'b (7.2)

For example, in the case of the machining operation,
the total hours expended in the manufacturing of unit 1 up
to and including unit 240 is given by:

240
YT (240) = 67.51 2

x=l

x‘°°148 = 8,420.5498

In column (2), the rectangular areas x yu are also

u
calculated (see Figure 4, Chapter IV). In column (3), the
MPF Hours are computed by subtracting the values in column

(2) from the corresponding values in column (1). Finally, the

220
MPF Cost figures in column (5) are calculated by multiplying
the MPF Hours in column (3) by the corresponding labor and
burden rates given in column (4). The totals for the product

are presented at the bottom of the table.

TABLE 7.5

MPF HOURS AND COST CALCULATION
(EXACT METHOD)

 

 

 

 

(1) (2) (3) (4) (5)

. MPF MPF

Operation yT (xu) xuyu Hrs. 'r Cost
Machining 8,420.5498 7,200 1,220.5498 10 12,205.50

Subassembly 3,321.0222 2,800 521.0222 9 4,689.20
Mech. Assy. 5,637.5737 4,020 1,617.5737 6 9,705.44
E1ect.Assy. 13,223.9939 10,200 3,023.9939 10 30,239.94
Testing 7,184.8316 3,960 3,224.8316 12 38,697.98

 

Totals 37,787.9712 28,180 9,607.9712 95,538.06

 

Approximate Method

 

The MPF Hours and Cost associated with the implementation
of product P may be calculated as exhibited in Table 7.6.For
each operation, the MPF Hours in column (4) are computed

according to formula (4.18), repeated below:

 

 

 

 

 

 

 

 

(xu + 0.5)1‘b - 0.51'b
MPF Hours 5 xu y - 1 (7.3)
u (1 - b) xi-b
_ J
TABLE 7.6
MPF HOURS AND COST CALCULATION
(APPROXIMATE METHOD)
(1) (2) (3) (4) (5) (6)
MPF MPF
Operation xu y b r
u Hrs. Cost
Machining 240 30 0.148 1,221.80 10 12,218.00
Subassembly 280 10 0.160 521.95 9 4,697.55
Mech. Assy. 670 6 0.291 1,618.56 6 9,711.36
Elect. Assy. 510 20 0.232 3,026.29 10 30,262.90
Testing 990 4 0.458 3,228.90 12 38,746.80
Totals ............ 9,617.50 95,636.61

 

For example, in the case of the Subassembly operation,

it follows that

0.840

0.840 5
840

280 x 10 (280 5
0.840 x 280

_ 1)

MPF Hours ' 8'

521.95

222
The MPF Cost figures (column 6) are calculated by mul-
tiplying the MPF Hours (column 4) by the corresponding labor
and burden rates given in column (5). The totals for the
product are presented at the bottom of the table. Note that
formulas (4.20) or (4.22) might have been used instead for
calculating the MPF Hours per operation.2 In any case the

approximate results exhibited in Table 7.6 compare well with

those from Table 7.5.

(e) Total Labor Requirements

Exact Method

 

The total labor hours associated with the implementation
of product P, considering a planning horizon of 24 months,
may be calculated as shown in Table 7.7. For each operation,
the total labor hours were machine-computed through exact
formula (7.2). Total labor hours for product P are presented

at the bottom of the table.

Approximate Method

 

The total labor hours associated with the introduction
of product P may be advantageously calculated by employing
aggregation formulas (4.43) and (4.44) derived in chapter IV.
These formulas are particularly useful when the number of
products and/or the number of operations in the manufacturhu;
of a given product are large. The calculations are carried

out as follows.

223
TABLE 7.7

TOTAL LABOR HOURS FOR PRODUCT P

(EXACT METHOD)

 

 

 

 

Operation Calculation
990
Machining yT = 67.51 2 x-0-148 = 28,229.92690
1 1
990
Subassy yT = 24.63 2 x-O'160 = 9,615.009714
2 1
, _ 99° -o.291 _
Jech.Assy yT — 39.86 2 x - 7,445.505004
3 1
99° -0 232
E1ect.Assy yT = 84.95 X x ' = 22,044.95553
4 l
990
Testing yT = 94.20 2 x‘0-458 = 7,184.831639
5 1
Total ...74,520.22878

 

(1) Aggregate Parameter Calculation

From (4.33), one must have

0<x52c

224

Also, from the data
xefl:l, 990]

Therefore, by taking the mid-point of the interval:

By using parameter E and b_values from Table 7.4 and
applying formulas (4.43) and (4.44) with c = 495, the fol-

lowing results are obtained:

67.51 x 0.148 24.63 x 0.160 39.86 x 0.291

+ + +
0.148 .
495 4950'160 4950 291

 

 

 

B = (

 

 

84.95 x 0.232 + 94.20 x 0.458 ;( 67.51 +_ 24.63 +_
0 232 001458 )' ‘“‘UTIEE “‘UTI66
495 495 495 495
39.86 4_ 84.95 4_ 94.20 ) = 14.5442 = o 2131
W W W ——...2623
and
A = cB D = 4950'2131 x 68.2623 = 256.0951 , where

D is the denominator of B.

(2) Aggregate Progress Function for Product P

In view of the above results the aggregate progress fung

tion for P in the given planning period (24 months) can be

225

written as follows:
y = 256.1 x‘0-2131 (7.4)

(3) Total Labor Hours Calculation

The total labor requirement associated with the introdug
tion of product P into the manufacturing system can now be
determined by using formula (4.4) as follows:

1-3 _ 256.1 1-o.2131_
11

___;L ___— -
yT - 1_B x 1_002131 x 990 74,089.28

The per cent deviation with respect to the exact result

given in Table 7.7 is -0.58%.

(f) Labor and Space Requirements On

A Periodic Basis

Exact Method

 

As an example, the labor hours and space required by the
Final Assembly (Mechanical and Electrical) and Testing operg
tions can be determined for year 1, on a monthly basis, as
exhibited in Tables 7.8, 7.9,and 7.10. In each table (column
3), the total labor hours yT from inception of production up
to and including the last unit of the ith monthly lot of
production are machine-calculated according to formula (7.2).
For example, the total hours expended in the Mechanical Assam

bly of product P from the beginning of production up to and

226
TABLE 7.8

LABOR AND SPACE REQUIREMENTS
(MECH. ASSY, EXACT METHOD)

 

 

 

 

(l) (2) (3) (4) (5) (6) (7) (8)
Cum. 5 (3')

Month Prod. yT yL 11 (n') (m2) (m2)
Jan 10 262.4548 262.4548 1 1.5 6 9.0
Feb 20 443.1955 180.7407 1 1.0 6 6.0
Mar 30 598.8934 155.6979 1 0.89 6 5.3
Apr 40 740.1337 141.2403 1 0.80 6 4.8
May 50 871.4692 131.3355 1 0.74 6 4.4
Jun 60 995.3966 123.9274 1 0.70 6 4.2
' Jul 80 1,226.7595 231.3629 1 1.3 6 7.8
Aug 100 1,441.7975 215.0380 1 1.2 6 7.2
Sep 120 1,644.6459 202.8484 1 1.2 6 7.2
Oct 160 2,023.2437 378.5978 2 2.2 12 13 2
Nov 200 2,375.0450 351.8013 2 2.0 12 12.0
Dec 240 2,706.8551 331.8101 2 1.9 12 11.4

 

I—‘
U1

Totals.... 2,706.8551 (15.4) 90 (92.5)

 

including the last unit of the "lot" produced in March (lot

3) are given by:

x-0.291

yT = 39.86 2 = 598.8934

227

TABLE

7.9

LABOR AND SPACE REQUIREMENTS

(ELECT.ASSY, EXACT METHOD)

 

 

 

 

 

(1) (2) (3) (4) (5) (6) (7) (3)
Month Cum. y y n (n') S (S‘)
Prod. T ' T (m2) (m2)

JAN 10 606.5843 606.5843 3 3.4 15 17

FEB 20 1,058.6636 452.0793 3 2.6 15 13
MAR 30 1,460.1249 401.4613 2 2.3 10 11.5
APR 40 1,831 5912 371.4663 2 2.1 10 10.5
MAY 50 2,182.1436 350.5524 2 2.0 10 10.0
JUN 60 2,516 8424 334.6988 2 1.9 10 9.5
JUL 80 3,150.4663 633.6239 4 3.6 20 18.0
AUG 100 3,748.1947 597.7284 3 3.4 15 17.0
SEP 120 4,318.7567 570.5620 3 3.2 15 16.0
OCT 16o 5,398.6586 1,079.9019 6 6.1 30 30.5
NOV 200 6,417.1926 1,018 5340 6 5.8 30 29.0
DEC 240 7,389 3200 972.1274 6 5.5 30 27.5
Totals... 7,389 3200 42 (41.9) 210 (209.5)

 

228

TABLE 7.10

LABOR AND SPACE REQUIREMENTS

(TESTING , EXACT METHOD)

 

 

 

 

 

(1 (2) (3) (4) (5) (6) (7) (8)

Cum. 5 (5')

Month Prod. YT yL 11 (n') (m2) (m2)
JAN 10 498.4215 498.4215 3 2.8 18 16.8
FEB 20 770.1003 271.6788 2 1.6 12 9.6
MAR 30 984.7455 214.6452 1 1.2 6 7.2
APR 40 1,168.8237 184.0782 1 1.1 6 6.6
MAY 50 1,332.9817 164.1580 1 0.93 6 5.6
JUN 60 1,482.7963 149.8146 1 0.85 6 5.1
JUL 80 1,751.6943 268.8980 2 1.5 12 9.0
AUG 100 1,991.3166 239.6223 1 1.4 6 8.4
SEP 120 2,209.8999 218.5833 1 1.2 6 7.2
OCT 160 2,602.0582 392.1583 2 2.2 12 13.2
NOV 200 2,951.3915 349.3333 2 2.0 12 12.0
DEC 240 3,269.9788 318.5873 2 1.8 12 10.8
Totals 3,269.9788 19 (18.6) 114 (111.5)

 

229
In column (4) of each focused table the labor hours yL
i
expended in the production of the ith lot (i.e., the lot

produced in the ith month) are computed as follows:

YT"

y
Li 1 T(1-1)

For example, the labor hours expended in Mechanical

Assembly of the lot produced in March are calculated astxflow:

yL3 = 598.8934 - 443.1955 = 155.6979
In column (5) of each table considered, the number of
personnel required on a monthly basis is computed assuming
that a person works 176 hours per month (one shift). Thus,
column (5) is calculated by dividing column (4) figures by
176 and rounding the results to the nearest integer. For
example, the Mechanical Assembly operation of product P

during March would require:

H

113 = Igé = 0.89 5 1 operator

As another example, the Electrical Assembly operation

in March would require

n = ——— = 2.3 5 2 operators

Column (6) in each table contains non-rounded figures

for‘n.

230

In column (7) of each focused table, the number of
square meters required on a monthly basis is computed assum-
ing the following coefficients of working area per person:

(1) Mechanical Assembly: 6 m2 per person; (ii) Electrical

2 per person and (iii) Testing: 6 m2 per person.

Assembly: 5 m
For example, the area required for the mechanical assembling

of product P during March is given by:

The same coefficients were applied to the non-rounded figures
of column (6) so as to obtain column (8) results in each

table considered.

The labor hours and space required by the Final Assem-
bly and Testing operations are summarized in Table 7.11. The
lot figures in column (3) were obtained by totalling the
corresponding lot figures in Tables 7.8, 7.9 and 7.10. For
example, the total hours expended in the Final Assembly and
Testing operations of product P during March are computed as

follows:

yL = 155.6979 + 401.4613 + 214.6452 = 771.8044
3

The total number of personnel required on a monthly
basis (column 4) is obtained by summing up the corresponding
figures in Tables 7.8, 7.9 and 7.10. Thus, the total number

Of personnel required for assembling and testing product P

TABLE 7.11

231

LABOR AND SPACE REQUIREMENTS

(FINAL ASSY.AND TESTING, EXACT METHOD)

 

 

 

 

 

(1) (2) (3) (4) (5) (6) (7)
Cum. S (8')

Month Prod. yL 11 (n') (m2) (m2)
JAN 10 1,367.4606 7 7.7 39 42.8
FEB 20 904.4988 6 5.2 33 28.6
MAR 30 771.8044 4 4.4 22 24.0
APR 40 696.7848 4 4.0 22 21.9
MAY 50 646.0459 4 3.7 22 20.0
JUN 60 608.4408 4 3.5 22 18.8
JUL 80 1,133.8848 7 6.4 38 34.8
AUG 100 1,052.3887 5 6.0 27 32.6
SE? 120 991.9937 5 5.6 27 30.4
OCT 160 1,850.6580 10 10.5 54 56.9
NOV 200 1,719.6686 10 9.8 54 53.0
DEC 240 1,622.5248 10 9.2 54 49.7
Totals...13,366.1539 76 (76.0) 414 (413.5)

 

232

during March is

n 1 + 2 + l = 4

Similarly, column (6) figures were obtained by summing
up the corresponding figures in Tables 7.8, 7.9 and 7.10.
For example, the total area required for assembling and

testing product P during March is given by:

S3=6+10+6=22m2

Columns (5) and (7) of Table 7.11 reflect the results
obtained when the calculations are carried out using the

non-rounded columns of Tables 7.8, 7.9 and 7.10.

Approximate Method

 

Column (3) of Tables 7.8, 7.9 and 7.10 might have been
calculated through approximate formulas (4.3), (4.4) or (4LED.
If formula (4.3) is used, the labor and space requirements on
a monthly basis can be quickly determined. For. example ,
Table 7.12 was computed for the case of the Testing operation.

The results compare well with those from Table 7.10.
(g) Aggregate Labor and Space Requirements
on a Periodic Basis

If management is just interested in knowing the total

labor hour and space requirements of the Final Assembly and

233
TABLE 7.12

LABOR AND SPACE REQUIREMENTS

(TESTING , APPROXIMATE METHOD)

 

 

 

 

 

(1) (2) (3) (4) (5) (6)
Cum. S
Month Prod. YT yL I1 (m2)
JAN 10 605.4 605.4 3 18
FEB 20 881.5 276.1 2 12
MAR 30 1,098.1 216.6 1 6
APR 40 1,283.4 185.3 1 6
MAY 50 1,448.4 165.0 1 6
JUN 60 1,598.9 150.5 1 6
JUL 80 1,868.6 269.7 2 12
AUG 100 2,108.9 240.3 1 6
SEP 120 2,327.9 219.0 1 6
OCT 160 2,720.7 392.8 2 12
NOV 200 3,070.5 349.8 2 12
DEC 240 3,389.4 318.9 2 12
Totals .... 3,389.4 19 114

 

234
Testing operations for aggregate planning purposes, the
procedure developed in (f) or (g) can be shortened by estab-
lishing an aggregate progress function of the three focused
operations in the time interval chosen . The following“

steps are carried out considering year 1 data:
(1) Aggregate Parameter Calculation
From (4.33), one must have
0<x_<,2c
Also, from the data
xefl:l, 240]

Thus, by taking the mid-point of the interval:

By using parameter 3 and E values from Table 7.4 and
applying formulas (4.43) and (4.44) with c = 120, the

following results are obtained:

39.86 x 0.291 84.95 x 0.232 94.20 x 0.458

 

 

B=( + + ).:.
120 .291 1200.232 1200.458
39 86 84.95 94.20 14.1861
(_——U_29I'+ ___0_232'+ ) = --—-- = 0.2932
120 ' 120 ' 120 ' 48:3877

and
A = CB D = 1200-2932 x 48.3877

196.9477, where

D is the denominator of B.

235
(2) Aggregate Progress Function for Final Assembly and

Testing Operations

The aggregate progress function for the Mechanical and
Electrical Assembly and Testing of product P can be written

as:

-0.2932

y = 196.9 x (7.5)

(3) Aggregate Labor and Space Requirements

The aggregate labor and space requirements for the Final
Assembly and Testing onerations can now be calculated on a

monthly basis as exhibited in Table 7.13.

The total labor hours yT in column (3) are calculated
through approximate formula (4.4). For example, the total
hours expended in the production of unit 1 up to and includ-

ing unit 40 are given by:

l-B 196.9 1-0.2932

_ A. - __ _
YT — l‘B x — 1_0.2932 X 40 "' 3,778.2

The total labor hours yL for each monthly lot i
i
(column 4) are calculated as follows:
Y - Y ‘ Y
Li Ti T(1-1)
For example, the total labor hours for the lot produced
during April is given by:

y = 3,778.2 - 3,083.0 = 695.2
L4

236
TABLE 7.13

AGGREGATE LABOR AND SPACE REQUIREMENTS
(FINAL ASSY AND TESTING,AGGREGATE PROGRESS FUNCTION METHOD)

 

 

 

 

(1) (2) (3) (4) (5) (6) (7)
Cum. 0 ' 3'

Months Prod. YT yL A n (m2)
JAN 10‘ 1,418.2 1,418.2 +3.7 8.0 46
FEB 20 2,314.8 896.6 -0.87 5.1 29
MAR 30 3,083.0 768.2 -0.47 4.4 25
APR 40 3,778.2 695.2 -0.23 4.0 23
MAY 50 4,423.7 645.5 -0.08 3.7 21
JUN 60 5,032.1 608.4 -0.00 3.5 20
JUL 80 6,166.7 1,134.6 0.06 6.5 37
AUG 100 7,220.2 1,053.5 0.10 6.0 34
SEP 120 8,213.2 993.0 0.10 5.6 32
OCT 160 10,065 1,851.9 0.06 10.5 60
NOV 200 11,785 1,719.5 -0.01 9.8 56
DEC 240 13,405 1,620.8 -0.10 9.2 52

 

Totals ... 13,405.4 76.3 435

 

237

The per cent deviations of the results in column (4)
with respect to the corresponding results in column (3) of
Table 7.11 are calculated for comparison purposes (see column
5). Finally, the number of personnel required (column 6) and
the necessary space (column 7) are calculated on a monthly
basis as was done for Tables 7.8, 7.9 and 7.10. It is

assumed an average coefficient of 5.7 m2

per person. The
results were not rounded to the nearest integer so that they
can be compared with those exhibited in Table 7.11. It can
be seen that the aggregate progress function method is quite

effective and expeditious in determining the aggregate labor

and space requirements for a given manufacturing program.

MULTINATIONALS AND THE COST OF LEARNING

Knowledge of the progress functions of similar producUs
in plants of the same multinational enterprise which are
located in different countries can be relevant for decision-

making purposes at parent company level.

As an example, suppose that the startups of the same
product P in plant B (country B) and plant C (country C) are
characterized by the following estimates of ultimate points,

respectively:

(xu, yu)B (910, 8.0)

)

and (xu, yu C

(1910, 16.0)

238

Parameter E values are also estimated from past re-

search:

bB = 0.174 and bC = 0.161

The corresponding MPF Hours and Total Hours up to unit

1910 are calculated as follows:

Product P in Plant B

 

 

 

(xu + 0.5)1‘b - 0.51"b
(MPF Hours) = x y — 1
B u u l-b
(1 - b) x
1.1
0.826 0.826
= 910 x 8.0 (910'5 ‘ 005826 -1y51520
0.826 x 910 '

z (MPF Hours)B+ x = 1520+1910}(8 = 16,800 (hours)

Y Y
TB u u

Product P in Plant C

 

. 0.8
1910.50 839-0.5 39

0.839 x 1910”839

 

(MPF Hours)C = 1910 x 16.0 ( —1)5*5,836

llz

yT (MPF Hours)C + xuyu = 5,836 + 1910 x 16.0

C

36,396 (hours)

239

Ceteris paribus, it would cost less to launch product

P in plant B than in plant C of the same multinational com—

pany.

A similar rationale is valid for operations within the
manufacturing cycle of the same product. For example, due to
plant advantage in implementation costs, plant C might have
possibly the machining and subassembly operations of product
P, while plant B would be in charge of the final assembly

and testing operations of the same product.

IMPLICATIONS AT NATIONAL LEVEL

Empirical estimates of progress functions can be used
on a firm as well as a national level for numerous purposes.
Progress function estimates can be useful for national plan-
ning in such areas as industry or nationwide forecasts of
labor requirements and cost, allocation of labor and materials

and similar determinations.

If aggregate progress functions are available (e.g.,
industry J, chapter VI), it is possible to use them as a
means of estimating future labor and space requirements for
sectors of an economy. If similar functions are available for
materials - - as Wright suggested in his pioneering article3

- - it is possible to allocate materials according to

planned requirements.

240
These potentialities may find even greater application
in developing countries that have macroeconomic planning
mechanisms of a more or less centralized nature. The neces-
sary calculations would be carried out in the same way as

explained in the last section.

Determining the Magnitude of Contracts

A problem of some interest is to determine the size of
a government contract which will furnish the proper produc-
tion base by day X in the future. The base of production for
product P is the cumulative output of P by day X. The problan
has some importance because of the interest in estimating
production bases for various products under rearmament pro-
grams. It has also a general interest for programs of varied

nature that are to be implemented by government funding.

The problem can be posed as follows, for the case of
airframes: Having a requirement of R units of airplane P to
be produced after day X with a fixed input of yT direct
manhours, what must be the size C of the contract which will
insure post day X unit costs low enough to make possible the

production of R units after day X?

Solution

If x1 is the serial number of the last plane produced

to date, we solve

241

x +C+R
yT = a f 1 x'b dx
x1+C

= 1%3 [Exl + C + R)1'b - (x1 + C)1-b]

for the value of C.

The preceding equation may be written as follows:

_ y (l-b)
f(C) = (x1+c+R)1'b — (xl+C)l b-—3————— = o (7.6)

a

Solving by Newton-Raphson's method, yields:

1-b l-b y1(1'b)

._ -b
f'(C1)=(l-b)(x1+C1+R) b - (1-b)(xl+C1)

- - y (l-b)
C -C (x1+C1+R)1 b--(xl+Cl)1 b "!LET__' (7 7)
2 1 , (l-b)[?x1+C1+R)'b - (x1+cl)'9] '

 

After some iterations of formula (7.7) it is possible

to find an approximation C* of C such that (7.6) is satisfied.

242
A COMPUTERIZED MPF SYSTEM

In this section an overall design of a computerized
Manufacturing Progress Function System is suggested by using
a system design and documentation technique known as HIPO
(Hierarchy plus Input-Process-Qutput). Because the functflnu;
of a system are described and not its organization and logic,
a HIPO description provides information on "what a system
does", and is thereby useful at most stages of planning,

development, and implementation.

The fundamental version of a HIPO package is devel
oped during the initial design phase and is called the initial

design package. This package is prepared by the design group

 

and gives the overall design of the proposed (or modified)
system. At the initial design level, the HIPO package is
lacking in details necessary for implementation but adequateLy
gives the scope of the project and can be used by management

for scheduling and cost estimation.

The following diagrams illustrate our initial concep-
tion of a computerized MPF System. The detail design phase
can be developed by translating the calculation procedures

exemplified in the previous sections into FORTRAN programs.

243

 

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CHAPTER VIII

SUMMARY AND CONCLUSIONS

The overall purpose of this study was to advance
knowledge on the subject matter of Manufacturing Progress
Functions. For the purpose of this dissertation the Manufac-
turing Progress Function was generally defined as the re—
lationship in which the labor input per unit used in the mag
ufacture of a product tends to decline by a constant percenE
age as the cumulative quantity produced is doubled. Where
this relationship is present, it may be represented by a

straight line in a double logarithmic scale.

The prime objective of the study was to contribute a
general symbolic-analytic model of the manufacturing progress

phenomenon.

Once the general model was established, an equally im-
portant objective was to respond to the need for a coherent
systematic approach to be used in predicting the develommaus
of the adaptation process in industrial concerns of almost

any kind.

The foregoing general statement of purpose was broken
down into a number of layers of investigation leading to the

following more specific subobjectives:

248

249

l. A review of the literature that is of relevance to the
objectives of the dissertation.

2. An investigation of the theory of the Manufacturing
Progress Function aiming at a systematization of the
existing body of knowledge, and at the derivation of
new theoretical results that will settle the question
of the parameters estimation of the general model.

3. The conception of a general symbolic-analytic model of
the system adaptation phenomenon, including the develop
ment of a method for using the model to predict the
course of future startups.

4. The testing of the model in a number of real world sip
uations by using data from diverse industrial operation;

5. The possibilities of implementing the model at firm

level and national level.

A brief recapitulation of the major points that have
emerged from each chapter of the dissertation will serve to

demonstrate thattfiuaaforementioned objectives were attained.

Chapter 11 contains a comprehensive review of the his-
torical development of the Manufacturing Progress Function
and a summary of the more important contributions to the prq;

ress curve literature that are relevant to this dissertation.

Although the progress curve was discovered in 1922, it
was largely unknown until World War II. T.P. Wright is given
credit for originating the formulation of the progress curve

theory in 1936. His statement, that cumulative average

250
manhours per unit decline by a constant percentage every time
the output is doubled, remains the most popular formulation

in existence.

From 1940-1949, a number of modifications of the origi
nal model were proposed. J.R. Crawford contributed the unit
learning curve and noticed that different rates of progress

might exist for different airframes.

Several relevant contributions were published in the
l950s.There was a pioneering effort by some researchers like
W.Z. Hirsch, S.E. Bryan, R.W. Conway and A. Schultz, Jr. to
extend the concept of the progress curve to labor-intensive
industries other than the airframe industry.In addition,the
studies by A. Alchian and H. Asher, in the airframe industry,
together with Hirsch's study in the machine-tool industry,
and Schultz and Conway's research in manufacturing of elec-
tronic and electro-mechanical products represent objective
and rigorous empirical investigations of the progress curve

concept.

The last fifteen years have seen some pioneering extep
sions of the Manufacturing Progress Function to machine-inflql
sive industries as well as to diverse labor-intensive indus-
tries. The studies carried out by N. Baloff constitute the
most comprehensive investigation of the progress function

during this time.

251

It is worthwhile to review the main hypotheses raised
bytfluaaforementioned authors: (i) Linearity, when in loga-
rithmic coordinates, between the labor per unit input and
cumulative output was generally confirmed except in Asher's
study; (ii) There is no one single progress curve that can
be universally applied to all types of operations involved h1
the manufacturing of a given product or to all products in a
given firm or industry; (iii) Assembly operations experience
higher rates of progress than machining operations; (iv) The
findings are contradictory with respect to the novelty hy-
potheses. Crawford, Schultz and Conway agree that the less
similar a new product is to a predecessor, the greater the
rate of progress experienced by the new product. Nevertheless,
Alchian and Hirsch found in diverse settings that the greater
the similarity of a product to a predecessor, the greater dug
rate of progress experienced by the new product; (v) Baloff
found a strong correlation between the parameters of the
startup model among startups that occurred in the same produg
tion facility whereas Asher noticed a strong correlation
among startups that occurred in different facilities; (vi)

Plateaux predictability continues to be a controversial issue.

The field of progress functions lacks notation uniform;
ty, precise definition of the variables and functional rela-
tionships involved, and formal mathematical proofs of several
assumed results. A coherent mathematical exposition can be
the basis for the derivation of new important results. Such

theoretical systematization is offered in Chapter III

252
Initially, four types of progress functions are identified.
Functional relationships for the four types are clearly and
compactly defined with recourse to set theory notation.Param
eters g and p are explained and interpreted. In a second sep
tion, four fundamental problems which users might have faced
consciously or unconsciously, are formally stated and solved,
at times by more than one method. Finally, parameter calculg
tion problems are classified and solved by exact or approxi-

mate formulas.

Chapter IV represents a continuation of the mathematical
exposition initiated in Chapter III. Two related topics of
practical relevance are approached: the integration of prog-
ress functions and the debatable problem of their aggregathL

Original approximations are proposed for both problems.

A general symbolic—analytic model of the manufacturing
progress phenomenon is offered in Chapter V. Some effort is
also expended in questioning the causes of the systematic
gains in productivity embodied in the progress curve. Thus,
a tentative concept of manufacturing progress is proposed
where the number of available hypotheses is reduced to a
minimum and the explanatory power of the remaining hypoth-

eses is greatly enhanced.

It is suggested that the manufacturing progress phenom
enon can be described and its course predicted by the follow

ing empirical equations:

253

y = ax'b (8.1)
and b = m + n 1n a, (8.2)
where

y = y/yu

a= a/yu
and

(xu, yu) is the estimated ultimate point.

In Chapter VI the manufacturing progress model pre-
sented in Chapter V is tested with real data from nine manp
facturers representing five different industries. A hundred
and fifty-nine separate cases of product and process startups
that occurred in four different countries and nine distinct
plants have been analyzed. In addition, aggregate data was
obtained for whole industries in one country, yielding nine

more startups.

The major conclusions that have emerged from the empip

ical research are the following:

(1) The descriptive efficiency of the manufacturing
progress model given by equation (8.1) is generally
supported by the results of regression analysis of
the startups from firms A,B,C,D,E,F,G,H,I, and

industry J. The coefficients of determination r2

(2)

(3)

254
vary from 0.59 to 1.000, with a median value of
0.969. In only ten per cent of the cases does the
regression fail to explain at least 81.7% of the
total variance in the dependent variable. The
t-ratios are generally impressive, ranging from
2.08 to 99.99, with a median value of 10.9. If
normality and common variance are assumed, the null
hypotheses p = 0 and B = 0 can be rejected at the
0.95 level of significance in 95% of the startups

analyzed.

The descriptive efficiency of the parameter model
given by equation (8.2) is generally supported by
the results of regression analysis of the startup
parameters from firms A,D,E,F,G,H,I and industry
J. The parameter model is supported among opera-
tions of the same product (firms A and G), among
products within the same plant (firms D,E,F,H and
I), and among groups of products of similar
technology within the same industrial sector of a

foreign economy (industry J).

The findings of this research and previous findings
by Asher and Baloff constitute adequate evidence
to suggest that the parameter model can be devel-
oped into an effective means of predicting the
parameter p of a new startup. However, additional

investigation and validation is necessary for

255

successful industrial application of the model.

The dissertation is concluded (Chapter VII) with a
discussion of the industrial implications of the findings
reported in Chapter VI. The importance of recognizing and
predicting the manufacturing progress phenomenon is related
to several decision—making functions that are encountered in
an industrial setting as well as in economic planning at the
national level. Finally, an overall design of a computerized

Manufacturing Progress Function (MPF) System is suggested.

APPENDICES

APPENDIX A

256

APPENDIX A

MATHEMATICAL PROOFS

(1) A FORMAL PROOF THAT lim 8:43 = + co

x+0

The above statement means that for each positive number
M it is possible to find a positive number 6 (depending on M

in general) such that

—a—b->M when 0<|x|<6
x

To prove this, note that

i%-> M when 0<<x <.§
x
1/b
a
or 0 < le < (171)
. a 1/b .
Ch0031ng 6 = (M) , the required result follows.
(2) A FORMAL PROOF THAT lim ax-b = O
x+oo

The above statement means that for any positive number
(s it is possible to find a positive number N = N (e) such

that

257

lax-'bl < 5 whenever x > N

To prove this, note that for

-b
lax |‘<s
one must have
1
x>(§. /b
e
l/b

Choosing N = (2) , the required result follows.

APPENDIX B

APPENDIX B

EQUATION OF THE ASYMPTOTE TO THE
CURVE REPRESENTING THE LOGARITHM
OF THE UNIT PROGRESS FUNCTION

The following theorem of Mathematical Analysis is used:

”If an infinite branch of a curve has an asymptote

y = cx + d, the coefficients p_and Q will be given by

NM

c = lim and d = lim (y - cx) (B.l)

X+°° x+oo

where the point M(x,y) remains on the branch of the curve.
And, conversely, if the limits in (3.1) exist when M(x,y)
moves to infinity along a branch of the curve, the straight

line y = cx + d will be an asymptote of that branch. ”

Let F [log(x):] = log { alEcl-b - (x-l)1-b:]} (B.2)
One must show that

f(log x) = log [a(l-bi] - b log x is asymptote to

F [log (xij given by (B.2)

Coefficient p pf the Asymptote

 

 

According to the forementioned theorem:

258

lim
x+oo

lim

259

log aLxl-b - (x-1)-§]

(108 X)+m 108 x

lim
x—>oo

E l-b l-b]
log a + log x - (x-l)

 

log x

Applying L'Hospital's rule, it follows that:

log

lim
x—mo

lim
x+oo

x+oo

 

 

 

 

log x
1 1 '7 -b b
' (l-b)l_x -(x-1) .]
1 ..
1n-0 [x1 b _ (x_1)1 b]
1. . I
lnlO x
(l-b) x [-bx b'l - igl- x'b‘z - ]
(l-b) b + (l-b)b -b-1 +
2!
«bx.b _ -b
-b
X

Coefficient d of the Asymptote

According to the previously stated theorem and using

the result c = -b, it follows that:

d = lim
(log x)+w

{log a [8

260

1"” - (x-1)1”b] + b 10g x}

= lim log {a [xl-b - (x-l)1-§] xb}

x+oo

= log a + lim

x+oo

x+oo

= log a + lim

x+oo

= log a + log

Equation of the

Since c

- -b
log {xb [x1 b - (x-l)l :1}

log {xb [(I-b)x‘b + “—51% bx‘b'l + ..J}

1.. [(1-8 +0538. 515+ ...]

118L145) +$lflb 1+ H]
x+w 2! x

(l-b) = log [a(l-bZ]

Asymptote to F [log(x)j

- b and d = log [8 (l-bX] , the equation

of the asymptote f(log x) to

F [log (x):]

= log {a [xl'b - (x-l)1-b:]}

261

is, according to the forementioned theorem:

f(log x) = - b log x + log Ea(l-b):] (q.e.d.)

APPENDIX C

APPENDIX C

PARAMETER FORMULAS DERIVATION

In chapter V it was suggested that the manufacturing
progress phenomenon can be described and its course predicted
by the following equations

y = ax-b (Cl)

and b=m+nln a (C2)
where y = XLEL
yu
_ EL
a - yu
and (xu,yu) is the estimated ultimate point.

If the best fit to the available data is achieved with
the cumulative-average progress function, the productivity

index y will have to be defined as

In this case, prediction of parameters g and p for a
new startup can be developed as follows.

262

263

Equation (Cl) yields for x = xu :

F( ) -
x“ = ax b (C3)
yu u

 

If the best fit is given by the cumulative-average prog
b

ress function §'= ax' , then:
§<x ) = ax’b (04)
u u
From Chapter II, equation (3.17)
y (x > % a<1-b> x‘b (CS)
u u
Dividing (C4) by (C5), yields:
$122; = 1%5 (C6)
' u

Substituting in (C3) the result obtained in (C6)

and solving for a, yields:

__Lb
a.- l-b xu (C7)

Substituting in (C2) the result obtained in (C7) it

follows that:

b = m + n 1n (I%B-x:) or

264

b

n ln ({¥%) - b + m.= 0 (C8)

Equation (C8) can be solved for pyby the Newton-Raphsan

Method as follows:
n [b 1n xu - 1n (l-b)j - b + m = O

f(bl) = nEbl ln xu - ln (l-blZJ- b1 + m = 0

I _. ___—1 - =
f (b1) n (1n xu + 1'b1) 1 O
nEa lnx -ln(1-b):'-b +m
b2 = bl _ 1 u 1 1 (c9)

1
n [1n xu + I:b:] - l

 

After some iterations of formula ( 9) it is possible
to obtain an adequate approximation b* of b. By substituting

b* in (07), it follows that:

_ 1 b*
a,— l-b xu

and b*

 

a = yut, = 1_b (c10)

APPENDIX D

APPENDIX D

STABILITY OF COEFFICIENTS 9
AND E OF THE PARAMETER.MODEL

The following is a check on the stability of coeffi-

cients m and n of the parameter model:
b = m + n ln 3

In the case of firm D (14 products), sample 1 was tdflfll
randomly from products D1 through D14. Sample 2 is formed by
the remaining products. Data tables are in page 268. In the
case of firm F (28 products), sample 1 was taken randomly
from products Fl through F28. Sample 2 is formed by the re-

maining products. Data tables are in pages 278-79

The following calculations were then carried out: (1)
Estimation of the regression of b on ln a for samples 1 and
2 of each firm; (2) Variances about regressions l and 2; (3)
Difference between the two regression coefficients n1 and
n2: variance, standard error and confidence limits; (4) Dif-
ference between the two intercepts m1 and m2: variance, stap
dard error and confidence limits; (5) Test of the hypothesis

that the variances about the two regressions are equal.

It is concluded that the difference between the two
.regression coefficients is not statistically significant.The
same goes for the difference between the two intercepts.

265

266

In the final section of Appendix D a comparison is made
between two approaches for predicting parameter b and §;values
of new startups, namely, by using average b and g values taken
from past data and by employing the parameter model approach
proposed in Chapter V. Sample 1 (Table D.5) was drawn randanLy
from firm F binding machines and simulataspast data. Sample 2
(Table D.6) is formed by the remaining types of binding ma-
chines produced by firm F. Predictions were then carried out
for parameters b and g of the products in Sample 2 by using
data available from Sample 1. The results of the comparison
are exhibited in Tables D.7 and D.8. The actual values of b
and 3 appear in column (2) of each table, respectively. The
average b's and 3's - - calculated from Sample 1 values - —
are in column (3). Per cent deviations of the predictionsrmde
by using average values with respect to the actual values are
in column (4). The predictions for b and §_made by using the
parameter model are in column (5). Detailed calculations pre-
cede Tables D.7 and D.8. Per cent deviations of the parameter

model predictions b and g ‘with respect to the actual values

P P
of b and g are in column (6) of each table.

The mean absolute deviation is defined as

= 2 [Prediction Errors]

 

M.A.D.
No. of Predictions

From the results in Tables D.7 and D.8, the following

M.A.D. values were calculated through the above formula:

267

(1) Predictions using averages of past data

0.0815
1.24

b parameter : M.A.D.

a parameter : M.A.D.

(2) Predictions using the parameter model

b parameter : M.A.D. 0.0497

g parameter : M.A.D. 0.502
Similar results were obtained with the data pertaining
to printing presses produced by firm F and with the data
available from other firms (e.g., firms G and I). The author
believes that the parameter model contributed in Chapter V has
proved to be superior to other existing methods for estimathu;

the parameters of new startups.

FIRM D

(14 products)

TABLE D.1

SAMPLE l, FIRM D

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PRODUCTS 3 1n a b
D3 1.341 0.2934 0.0936
D11 3.521 1.259 0.220
D4 2.483 0.9095 0.275
D14 1.333 0.2874 0.0533
D5 1.230 0.2070 0.0543
D7 5.252 1.658 0.333
D12 7.002 1.946 0.406

TABLE D.2
SAMPLE 2, FIRM D

PRODUCTS 3 1n 3' b
D10 6.945 1.938 0.409
Dl 2.649 0.9742 0.368
D2 3.230 1.173 0.297
D6 2.882 1.059 0.205
D9 1.625 0.4855 0.0929
D8 1.156 0.1450 0.0226
D13 6.683 1.900 0.409

 

269
l- ESTIMATION OF THE REGRESSION OF b ON In E
(LET b = y and 1n 3 = x)

REGRESSION 1 (sample 1)

 

n = no. of observations = 7
2x = 6.5603; i = 5% = 0.9372
n
2(x - §)2 = x2 - (2x)2/n' = 9.1597 - 6.56032/7 = 3.0115
2y = 1.4352; y = §X = 0.2050
n’
— 2 2 2 . 2
Z(y - y) = 2y - (1y) /n = 0.4143 - 1.4352 /7 = 0.1200

 

2(x—§)(y-§) = zxy - zxzy/n' = 1.9233 - 6-56°3’7‘1°‘*352==0.5783

The regression coefficient n is given by

l

= Z(x - §)(y -4§) = 0.5783 _

 

0.1920

:3
H
I

2
2(x - R) 3.0115

The regression equation of y on x is then:

I-<
II

§’+ n1 (x — i) = 0.2050 + 0.1920 (x - 0.9372)

or Y 0.0251 + 0.1920 x

270
or B = 0.0251 + 0.1920 1n 3

where: ml= 0.0251(intercept) and n1 = 0.1920 (slope)

REGRESSION 2 (sample 2)

 

2x 7.6747; R =sz/n" = 1.0964

_ 2 2
2(x - X) = 11.0691 - 7.6747 /7 = 2.6547
Zy = 1.8035; y = Ey/n" = 0.2567

£(y - §)2 = 0.6094 - 1.80352/7 = 0.1447

(x - §)(y - y) = 2.5421 - 7 6747 g 1-3035 8 0.5648

 

n2 = %f%%%% = 0.2128 (regression coefficient)

The regression equation of y on x is then:

Y = y + n2(x - §) = 0.2576 + 0.2128 (x - 1.0964)

or Y 0.0244 + 0.2128 x

'or B 0 0244 + 0.2128 1n 3

271

where: 012 = 0.0244(intercept) and n2 = 0.2128 (slope)

2 - VARIANCE ABOUT THE REGRESSION

VARIANCE ABOUT REGRESSION l

 

The sum of squares about the regression is given by

2(y-Y)2=z(y-§)2-z<Y-§)2 (D.1)

Sum of squares of observations y about their mean

£(y - §)2 = 0.1200 (already calculated)

Sum of squares due to the regression

_ 2 _ 2 _ ._ 2 ..2
2(Y - y) b 2(x-X) = [2(x-X)(y-y):] /2(x-X)

0.57832/3.0115

0.1111
Substituting in (D.1) yields:
2(y - Y)2 = 0 1200 - 0.1111 = 0.0089

Hence, the variance about the regression is estimated

272

82 = 9899§3 = 0.00178
1 5

(Since the sum of squares about the regression is based

on n' — 2 = 7 - 2 = 5 degrees of freedom.)

VARIANCE ABOUT REGRESSION 2

 

Similarly, we have:

2(y - §)2 0.1447

2(Y - §)2 = 0.56482/2.6547 = 0.1202

2(y - Y)2 0.1447 - 0.1202 = 0.0245

Hence, the variance about the regression is estimated

3% = 0.0245/5 = 0.00490
THE DIFFERENCE BETWEEN THE TWO REGRESSION COEFFICIENTS
111 AND n2.

Let 02 be the error variance, which is usually esti-

mated by combining the mean squares about the two regressions,

i I

' 2

2
if S1 = variance about regression 1 (slope n1)

with 01 degrees of freedom

273

2 . .
and S = variance about regre331on 2 (slope n

2 2)

with 02 degrees of freedom

then 2
O is estimated by

32 = (01 8% + <12 s§>/<<I>1 + «12)

The variance of the regression coefficient is given by:

2
O

 

V (n) = (D.2)

2
2(x - 35)

Substituting S for O in (D.2) yields:

_ 2
V (n1) = 32/ 21(x — x)

and

V (n )

2 _—2
2 S/22(x x)

where 21(x - i)2 is based on the observations from which 111

was calculated and similarly for 22(x - §)2. Therefore, since
I11 and 112 are independent estimates:

v (nl-n ) = $2 1 + 1 (D.3)

2 21 (x3102 22 (x-i’)2

 

 

 

 

_ 1 1
and S.E.(bl-bz) - s _ 2 + _ 2 (DA)
21(x-x) 22(x-x)

274

These enable confidence limits for the difference to be
calculated, using the value of t with (01 + 02) degrees of
freedom. If the variances about the two regressions cannot be
assumed equal, then the confidence limits must be calculated
by another method. We assume that the variances are equal

(this assumption will be tested in a later section).

Thus, 02 is estimated by

$2 = 0.5 (0.00178 + 0.00490) = 0.00334

The variance and the standard error of the difference

between I11 and 112 are calculated as follows:

2 1 l 1 1
- =S + =0.0034 -————-+————
V(n1 n2) [21(x-x)2 22(x-x)2:l 3 [3.0115 2.6547]
= 0.002367
%
S.E. (n1 - n2) = (0.002367) = 0.0487

Confidence Limits

 

Using t with (01 + ¢2) = 10 degrees of freedom, the 95%

confidence limits are:

.l.
(nl - n2) - 2.23 S.E. (nl-nz) = (0.1920-0.2128) f2.23x0.0487

- 0.1294 to 0.0878

275
Since the confidence limits include zero, the difference
between the regression coefficients 111 and n2 is not statisti

cally significant.

4 - THE DIFFERENCE BETWEEN THE TWO INTERCEPTS m1 AND m2

The variance of the intercept m is given by:

2 i2

V(m) = O (D-5)

504
+

2(x-i)

Substituting S for O in (D.5) yields

_2
x
V(ml) = 82 8L.+ 1
I 2
n 2 (x-x)
l
2 1 £2
V(m2) = S _— + ___—2-
n" 2 (x-i)
2
Since 1111 and m2 are independent estimates and n' = n"
8.11-1.2) = 32 .2. + ___.1— + 2

n 21(x-§)2 22(x-i)2

2 2
0.00334 (.3. + MALL + _1_-_0_9_6_‘L_)

3.0115 2.6547

0.003441

276 }
S.E. (m1 - 82) = (0.003441)2 = 0.0587

Confidence Limits

 

Using t with (01 + 02) = 10 degrees of freedom, the 95%

limits are

(ml - m2) T 2.23 x S.E.(ml-mz) = (0.0251-0.0244) f 2.23x0.0587
=-0.1302 to 0.1316

Since the confidence limits include zero, the difference
between the intercepts m1 and 102 is not statistically

significant.

Note. If the variances about the two regressions cannot be
assumed equal, then the confidence limits must be calculated
by another method. In the following section the hypothesis

2 2 2 2
Hozol = 02 is tested w1th H¢ : 01 ¢ 02 .
5 — TEST OF THE HYPOTHESIS THAT THE VARIANCES

ABOUT THE TWO REGRESSIONS ARE EQUAL.

When the populations are normally distributed and the
samples are independent, the ratio of two samples variances
is distributed according to the F distribution and has the

test statistic

2
Sllo

l-‘N

 

F:

S

2
lo2

2
2

277

2 2

where S and 8: are the sample variances and o and Oi
are the population variances from which the samples were takem
The hypothesis is that 0% = 0%, and the estimates of these an)
parameters are represented by their unbiased estimates Si and

8%. Consequently, the test statistics reduces to

_ 2 2

Since Si = 0.00490 and 8% = 0 00178,

F = 0.00490/0.00178 = 2.76

We have also:
01 = 02 = 5 degrees of freedom

From the table of the variance ratio (F - Distribution),

the 5% and 1% points are 5.05 and 11.0, respectively.

02 = O: is accepted.

Thus, the hypothesis HO : 1

FIRM F

(28 products)

TABLE D . 3

SAMPLE 1, FIRM F

 

 

PRODUCTS 3 1n 3 b
F24 4.74 1.556 0.228
F18 2.70 0.9933 0.176
F11 1.48 0.3920 0.0758
F12 1.80 0.5878 0.148
F21 2.02 0.7031 0.123
F17 3.42 1.230 0.237
F7 1.77 0.5710 0.123
F20 4.47 1.497 0.235
F28 3.16 1.151 0.227
F19 2.71 0.9969 0.186
F13 3.85 1.348 0.271
F26 1.57 0.4511 0.0731
F25 2.41 0.8796 0.120
Fl 3.57 1.273 0.238

 

279
TABLE D.4

SAMPLE 2, FIRM F

-,—
___

PRODUCTS a_ 1n a b

 

 

 

 

F5 2.01 0.6981 0.147
F6 1.99 0.6881 0.150
F4 1.97 0.6780 0.184
F27 1.95 0.6678 0.131
F9 1.93 0.6575 0.136
F3 1.83 0.6043 0.110
F10 1.80 0.5878 0.128
F2 1.49 0.3988 0.0604
F8 1.36 0.3075 0.0551
F16 4.94 1.597 0.246
F15 3.22 1.169 0.219
F22 3.17 1.154 0.201
F23 2.80 1.030 0.133
F14 2.59 0.9517 0.172

 

280

1 - ESTIMATION OF THE REGRESSION OF b ON In g

011'

or

(LET b = y AND 1n 3 = x)

REGRESSION 1

no. of observations = 14

:3
II

Xx 13.6298; i = Zx/n' = 0.9736

2(x-§)2 = 2x2 - (Zx)2/n'

2y = 2.4609; § = Zy/n' = 0.1758

2(y-7)2 = Eyz - (Zy)2/n'

15.2147-13.62982/14 = 1.9453

O.4882-2.46092/14= 0.05563

13.6298 x 2.4609

 

 

 

Z(x-§)(y-y) = 2xy - Zny/n' = 2.6998 -
= 0.3040
The regression coefficient n1 is given by
n1 = Z(x-x)(yey) = 0.3040 = 0 1563

)3 (x-§) 2 1. 9453

The regression equation of y on x is then:

14

Y = y + nl (x - E) = 0.1758 + 0.1563 (x - 0.9736)

F<
ll

0.0236 + 0.1563 x

00
ll

0 0236 + 0.1563 In 3

231

where: m = 0.0236 (intercept) and n = 0.1563 (slope)

or

or

1 1

REGRESSION 2

n" = 14

11.1896; R = Ex/n" = 0.7993

Xx

2(x-32)2 = 10.4784 - 11.18962/14 = 1.5350

2y = 2.0725; § = Zy/n” = 0.1480

2(y-§)2 = 0.3449 - 2.07252/14 = 0.03810

11.1896 x 2.0725

 

2(x-§)(y-§) = 1.8717 - = 0.2152
14
n2 = 2.2—1.5.; = 01402
1.5350

The regression equation of y on x is then:

Y = y:+ n2 (x-i) = 0.1480 + 0.1402 (x - 0.7993)
Y = 0.0359 + 0.1402 x
B = 0.0359 + 0.1402 1n a

where m = 0.0359 (intercept) and n = 0.1402 (slope)

2 2

282
2 - VARIANCE ABOUT THE REGRESSION

VARIANCE ABOUT REGRESSION 1

 

The sum of squares about the regression is given by

2(y-Y)2 = z<y-§>2 - 208302 (D.6)

Sum of squares of observations y about their mean

Z(y - §)2 = 0.05563

Sum of squares due to the regression

2 (Sr-37) 2

_ _ _ 2 _
b2 >2<x-x>2 = [2(x-x)(y-y) ] /2:<x-x>2

0.30402/1.9453

0.04751

Substituting in (D.6) yields:

2
2(y—Y) = 0.05563 - 0.04751 = 0.00812

Hence, the variance about the regression is estimatedtnr

52 _ 0.00812

1 - 12 = 0.000677

(Since the sum of squares about the regression is based on

n' — 2 = 14 - 2 = 12 degrees of freedom.)

283
VARIANCE ABOUT REGRESSION 2

 

Similarly, we have:

2(y - §)2 = 0 03810

_ 2 2
2(Y - y) = 0.2152 /1 5350 = 0.03017
2(y - Y)2 = 0.03810 - 0.03017 = 0.00793

Hence, the variance about the regression is estimatedtun

3% = 0.00793/12 = 0.000661

3 - THE DIFFERENCE BETWEEN THE TWO REGRESSION COEFFICIENTS

2 . . . . .
O , the error variance Is estimated by combining Si and

Si as follows:

(D
II

0.5 (0.000677 + 0.000661)

0.000669

The variance and the standard error of the difference

between nl and 112 are calculated as follows

2 1 1 1 1
= . +
21(x-i)21-22(x-§)2 0 000669 1.9453 1.5350

 

I
U)

 

V(n1-n2) —

0.001463

284
S.E. (n1 - n2) = (0.001463)% = 0.0382

Confidence Limits

 

Using t with (¢1 + ¢2) = 24 degrees of freedom, the 95%

confidence limits are

(nl-nz) T 2.06 S.E.(n (0.1563-0.1402)T 2.06 x 0.0382

1‘n2)

-0.0626 to 0.0948

Since the confidence limits include zero, the diffemence
between the regression coefficients n1 and n2 is not statisti

cally significant.

4 - THE DIFFERENCE BETWEEN THE TWO INTERCEPTS m AND m .

l 2

The variance and the standard error of the difference

(m1 - m2) are calculated as follows:

x x
V(ml-m2) = 82 _;,+ ————l——— + 2

n 21(x-§)2 22(x-§)

2 2
g; + 0.9736 + 0.7993

14 1.9453 1.5350

= 0.000669
= 0.0007000

}«
S.E.(ml - m2) = (0.0007000)2 = 0.0265

285

Confidence Limits

 

Using t with (01 + ¢2) = 24 degrees of freedom, the 95%

limits are
(ml-m2) f 2.06 S.E.(ml—m2)=(0.0236-0.0359) t 2.06 x 0.0265
= -0.0669 to 0.0423

Since the confidence limits include zero, the difference
between the intercepts m1 and m2 is not statistically

significant.

5 - TEST OF THE HYPOTHESIS H :02 = 02

0 1 2
. 2 2
Since S1 = 0.000677 and 82 = 0.000661,
_ 0.000677 =
F ’ 0.000661 1'02

We have also:

01 = 02 = 12 degrees of freedom.

From the table of the variance ratio (F-Distribution),

the 5% and 1% points are 2.69 and 4.16, respectivelly.

Thus, the hypothesis H oi = o: is accepted.

02

286
PREDICTING PARAMETER b AND a VALUES
OF NEW STARTUPS (FIRM F, BINDING MACHINES)

TABLE D.5

SAMPLE l (SIMULATES PAST DATA)

 

 

 

 

 

 

 

 

 

 

 

PRODUCT a In a b
F15 3.22 1.169 0.219
F12 1.80 0.5878 0.148
F14 2.59 0.9517 0.172
F9 1.93 0.6575 0.136
Fl 3.57 1.273 0.238
F6 1.99 0.6881 0.150

TABLE D.6
SAMPLE 2 (SIMULATES NEW STARTUPS)

PRODUCT a 1n a b
F7 1.77 0.5710 0.123
F8 1.36 0.3075 0.0551
F10 1.80 0.5878 0.128
Fll 1.48 0.3920 0.0758
F13 3.85 1.348 0.271
F16 4.94 1.597 0.246

 

 

287

REGRESSION EQUATION (FROM SAMPLE 1)

b = 0.0518 + 0.141 1n 3

 

 

 

"ULTIMATES"

PRODUCT xu yu
F7 37 236
F8 32 241
F10 33 168
F11 39 215
F13 88 295
F16 110 169

PREDICTIONS

Product F7

 

Parameter b

Parameter b is calculated through equation (C.9), as

follows:

_ n [b1 1n xu - 1n (l-b1)_l - bl + m

 

b2 = bl

Set b1 = 0.100

288

b = 0 100 _ 0 141 [0.100 ln 37-1n(1-0.100):l- 0.100+0.0518

2 . 1
0.141 (1n 37'*1-o.100) - 1

0.0176 _

-o.334 ‘ 0'153

0.100 -

Set b2 = 0.153

b _ o 153 _ 0.141 [0.153 In 37-ln (1—o.153):J- 0.153+0.0518

3 1
0.141(11‘1 37 + m) - 1

0.000112
-0.324

Ilz

0.153 - 0.153.

Parameter a

Parameter a is calculated through equation (C.10) as

follows:
b
a = 1&1 _ 370.153 _ 2 05
- l-b l-0.153 '
Product F8

 

Parameter b

Set 61 = 0.100

0.141 [0.100 In 32 - ln (140.100):]-»0.100+0.0518_

1
0 141 (1n 32 1-0.100)

0.100 -

b2

0.364

S€t b2 = 0.364

289

0 141 [0.364 1n 32-1n(1-0.364:fl - 0 364+0.0518

 

0.141 (In 32 +

Set b = 0.121

1

1-0.364)

- 1

_ 0 121 0 141 [0.121 In 32-—1n(1-0.121):]- O.121+0.0518

0.141 (In 32 + ITELIET) - 1
= 0.144
Set b4 = 0.144
b5 a 0.144

Product F10
Similarly we have:

Parameter b

b = 0.146

Parameter a

b
xu 330.146

e=1t6=mz6=1-95

290

Product Fll

 

Parameter b

b = 0.157

Parameter a

Product F13

 

Parameter b

b = 0.251

Parameter a

Product F16

 

Parameter b
b = 0.306

Parameter a

a = iii = llQELEES = 6 07
_ l-b 1-0.306 °

291

 

 

 

 

 

 

 

 

 

 

 

 

 

RESULTS
TABLE D.7
PREDICTED PARAMETER b VALUES
(1) (2) (3) (4) (5) (6)
PRODUCT b B A% bp A%
(actual)
F7 0 123 0.177 44 0.153 24
F8 0.0551 0 177 221 0.144 161
F10 0.128 0.177 38 0.146 14
Fll 0.0758 0.177 134 0.157 107
F13 0.271 0.177 -35 0.251 -7.4
F16 0 246 0.177 -28 0.306 24
TABLE D.8
PREDICTED PARAMETER a VALUES
(1) (2) (3) (4) (5) (6)
PRODUCT i} E A% 3p A%
(actual)
F7 1.77 2.52 42 2.05 16
F8 1 36 2.52 85 1.92 41
F10 1 80 2.52 40 1.95 8.3
F11 1.48 2.52 70 2.11 42
F13 3.85 2.52 -35 4.11 6.7
F16 4.94 2.52 -49 6.07 23

 

APPENDIX E

APPENDIX E

xu
A CODE FOR CALCULATING ail WITH AN
1

HP-25 SCIENTIFIC PROGRAMMABLE POCKET CALCULATOR

Switch to PRGM mode and press f PRGM to clear program
memory and display step 00. Then key in the list of keys
below:

Kevs Comments

 

b Enter parameter b_value

a Enter parameter a value

x Enter xu value

293
To run the program switch to automatic RUN mode and
press f PRGM so that the calculator will begin execution
from step 00. Press R/S to start execution. When execution

stops press RCL 2 to retrieve the result stored in register

R2

10
It takes approximately 20 seconds to compute a 22 x-b

l

APPENDIX F

DATA TABLES (CHAPTER VI)

294
TABLE 6.3

FINAL ASSEMBLY OF THE CPU OF A
THIRD GENERATION COMPUTER (FIRM A)

 

 

 

 

 

Startup Productivity Index (y) at Cum. Unit
Code 1 2 3 4 5
A1 1.87 1.68 1.65 1.49 1.39
A2 1.68 1.59 1.49 1.38 1.30
A3 2.70 2.18 1.94 1.76 1.63
A4 2.40 2.26 1.95 1.80 1.67
A5 2.12 1.92 1.72 1.58 1.47
TABLE 6.4

MANUFACTURING OF CARD PUNCH X (FIRM A)

 

 

 

 

Startup Productivity Index (y) at Cumulative Unit
Code 94 183 277 421 584 757
A6 1.53 1.39 1.34 1.29 1.21 1.16
A7 1.22 1.05 1.00 1.00 1.00 1.00
A8 1.24 1.13 1.09 1.15 1.11 1.08
A9 1.71 1.65 1.63 1.53 1.38 1.29
A10 1.89 1.68 1.60 1.42 1.31 1.24
A11 1.81 1.67 1.61 1.47 1.34 1.26

 

295
TABLE 6.5

ASSEMBLY OF MAJOR UNITS
OF CARD PUNCH Y (FIRM A)

 

 

 

 

 

Startup Productivity Index (y) at Cumulative Unit
Code 111 155 214 287 394 575
A12 1.34 1.31 1.29 1.27 1.21 1.16

TABLE 6.6

FINAL ASSEMBLY & TESTING OF
CARD PUNCH X (FIRM B)

 

 

 

 

 

Startup Productivity Index (y) at Cumulative Unit
Code 15 40 52 358 1007 1912
B1 2.44 2.52 2.37 1.61 1.38 1.22
TABLE 6.7

FINAL ASSEMBLY OF CARD PUNCH Y (FIRM C)

 

 

Startup Productivity Index (y) at Cumulative Unit
Code 40 84 134 195 264 340 410 490 580 680 800
Cl 3.07 2.73 2.52 2.33 2.18 2.06 1.99 1.91 1.84 1.76 1.70

 

920 1050 1180 1320 1470 1610 1780 1940 2140

 

1.64 1.59 1.54 1.50 1.46 1.43 1.39 1.37 1.33

296
TABLE 6.8

ASSEMBLY OF 2nd GENERATION COMPUTER
UNITS (FIRM D, PROGRAM # 1)

 

 

 

 

Startup Productivity Index (y) at Cumulative Unit
Code 1 2 3 4 5 6 7 8 9 10 11
D1 2.68 2.02 1.75 1.60 1.48
D2 3.12 2.59 2.38 2.26 2.06 1.92 1.81 1.73 1.66 1.59 1.54
D3 1.31 1.28 1.23 1.19 1.16 1.13 1.11 1.09 1.09 1.08
D4 2.47 2.05 1.89 1.66
D5 1.23 1.16 1.19 1.16 1.12 1.10
D6 2.86 2.57 2.23 2.23 2.05 1.99 1.89 1.83 1.84 1.82 1.79
(cont'd)

12 13 14 15 16 17 18 19 20 21 22

 

D6 1.

74

1.69 1.69 1.68 1.66 1.64 1.60 1.58 1.54 1.52 1.49

297

TABLE 6.9

ASSEMBLY OF SMALL COMPUTER

COMPONENTS (FIRM D, PROGRAM # 2)

 

 

 

 

Startup Productivity Index (y) at Cumulative Unit
Code 5 10 15 20 23 30 37 38 39 40
D7 3.07 2.44 1.95 1.70 '1.57

D8 1.13 1.07 1.09 1.08 1.06

D9 1.38 1.33 1.25 1.18 1.14
D10 3.51 2.82 2.05 1.70

D11 2.41. 2.12 2.03 1.53

D12 3.56 2.83 2H39 1.90

D13 3.37 2.73 1.96 1.64

D14 1.23 1.16 1Jl6 1H14 1.12 1.09

 

298

TABLE 6.10

ASSEMBLY OF COMPUTER COMPONENTS
AND DATA STORAGE UNITS (FIRM E)

 

 

 

 

Startup Productivity Index (y) at Cumulative Unit
Code 5 10 20 30 40 50 60 70 80 90 100 110
E1 4.35 4.58 2.86 2.31 2.03 1.94 1.82 1.73 1.65

E2 1.89 1.80 1.71 1.55 1.47 1.31

E3 2.59 2.32 2.25 2.09 1.92 1.80 1.70 1.59

E4 2.73 2.28 2.06 1.91 1.73 1.61 1.51 1.43

E5 2.89 2.57 2.32 2.33 2.15 2.00 1.86 1.74 2.43 2.30 2.18 2.07
E6 3.34 3.24 2.96 2.85 2.70 2.59 2.55 2.50

E7 2.94 2.19 1.79 1.77 1.60 1.49

E8 2.03 2.00 1.56 1.48 1.35 1.29

E9 5.14 4.08 3.47 2.87 2.53 2.31 2.10 1.94

E10 2.23 2.14 1.75 1.72 1.59 1.53 1.44

E11 3.79 4.03 3.28 3.07 2.76 2.59 2.52 2.38 2.21

E12 2.98 2.53 2.04 1.99 1.85 1.65 1.56

 

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303
TABLE 6.24

MANUFACTURING OF BINDING MACHINES AND PRINTING PRESSES
FIRM F (U.S.)

 

 

 

 

 

 

Startup
x = cumulative production y = productivity index
Code i
F1 x... 10 13 19 23 30 36 51 56
y...2.03 1.94 1.78 1.71 1.62 1.53 1.39 1.35
F2 x... 1 4 12 17 25
y...1.43 1.45 1.32 1.25 1.17
F3 x... 7 11 16 23 27
y...1.44 1.44 1.37 1.29 1.25
F4 x... 2 3 4 5 8
y...1.69 1.63 1.57 1.49 1.31
F5 x... 5 8 10 13 16 18 21
y...1.57 1.48 1.45 1.41 1.35 1.31 1.26
F6 x... 6 10 18 28 37
g...l.53 1.40 1.29 1.21 1.16
F7 x... 6 16 24 32 41

 

y...1.42 1.26 1.19 1.16 1.12

Startup

Code

F8

F9

F10

F11

F13

F14

F15

304
TABLE 6.24 (cont'd)

 

 

 

 

 

 

 

 

cumulative production y = productivity index
9 17 22 3O 39 48 57 66
.1.19 1.16 1.15 1.15 1.11 1.09 1.08 1.07
16 29 4O 48 57
.1.33 1.22 1.16 1.13 1.13
12 18 25 39 50
.1.32 1.24 1.18 1.11 1.11
2 13 25 33 43
.1.40 1.22 1.16 1.13 1.11
3 4 8 15 22
.1.54 1.46 1.32 1.21 1.14
12 42 72 102
.2.01 1.32 1.20 1.14
10 25 4O 55 7O 85
.1.71 1.54 1.39 1.29 1.24 1.20
. 20 4O 70
..1.66 1.46 1.26

Startup

Code

F16

F18

F20

F21

F22

cumulative production

305

TABLE 6.24 (cont'd)

40

y = productivity index

60 80 100 120

 

2.06

15

1.85 1.70 1.57 1.47

30 45 6O

 

. 10

1.82

30

1.54 1.38 1.29

50 7O 90 110

 

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. 10

1.50

20

1.36 1.27 1.22 1.18

35 50 65 80

 

..1.73

10

1.59

35

1.41 1.30 1.24 1.19

60 85 110 135

 

..2.54

1.99

14

1.75 1.59 1.46 1.37

34 54 74 94

 

..1.64

1.55

15

1.31 1.22 1.18 1.14

30 5O 7O 9O

 

...2.28

1.85

1.61 1.46 1.35 1.27

306

TABLE 6.24 (cont'd)

 

 

 

 

 

 

 

Startup
x cumulative production y = productivity index
Code
F23 x. 10 30 55 85 111 140 167 195
y...2.03 1.80 1.67 1.57 1.50 1.44 1.40 1.38
(cont'd) x...225 253 283 313 343 373 403 433
y...1.36 1.34 1.33 1.32 1.30 1.29 1.28 1.26
(cont'd) x...463 493 523 553 581 611 641 671
y...1.24 1.23 1.22 1.20 1.20 1.19 1.17 1.17
F24- x... 6 16 36 66 96 136 186 236 286
g... 2.99 2.55 2.17 1.89 1.72 1.55 1.41 1.32 1.27
'F25 x... 10 20 40 70 90 115 145 175 205
g... 1.76 1.70 1.51 1.52 1.45 1.39 1.34 1.31 1.29
(cont ' d) x. . . 235 255 270 285 295 305 315 327 339
g... 1.26 1.24 1.23 1.21 1.20 1.20 1.19 1.18 1.18
F26 x... 10 20 35 50 60 80
y...1.31 1.24 1.18 1.35 1.13 1.12
(cont'd) x...100 120 150 168

 

y...1.11 1.10 1.08 1.07

307
TABLE 6.24 (cont'd)

Startup

x = cumulative production y = productivity index
Code
F27 x... 8 12 30 43 57 72

 

y...1.50 1.44 1.26 1.17 1.11 1.07

(cont'd) x... 86 103 122 140 156 171

 

y...1.05 1.03 1.03 1.04 1.04 1.04

F28 x... 5 7 18 20 27 33

 

y...2.17 2.03 1.66 1.62 1.50 1.43

(cont'd) x... 38 39 42 43 45 49
y...1.38 1.37 1.35 1.34 1.33 1.30

 

308
TABLE 6.29

PARAMETERS a AND b
(FIRM F, BINDING MACHINES)

 

 

 

 

Startup

a b
Code
F16 4.94 0.246
F13 3.85 0.271
F1 3.57 0.238
F15 3.22 0.219
F14 2.59 0.172
F5 2.01 0.147
F6 1.99 0.150
F4 1.97 0.184
F9 1.93 0.136
F3 1.83 0.110
F12 1.80 0.148
F10 1.80 0.128
F7 1.77 0.123
F2 1.49 0.0604
F11 1.48 0.0758
F8 1.36 0.0551

 

309

TABLE 6.30

PARAMETERS a.AND b
(FIRM F, PRINTING PRESSES)

 

 

 

 

Startup

a b
Code
F24 4.74 0.228
F20 4.47 0.235
F17 3.42 0.237
F22 3.17 0.201
F28 3.16 0.227
F23 2.80 0.133
F19 2.71 0.186
F18 2.70 0.176
F25 2.41 0.120
F21 2.02 0.123
F27 1.95 0.131
F26 1.57 0.0731

 

310
TABLE 6.38

SHIPBUILDING

FIRM G (BRAZIL)

 

 

Productivity Index

(y) at Cumulative Unit

 

 

Startup Code 1 2 3 4 5
G1 1.34 1.31 1.25 1.21 1.16
G2 1.74 1.74 1.65 1.57 1.45
G3 1.23 1.19 1.15 1.13 1.10
G4 1.41 1.39 1.31 1.24 1.19
G5 1.27 1.20 1.20 1.15 1.12
G6 1.28 1.20 1.14 1.14 1.11
G7 1.09 1.10 1.06 1.06 1.04
G8 1.08 1.11 1.09 1.09 1.07
G9 1.42 1.38 1.29 1.23 1.19
G10 1.84 1.92 1.77 1.59 1.47
G11 1.34 1.31 1.21 1.18 1.15
G12 1.02 1.04 1.08 1.08 1.07
G13 1.02 1.21 1.18 1.17 1.14
G14 1.93 1.70 1.50 1.37 1.30
G15 1.51 1.78 1.61 1.48 1.38
G16 2.10 1.77 1.57 1.46 1.37
G17 1.35 1.27 1.21 1.17 1.14
618 1.29 1.21 1.19 1.14 1.11
G19 1.15 1.11 1.09 1.07 1.06

 

311
TABLE 6.42

OIL DRILLING TOOLS & EQUIPMENT
FIRM H (BRAZIL)

 

 

Startup
x = cumulative production y = productivity index
Code
H1 x... 24 44 45 63 65
y...1.15 1.12 1.13 1.08 1.07
H2 x... 1 21 61 71 101 131 132
y...2.56 1.51 1.30 1.37 1.31 1.18 1.19
H3 x... 3 5 13 20 33

 

y...2.42 2.12 1.45 1.47 1.35

312
TABLE 6.47

PARAMETERS 3 AND b
(FIRM I , SPARE PARTS)

 

 

 

 

Startup
a b
Code
110 843.79 1.461
19 247.29 1.196
17 96.75 0.993
I37 46.02 0.8315
IS 12.80 0.554
125 5.03 0.351
117 4.87 0.344
128 4.84 0.342
I27 4.385 0.321
I62 4.31 0.317
168 4.31 0.317
I61 3.89 0.295
165 3.78 0.289
I46 3.74 0.2865
157 3.70 0.284
154 3.645 0.281
156 3.64 0.281
16 3.438 0.268
.129 3.10 0.246

3

TABLE 6.47 (cont'd)

13

 

—;
_

 

 

 

 

 

Startup
a b
Code —
141 3.01 0.240
138 2.825 0.226
166 2.81 0.2245
18 2.705 0.216
148 2.61 0.208
115 2.47 0.196
130 2.17 0.168
122 2.11 0.162
159 2.06 0.157
169 1.98 0.148
158 1.91 0.141
167 1.65 0.109
160 1.62 0.104
164 1.62 0.104
163 1.53 0.0928
121 1.44 0.0799
119 1.44 0.0790
116 1.44 0.0786
153 1.28 0.0542
150 1.25 0.0485
118 1.23 0.0442
1 0

I 149

.20

.0396

314
TABLE 6.47 (cont'd)

 

 

 

 

Startup
a b
Code
145 1.19 0.0379
133 1.16 0.0318
123 1.14 0.0290
112 1.13 0.0264
134 1.06 0.0131
143 1.05 0.0116
144 1.05 0.0103
124 1.03 0.0071
120 1.02 0.0053
136 1.00 0.00
126 1.00 0.00
135 1.00 0.00
111 1.00 0.00
131 1.00 0.00
113 1.00 0.00
132 1.00 0.00
114 1.00 0.00
139 1.00 0.00
140 1.00 0.00
142 1.00 0.00
147 1.00 0.00

315
TABLE 6.47 (cont'd)

 

—
——‘

 

 

 

 

Startup
a b
Code _
151 1.00 0.00
152 1.00 0.00

155 1.00 0.00

 

Startup
Code
J1

J2

J3

J4

J5

J6

J7

J8

J9

316

TABLE 6.51

MECHANICAL AND ELECTRICAL INDUSTRY

(BRAZIL :

1960-1964)

x = cumulative output (m.t )

y= productivity index

 

 

 

 

 

 

 

 

 

x...49,500 101,100 155,100 216,600 287,100
y.. 1.19 1.14 1.12 1.08 1.06
x...43,055 83,985 136,605 187,425 240,290
y.. 1.12 1.10 1.06 1.05 1.04
x...41,800 86,600 133,100 179,600 231,800
g. 1.16 1.11 1.10 1.08 1.06
x...67,400 151,800 249,600 350,800 437,600
y.. 1.27 1.16 1.10 1.08 1.06
x...60,500 125,000 172,600 273,600 388,200
y.. 1.19 1.14 1.10 1.07 1.04
x...46,300 52,000 74,500 108,900 158,600.
y.. 1.10 1.08 1.04 1.03 1.02
x...14,100 39,400 73,350 113,050 153,750
y.. 1.38 1.34 1.25 1.17 1.12
x. . 2,240 4,190 11,190 34,390 52,540
g... 1.70 1.64 1.32 1.10 1.07
x... 2,800 9,700 36,700 72,300 116,900
g... If51 1.50 1.33 1116 1.10

APPENDIX G
REGRESSION ANALYSIS OF LOG-
TRANSFORMED DATA -- A FORTRAN IV PROGRAM

317

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BIBLIOGRAPHICAL NOTES

BIBLIOGRAPHICAL NOTES

Chapter I

 

1
Of special interest for the purpose of this study were:

T.P. Wright, "Factors Affecting the Cost of Airplanes",
Journal of Aeronautical Sciences 3, no.4 EFebruary 1936]:
122-8; A. Alchian, "An Airframe Production Function", The
RAND Corporation, P-108, October 10, 1949, and "Reliability
of Progress Curves in Airframe Production", The RAND Corpora
tion, RM-206-1, February 3, 1950; F.J. Andress,"The Learning

Curve as a Production Tool", Harvard Business Review 32,no.1,

 

Egan-Feb 195E]: 87-97; K.J. Arrow and S.S. Arrow, "Methodolg
gical Problems in Airframe Cost-Performance Studies", The
RAND Corporation, RM-456, September 20, 1950; K.J. Arrow and
H. Bradley, "Cost Quality Relations in Bomber Airframes",The
RAND Corporation, RM-536, February 6,1951; H. Asher, "Cost-
Quantity Relationships in the Airframe Industry", The RAND
Corporation, R-291, July 1, 1956; N. Baloff, "Manufacturing
Startup: A Model", Unpublished Ph.D. dissertation, Stanford
University, 1963; "Startups in Machine-intensive Production

Systems", The Journal of Industrial Engineering 17, no.1

 

[Jan—Feb 1966]: 25-32; "The Learning Curve - Some Controver

sial Issues", Journal of Industrial Economics,[§uly 1966] :

 

275-82; "Estimating the Parameters of the Startup Model — An

Empirical Approach", The Journal of Industrial Engineering

 

18, no.4 [April 1967] : 248-53, and "Extension of the Learn

ing Curve - Some Empirical Results", Operations Research
319

 

Notes pages 2 - 3
Quarterly 22, no.4 [December 1971] : 329-40; N. Baloff and

 

S. Becker, "A Model of Group Adaptation to Problem Solving

Tasks", Human Performance and Organizational Behavior 3,

 

[August 1968] : 217-38; S.E. Bryan, "A Melhoria da Producfio

no Brasil", Revista de Administracao de Empresas 1, no.2

 

[Sep.-Dec. 1961] : 27-55, and "Fair Value and the Learning

Curve", Purchasing, no. , [September 1954:]: ; R. w.

 

Conway and A. Schultz, Jr., "The Manufacturing Progress

Function", The Journal of Industrial Engineering 10, no. 1

 

[gan-Feb 1959] : 39-53; W. Z. Hirsch, "Manufacturing Progress

Functions", The Review of Economics and Statistics 34, no. 2

 

[May 1952] : 145-55, and "Firm Progress Ratios",Econometrica

 

24, no.2 [April 1956] : 136-43; W.B. Hirschman, "Profit from

the Learning Curve", Harvard Business Review 42, no. 1

 

[January 1964:]: 125-39; M.D. Kilbridge, "Predetermined Learn

ing Curves for Clerical Operations", The Journal of Industrflfl.

 

Engineering 10, no.3 [May-June 1959] : 206- ; F.K. Levy,

 

"Adaptation in the Production Process", Management Science 11,

 

no.4 [April 1965:]: 136-54; Stanford Research Institute, An

Improved Rational and Mathematical Explanation of the Progress

 

Curve in Airframe Production. Stanford: SRI, August 10, 1949,

 

and Relationships for Determining the Optimum Expansibility

 

of the Elements of a Peacetime Aircraft Procurement Program.

 

Stanford: SRI, December 31, 1949; P.F. Williams, "The Appli-
cation of Manufacturing Improvement Curves in Multi-Product

Industries", The Journal of Industrial Engineering 12, no.2

[Mar-Apr 1961] : 108-112.

 

320

2 Notes pages 4 - 7
Baloff, "Manufacturing Startup: A Model", pp.5-6.

3
Asher, "Cost-Quantity Relationships in the Airframe

Industry"; Baloff, "Startups in Machine-intensive Production
Systems"; Conway and Schultz, Jr. , "The Manufacturing Progre$
Function", and R.P. Zieke, "Progress Curve Analysis in the
Aerospace Industry", The Boeing Co., 1962.

l;
Asher, "Cost-Quantity Relationships in the Airframe

Industry" ; Baloff, "The Learning Curve - Some Controversial
Issues"; Conway and Schultz, Jr., "The Manufacturing Progress
Function"; Hirsch, "Firm Progress Ratios", and "Manufacturing
Progress Functions", and Levy, "Adaptation in the Production
Process".

5
Baloff, "Startups in Machine-intensive Production

Systems", and "Extension of the Learning Curve - Some Empiri
cal Results"; Bryan, "Fair Value and the Learning Curve";
Conway and Schultz, "The Manufacturing Progress Function" ;
Hirsch, "Firm Progress Ratios", and "Manufacturing Progress
Functions", and Wright, "Factors Affecting the Cost of
Airplanes".

6
Alchian, "An Airframe Production Function", and "Relgg

bility of Progress Curves in Airframe Production"; Asher ,
"Cost-QUantity Relationships in the Airframe Industry" ;
Baloff, "Manufacturing Startup: A Model", "Estimating the Pa
rameters of the Startup Model — An Empirical Approach", and
"Extension of the Learning Curve - Some Empirical Results";

321

Notes pages 7 - 11
Conway and Schultz, Jr., "The Manufacturing Progress Func-
tion"; Hirsch, ”Firm Progress Ratios", and "Manufacturing
Progress Functions".

7
Andress, "The Learning Curve as a Production Tool";

Asher, Cost-Quantity Relationships in the Airframe Industry";
Baloff, "The Learning Curve - Some Controversial Issues", and
"Extension of The Learning Curve - Some Empirical Results" ,

and Conway and Scultz, Jr.,"The Manufacturing Progress Func-

tion".

8
Kaplan, A., The Conduct of Inquiry: Methodology for

Behavioral Science . San Francisco, Calif. : Chandler Pub—
lishing Co., 1965.

9
Ibid.

Chapter II

 

1
Wright, "Factors Affecting the Cost of Airplanes".The

original formulation of the progress curve theory is attrib-
uted to T.P. Wright. Nevertheless, at least two authors
believe that Leslie McDill should get the credit for its
earliest idealization. According to Reguero (see M. A.
Reguero, "An Economic Study of the Airframe Industry", Unpub
lished Ph.D. dissertation, Department of Economics, New York
University, October, 1957, p.213), the credit for the first
investigation which led to the formulation of the progress

curve theory was given to Leslie McDill, who was commanding
322

Notes pages 12 - 14
officer at McCook Field (the predecessor of Wright-Patterson
Air Force Base, near Dayton, Ohio) in 1925. Also, Max Stupar
(see Max Stupar, "Forecasting of Airplane Man-Hours", Unpub-
lished manuscript, Headquarters Air Materiel Command, 1942,
p.7) says that about 1932 Leslie McDill studied costs of
airplanes in various quantities and arrived at the conclusion
that doubling cumulative airframe output was accompanied by
an average reduction in man-hour consumption of about 20 per
cent. This meant that the average labor requirement after
doubling quantities of output was about 80 per cent of what
it had been before. Neither of the above mentioned authors
give the source of their information. Whether or not Wright
originated the learning curve concept cannot be determined
with certainty. Since McDill did not publish his findings,
one will never know precisely what was the character and
extension of his contribution. Nonetheless, it is granted
that Wright was the first to make the theory known.

2
This latter term is misleading since it is mathemati-

cally illogical - it is neither the slope of y nor of log y:

3
Wright, ”Factors Affecting the Cost of Airplanes" ,
p. 124.
4
Ibid., p. 124
5
Ibid., p. 125.
6

See J.M. White, "The Use of Learning Curve Theory in

Setting Management Goals", The Journal of Industrial
323

 

Notes pages 14 - 18

 

Engineeripg 12, no.6 [November-December 1961:]: 409-411,
p. 410; see also Zieke, "Progress Curve Analysis in the Aerg

space Industry", p.2.

7
Asher, "Reliability of Progress Curves in Airframe

Production", p.16.

8
Crawford's notation is slightly different. However,the

present dissertation requires standardization of all symbols
employed in order to make the discourse coherent to the

reader, particularly in the chapters ahead.

9
J.R. Crawford, Learning Curve, Ship Curve, Ratios,

 

Related Data, Lockheed Aircraft Corporation, Burbank, Cali-

 

fornia (n.d.), p.52.

10
,Estimating, Budgeting, and Scheduling,

 

 

Lockheed Aircraft Corporation, Burbank, California, 1945,ph5L

11
Ibid., p. 26

12
Ibid.

13
F.J. Montgomery, "Increased Productivity in the Con-

struction of Liberty Vessels", Monthly Labor Review, no.
[November 1943] : 861-64.

14
A.D. Searle, "Productivity Changes in Selected Wartime

 

Ship Building Programs”, Monthly Labor Review 61, no.6
[December 1945] : 1132-47.

15
Ibid., p. 1144.

 

324

Notes pages 18 - 25
16
K.A. Middleton, "Wartime Productivity Changes in the

Airframe Industry", Monthly Labor Review 61, no.2 [August
1945:]: 215-25.

17
The idea of working with this new function, i.e., man-

 

-hours per pound of airframe versus cumulative production 32

 

pounds is due to A.B. Berghell. See Berghell. Production

 

Engineering in the Aircraft Industry. New York: McGraw-Hill

 

Book Company, Inc., 1944, Chap.12.

l8
K.A. Middleton, "Wartime Productivity Changes in the

Airframe Industry", p.221.

19
E.A. Rutan, Theory of Learning Curves, Chance Vought

 

Aircraft Incorporated, Dallas, Texas, October, 1948.

20
Gordon W. Link and Don A. Ellis, The Experience Curve,

 

Boeing Aircraft Company, Wichita, Kansas, December, 1945.

21
"Source Book of World War 11 Basic Data”, Airframe

Industry, Vol.1, Air Materiel Command, Dayton, Ohio, 1947.

22
J.R. Crawford, and E. Strauss, World War II Accelera-

 

tion of Airframe Production, Air Materiel Command, Dayton ,

 

Ohio, 1947.

23
"Source Book of World War 11 Basic Data”, Air Materiel

Command.

24
J.R. Crawford, and E. Strauss, World War II Accelera-

 

tion of Airframe Production, p. 13.

 

325

Notes pages 26 - 30
25
Relationships for Determining the Optimum Expansibi-

lity pf the Elements gf‘g Peace-time Aircraft Procurement

 

 

Program, Stanfors Research Institute, prepared for Air Mate-
riel Command, USAF, December 31, 1949.

26
A.Garg and P. Milliman, "The Aircraft Progress Curve

- Modified for Design Changes", The Journal of Industrial

 

Engineering 12, no.1 [Jan-Feb 1961] : 23-28.

 

27
Eric Mensforth, "Airframe Production", Parts I and II,

Aircraft Production 9, [September 1947] :343-50, and @ctober
1947] : 388-95.

 

 

28
Ibid., p.392
29
P. Guibert. Le Plan de Fabrication Aeronautique.Parflm
Dunod, 1945.
30

 

. Mathematical Studies of Aircraft Construc-

 

tion, Central Air Documents Office, Wright-Patterson Air

Force Base, Dayton, Ohio.

31
Ibid.,p.64.
32
For example, Guibert found that m_may be approximated
by
- 1 __£Q__ h = t f r duction
m — - a + 81 w ere a ra e o p o

Other empirical equations were derived for A and a, as well.

See Guibert. Le Plan de Fabrication Aeronautique.

 

326

Notes pages 31 - 34
33
A.B. Berghell. Production Engineeringig the Aircraft

 

 

Industry. New York: McGraw-Hill Book Company, Inc., 1944,
Chap. 12.

34
Ibid., p.177.

35
Berghell does not state whether this common slope is

80 per cent or some other value.

36
I.M. Laddon, "Reduction of Man-Hours in Aircraft pro-

duction", Aviation [May 1943] : 170-73.

37
G.W. Carr, "Peacetime Cost Estimating Requires New

Learning Curves", Aviation 45, no.4 [April 1946] : 76-77.

38
Boeing Transport Division, Improvement Curve Handbook,

 

 

Renton, Washington: Boeing Airplane Company, 1956.

39
Relationships for Determining the Optimum Expansibili-

ty o§_the Elements of a Peace-time Aircraft Procurement Pro—

 

gram, Stanford Research Institute.

40
Guibert, Lg Plan d3 Fabrication Aéronautique.

41
Werner Z. Hirsch, "Manufacturing Progress Functions",

 

 

 

The Review of Economics and Statistics 34, no.2 [May 1952] :
145—55, and "Firm Progress Ratios", Econometrica 24, no.2
[April 1956:] : 136-43.

42
Hirsch, "Firm Progress Ratios".

 

327

Notes pages 34 - 39
43
Hirsch's terminology is suggestive: The exponent (-b)

of the progress function y = ax’b is called "progress e1astig

° N

ity . The complement to one of Wright's "slope" is defined.as
the ”progress ratio". It is easy to see that Hirsch's "prog-

ress ratio” and the exponent b vary in the same direction.

44
Computed for eight products only.

45
A. Alchian, ”An Airframe Production Function", The

RAND Corporation, P-108, October 10, 1949, p.12.

46
Stanley E. Bryan, "Fair Value and the Learning Curve",

Purchasing [September 19543:

47
Ibid., p. 97.

48
This account of the Schultz and Conway research is

 

based on three main sources: (1) R.W. Conway and A. Schultz,

Jr.,"The Manufacturing Progress Function", The Journal of

 

Industrial Engineering 10, no.1 [Jan-Feb 1959] : 39-53; (2)

 

Schreiner, D.A., "The Manufacturing Progress Function, Its
Application to Operations at IBM, Endicott", Paper presented
at 12th Annual ASQC Convention, Boston, Mass., May, 1958,and
(3) information independently gathered by this author.

49
Rate of progress here has the same meaning as Hirschds

progress ratio. See note # 43.

50
R. W. Conway and A. Schultz, Jr., "The Manufacturing

Progress Function", p.42.

328

51 Notes pages 39 - 47

Ibid., p.43.

52
Schreiner, D.A., "The Manufacturing Progress Function,

Its Application to Operation at IBM, Endicott".

53
The model y = ax"b describes the progress phenomenon

only up to point (xu, y After that a plateau is assumed.

u)'
Recall that yu is pratically defined as the point in cumula-
tive production at which the reduction in manufacturing hours
per unit from the first unit produced in the month to the

last unit of the month is approximately 2-3%. See p.

54
A.W. Morgan, Experience Curves Applicable 59 the Air-

 

craft Industry, The Glenn L. Martin Company, Baltimore,

 

Maryland, 1952.

55
W.A. Rayborg, Mechanics pf the Learning Curve,Northrop

 

 

Aircraft, Inc., Hawthorne, California, March, 1952.

56
E.J. Blume and D. Peitzke, Purchasing with the Learn-

 

 

ing Curve, North American Aviation, Inc., Inglewood, Califop

 

nia, August, 1953.

57
William F. Brown, The Improvement Curve, Boeing Air-

 

plane Company, Wichita, Kansas, March, 1955, and Roy W.Smith,

William C. Lansing, and Henry G. Horton, Improvement Curve

 

Handbook, Boeing Airplane Company, Seattle, Washington, 1956.

58
William F. Brown, The Improvement Curve, p.6.

 

329

Notes a es 47 - 52
59 P g

The Experience Curve and Improvement Curve Study,
Boeing Airplane Company, Wichita, Kansas (n.d.), p.9

60
A. Alchian,"Re1iabi1ity of Progress Curves in Airframe

Production".
61
H. Asher, "Cost-Quantity Relationships in the Airframe
Industry".
62
Ibid., p.129.
63

F.T. Koen, "Dynamic Evaluation", Factory 117, no.9
[September 1959] : 98-103.

64
F.J. Andress, " The Learning Curve as a Production

Tool", Harvard Business Review 32, no.1 [Jaaneb 1954]:87-97.

65
N. Baloff, "Manufacturing Startup: A Model",Unpub1ishai

 

Ph.D. dissertation, Stanford University, 1963; "Startups in

Machine-Intensive Production System", The Journal of Indus-

 

trial Engineering 17, no.1 [gan-Feb 1966] : 25-32, and

 

"Estimating the Parameters of the Startup Model - An Empiri-
cal Approach", The Journal of Industrial Engineering l8,no.4
[April 1967] : 248-53.

66
N. Baloff, "Extension of the Learning Curve - Some Em

 

pirical Results", Operations Research Quarterly 22, no. 4

 

[December 197L] : 329-40.

67
Baloff, "Manufacturing Startup: A Model", and "Startums

in Machine-Intensive Production Systems". Baloff's model is

330

Notes pages 52 - 54
the power function y = axb where y is per cent performance
efficiency (PPE), x is cumulative output (in appropriate
units) and a and p are parameters of the model. The relatiop
ship is very similar to those previously used in labor-intep
sive manufacture. The per cent performance efficiency (PPE)
is an index that relates the actual performance of the pro-
cess on each product to predetermined, engineered standards
of performance. For example, for a given product, a 70% PPE
indicates that the process has performed at 70% of the

standard output level for that product.

68
Baloff, "Extension of the Learning Curve - Some Empip

ical Results".

69

Hirschmann, "Profit From the Learning Curve".

7O
Baloff, "Manufacturing Startup: A Model", and "Start-

ups in Machine-Intensive Production Systems.

71
Alchian, "Reliability of Progress Curves in Airframe

Production", and Asher, "Cost-Quantity Relationships in the

Airframe Industry".

72
Baloff, "Manufacturing Startup: A Model", and "Start-

ups in Machine-Intensive Production Systems".

73
Alchian, "Reliability of Progress Curves in Airframe

Production", and Asher, "Cost-Quantity Relationships in the
Airframe Industry".

74
Conway and Schultz,"The Manufacturing Progress Functiaf.

331

75 Notes pages 54 — 58
Baloff,"Estimating the Parameters of the Startup Model
— An Empirical Approach", p.250.

76
Asher, "Cost-Quantity Relationship in the Airframe

Industry", pp 74-79.

77
Baloff, "Manufacturing Startup: A Model".

78
Baloff and Becker, "Group Learning Curves: A Power

Function?", Working Paper, Graduate School of Business, Uni-
veristy of Chicago.

79
S. Becker and N. Baloff, ”Organization Structure and

Problem Solving Behavior", Administrative Science Quarterly
14, no.6 [June 1969] : 260-71.

80
Baloff, "Extension of the Learning Curve - Some Empip

 

ical Results".

81
Philip B. Metz, "A Manufacturing Progress Function

Nomograph", The Journal of Industrial Engineering 13, no.4
[July-Aug 1962] : 253-56.

82
J.G. Kneip, "The Maintenance Progress Function", The

 

Journal of Industrial Engineering16, no.6 [Nov-Dec 1965] :
398-400

83
J.H. Russell, "Progress Function Models and their

 

Deviations", The Journal of Industrial Engineering19, no.1

[January 1968] : 5-10.

 

332

Notes pages 58 - 72

84
J.H. Russell,"Predicting Progress Function Deviathxuffi
IBM TR 22.4 , 1967.
85

J.M.White, ”The use of Learning Curve in Setting Man-

agement Goals",The Journal of Industrial Engineering 12,no.6

 

[Nov-Dec 196K] : 409-411.

86
P.F. Williams, "Application of Manufacturing Improve-

ment Curves to Multi-Product Industries", The Journal of

 

Industrial Engineering 12, no.2]:March-April 1961] :108-112.

87
W.B. Hirschmann, "Profit From the Learning Curve”,

 

Harvard Business Review 42, no.1]:Jan-Feb 1964] : 125-139,
p. 125.

88
W.J. Abernathy and K. Wayne,”Limits of the Learning

Curve", Harvard Business Review no.5[:Sept—Oct 1974]:109-119.

 

Chapter III

 

1
Allen, R.G.D. Mathematical Analysis for Economists

 

London: MacMillan & Co. Ltd., 1962, pp 251-55.

2
For example, in the airframe industry the independent

variable is sometimes the weight of the airplane.In petrolemn
refining Hirschmann plotted a progress curve of performanmeof
cracking units (expressed in days per 100,000 barrels) versus
cumulative throughput (expressed in million barrels). In

steel-making (Baloff),outputs are expressed in thousands of

333

Notes pages 72 - 100

tons, millions of lineal feet, and so on. See Chapter 2.

3
A formal proof that 1im (ax'b) = 0 and that
x+oo
1im (ax-b) = m is given in Appendix 1A.
x+0
4

See, for example, Marlin U. Thomas, "Developing More
Accurate Cubic Learning Curves", Manufacturing Engineering

and Management, [guly 1972] : 29-30.

 

5
Franklin, P. Differential and Integral Calculus . N.

York: McGraw-Hill Book Company, Inc., 1953, p.366.

6
. Methods of Advanced Calculus. N.York :

 

McGraw-Hill Book Company, Inc., 1944, pp 26-28.

7
An initial value B1 may be found by some means, e.g.

by a graph or by tabulation.

8
If B is a first approximation to the root of the

1
equation f(B)=0, the Newton-Raphson formula for a better
approximation B2 is B2= B1 - f(B1)/f'(B1). If a is the error

in B1, i.e., B1 = B + a, one can write:

B2 = B + e- f(B + €)/f'(B + a)

By suitably expanding f(B + e) and f'(B + a) one can show
that, if e is small, the error in B2 is approximately

% £2 f"(B)/f'(B). This means that if the error in the first
approximation is e, the error in the next approximation is

2

roughly proportional to e , a result which is used when consyi

ering the rate of convergence of the Newton-Raphson formula.
334

Notes pages 101 - 143
Chapter IV

 

1
For the derivation of Euler-MacLaurin' formula see

de La Vallée-Poussin, Ch-J. Cours d'Analyse Infinitésimale,

 

Paris: Gauthier-Villars, 1957, volume 2, Ch 11,pp 374-378.

2
Ibid. Volume 2, Chapter 3, pp 78-80 and Ch. 11, pp

368-69.

3
C.V.L. Charlier. Mechanik des Himmels, II,pp.13-16.

 

See also Carl-Erik Fr5berg, "Introduction to Numerical

 

Analysis", Addison-Wesley Publishing Company, Inc., 1966,

 

p.211 (This book is an English translation of Larobok i

 

numerisk analys by C E Fr6berg, Lund, Sweden: Svenska

 

Bokforlaget/Bonniers, 1962.

4
Scarborough, J.B., Numerical Mathematical Analysis.

Baltimore: The John Hopkins Press, 1966, p.193.

5
Conway and Schultz, "The Manufacturing Progress

Function", p.41 and pp 43-44.

6
Franklin, P. Differential and Integral Calculus. New

York: McGraw-Hill Book Company, Inc., 1953, p.443.

Chapter V

 

1
See, for example Andress, "The Learning Curve as a

Production Tool",p.89. See also Hirschmann,"Profit From the

Learning Curve", p.126.
335

Notes pages 143 - 153
2

Hirsch, "Manufacturing Progress Functions". See also

{irsch, "Firm Progress Ratios".

3
Schultz and Conway, "The Manufacturing Progress

Function", p.49.

4
Baloff, "Manufacturing Startup: A Model", p.196. See

also Schultz and Conway, "The Manufacturing Progress
Function", p. 46.

5
Baloff, "Manufacturing Startup: A Model", p. 16.

6
For a brief account of work measurement systems that

make use of predetermined times, see J.L. Riggs. Production

 

Systems: Planning, Analysis, and Control. New York: John

 

Wiley and Sons, Inc., 1970, Ch.9. The original M.T.M. is
described in Maynard, H.B., G.J. Stegemerten,and J.L.Schwab.

Methods-Time Measurement. New York: McGraw-Hill Book Company,

 

Inc., 1948.

7
Implicit in the derivation of formulas (5.4) and

(5.6) is the assumption that the best fit to the available
data is given by the unit progress function or by the lot-
average progress function. In Appendix C parameter formu-
las are derived for the case where the best fit to the data
is achieved with the cumulative-average progress function.

8
The data for firm D was obtained from S.A. Billon,

"Industrial Time Reduction Curves As Tools for Forecasting",
lUnpublished Ph.D. dissertation, Michigan State University,

1960. 336

9 Notes pages 153 - 222
Ibid. as to firm E.

10
Ibid. as to firm F.

11
Aggregate data for whole Brazilian industries was

obtained from: Ministério do Planejamento e Coordenacao Eco-
nomica, Escrit5rio de Pesquisa Economica Aplicada - EPEA,
Plano Decenal de Desenvolvimento Economico e Social-Diagnos-

tico Preliminar da Industria Mecanica e Elétrica, May 1966.

 

Chapter VI
1

Hirschmann, "Profit From the Learning Curve",pp 133-34.

2
Ministério do Planejamento e Coordenacao Ecoanica,

EPEA, Plano Decenal de Desenvolvimento Econ5mico e Social -
Diagnéstico Preliminar da Industria Mecanica e Elétrica, pp
59-60.

3
The findings by Asher and Baloff were already discussed

in Chapter II.

Chapter VII
1
The calculations were carried out with an HP-25 Sciep
tific Pocket Calculator. The program used is in Appendix E.

2
The relative accuracy of each approximate formula was

previously discussed in chapter IV.

337

Notes pages 223 - 258

3
Wright, "Factors Affecting the Cost of Airplanes".

4
For a comprehensive introduction to the HIPO method,

see Katzan, H. Jr., Systems Design and Documentation - - Ag

Introduction £9 the HIPO Method. New York: Van Nostrand

 

Reinhold Company, 1976. See also HIPO - A Desigp Aid and

Documentation Technigpe, White Plains, New York, IBM

 

Corporation, 1974, Form GC 20-1851.

Appendix B

1
De La Vallée Poussin, Ch. J. Cours d'Analyse

Infinitésimale, Paris: Gauthier-Villars, 1957, Volume 2,

pp. 396-97.

338

BIBLIOGRAPHY

BIBLIOGRAPHY

ARTICLES

Abernathy, W.J., and Wayne, K."Limits of the Learning Curmy'.
Harvard Business Review, no.5 (September-October 1974)
109-19.

Andress, Frank J. "The Learning Curve as a Production Tool".
Harvard Business Review 32, no.1 (January-February 1954)

 

Baloff, N. "Startups in Machine— intensive Production Systems"
The Journal of Industrial Engineering 17, no. 1 (January-

233m mary 1966): 25: 32.

 

. "The Learning Curve - Some Controversial Issues"
Journal pf Industrial Economics (July 1966) : 275-82.

 

"Estimating the Parameters of the Startup Model -
An Empirical Approach". The Journal of Industrial Engineer-
ing 18, no. 4 (April 1967) 248- 53.—

 

"Extension of the Learning Curve - Some Empirical
Results". Operations Research Quarterly 22, no.4 (December
1971) : 329-40.

 

, and Becker, S. "A Model of Group Adaptation to
Problem Solving Tasks". Human Performance and Organization
Behavior 3, (August 1968) : 217-38.

 

Becker, 8., and Baloff, N. "Organization Structure and Prob-
lem Solving Behavior". Administrative Science Quarterly 14,
no.6 (June 1969) : 260-71.

 

Bryan, Stanley E. "Fair Value and the Learning Curve"
Purchasing (September 1954)

 

 

"A Melhoria da Producao no Brasil". Revista

%§.Administracao de Empresas 1, no. 2 (setembro-dezemEro
6I) 27 55

'Carr, G.W. "Peacetime Cost Estimating Requires New Learning
Curves". Aviation 45, no.4 (April 1946) : 76-77.

339

CO‘.

C;

340

Conway, R.W., and Schultz, A. "The Manufacturing Progress
Function". The Journal pf Industrial Engineering 10, no.1
(January-February I959) : 39-531

 

Garg, A., and Milliman, P. "The Aircraft Progress Curve -
Modified for Design Changes". The Journal of Industrial
Engineering 12, no.1 (January-February 196T) : 23-28.

 

 

 

Hirsch, Werner Z. "Manufacturing Progress Functions". The
Review gprconomics and Statistics 34, no.2 (May 1952) :

 

 

. "Firm Progress Ratios". Econometrica 24,
no.2 (April 1956) : 136-143.

 

Hirschmann, W.B. "Profit from the Learning Curve". Harvard
Business Review 42, no.1 (January 1964) : 125—39.

 

Kilbridge, M.D. "Predetermined Learning Curves for Clerical
Operations". The Journal pf Industrial Engineering10,no.3
(May-June 1959) .

 

Kneip, J.G. "The Maintenance Progress Function". The Journal
of Industrial Engineering16, no.6 (November—December

‘I965)_T_398:400.

Koen, F.T. "Dynamic Evaluation". Factory 117, no.9, (Septem
ber 1959) : 98-103.

 

 

Laddon, I.M. "Reduction of Man-Hours in Aircraft Productflxflfl
Aviation (May 1943) : 170-73.

 

Levy, F.K. "Adaptation in the Production Process". Manage-
ment Science 11, no.4 (April 1965) : 136-54.

 

Mensforth, E. "Airframe Production", Parts I and II, Air-
craft Production 9, (September 1947) : 343-50 and

(October I947) : 388-95.

Metz, P.B. "A Manufacturing Progress Function Nomograph".

The Journal 9: Industrial Engineering13, no.4 (July-
August I962) : 253-56.

Middleton, K.A. "Wartime Productivity Changes in the Air-
frame Industry". MonthlyLabor Review 61, no.2 (August
1945) : 215-25.

 

 

Montgomery, F.J. "Increased Productivity in the Construction
of Liberty Vessels". Monthly Labor Review, (November 1943):
861-64.

 

.Russel, J.H. "Progress Function Models and their Deviationsfl
The Journal of Industrial Engineering19, no.1 (January

I938) : 531:0:—

 

 

341

Searle, A.D., and Cody, G.S. "Productivity Changes in Se-
lected Wartime Ship Building Programs". Monthly Labor
Review 61, no.6 (December 1945) : 1132-47.

 

Thomas, M.U. "Developing More Accurate Cubic Learning Curva§'.
Manufacturing Engineering and Management, (July 1972)
29-30.

 

White, J.M. ”The Use of Learning Curve Theory in Setting
Management Goals". The Journal of Industrial Engineering
12, no.6 (November-December 196T) : 409-11.

 

 

Williams, P.F. "The Application of Manufacturing Improvement
Curves in Multi-Product Industries". The Journal of Indus-

trial Engineering 12, no.2 (March-ApriI I96I) : I08-I2.

 

Wright, T.P. "Factors Affecting the Cost of Airplanes".
Journal 9f Aeronautical Sciences 3, no.4 (February 1936):

 

BOOKS

Allen, R.G.D. Mathematical Analysis for Economists. London:
MacMillan & CO LTD, 1962.

 

Berghell, A.B. Production Engineeringip the Aircraft Indus-
trv. New York: McGraw-Hil Book Company, Inc., 1944.

 

 

 

de La Vallée-Poussin, C.J. Cours d'Analyse Infinitésimale.
Paris: Gauthier-Villars, 1959, Volume I.

. Cours d'Analyse Infinitésimale.
Paris: Gauthier-Villars, 1957, Volume II.

 

 

 

 

 

Franklin, P. Differential and Inte ral Calculus. New York:
McGraw-Hill BoOk Company, Inc., I953.

. Methods gf_Advanced Calculus. New York: McGraw-
Hill Book Company, Inc., 19447

 

 

 

 

Froberg, C.E. Introduction pp Numerical Analysis. New York:
Addison-Wesley Publishing Company, Inc., .

 

 

. Larobok i numerisk analys. Lund, Sweden:
Svenska Bokf3rIaget/Bonniers, 1 .

Kaplan, A. The Conduct pf In uir : Methodology for Behavior-
al Science. San Francisco, Ca 1 .: Chandler PuEIisEing 00.,
I965.

 

 

342

Maynard, H.B., Stegemerten, G.J., and Schwab, J.L. Methods-
Time Measurement. New York: McGraw-Hill Book Company, Inc.,

Riggs, J.L. Production Systems :Plannin , Anal sis, and
Control. Nengork: John Wiley and Sons, Inc., I970.

Scarborough, J.B. Numerical Mathematical Analysis. Baltimore:
The John Hopkins Press, 1966.

 

 

 

REPORTS, PAPERS, AND OTHER PUBLICATIONS

Alchian, A. "An Airframe Production Function". The RAND
Corporation, P-108, October 10,1949.

 

. "Reliability of Progress Curves in Airframe
Production". The RAND Corporation, RM-206-1, February 3,
1950.

Arrow, K.J., and Arrow, S.S. "Methodological Problems in
Airframe Cost-Performance Studies". The RAND Corporation,
RM-456, September 20,1950.

 

, and Bradley, H. "Cost Quality Relations in
Bomber Airframes". The RAND Corporation, RM-536, February
6, 1951.

Asher, H. Cost-Quantipy Relationships ip_the Airframe Indus-
try. The RAND Corporation, R—291, July 1, I956.

 

 

Boeing Airplane Company. The Experience Curve and Improve-
ment Curve Study. Wichita, Kansas (n.d.).

 

 

Boeing Transport Division. Improvement Curve Handbook. Renton,
Washington: Boeing Airplane Company,

 

Brazil, Ministério do Planejamento e Coordenagao Economica,
EPEA,_P1ano Decenal de Desenvglvimento Economico g Social -
Diagnostico Preliminar da Industria Mecanica e Eletrica,
Brasilia, May 1966.

Brown, W.F. The Improvement Curve. Boeing Airplane Company,
Wichita, Kansas, MarCh 1955.

 

 

Blume, E.J. and Peitzke, D. Purchasing with the Learnin
Curve. North American Aviation, Inc., Inglewood, CaIifornfla,
August, 1953.

_Crawford, J.R. Learning Curve, Ship Curve, Ratios, Related
Data. Lockheed Aircraft Corporation, Burbank, California

(573.).

 

 

 

343

Crawford, J.R. Estimating, Budgeting, and Scheduling.
Lockheed Aircraft Corporation, Burbank, California, 1945.

 

, and Strauss, E. World War II Acceleration pf

Airframe Production. Air Materiel Command, Dayton, Ohio,
1947.

Guibert, P. Mathematical Studies of Aircraft Construction.
Central Air Documents Office, Wright-Patterson Air Force
Base, Dayton, Ohio.

 

 

Link, G.W., and Ellis, D.A. The Experience Curve. Boeing
Aircraft Company, Wichita, Kansas, December, 1945.

 

Morgan, A.W. Experience Curves Applicable to the Aircraft
Industr . The Glenn L. Martin Company, BaIEimore, Maryland,
I952.

 

 

Rayborg, W.A. Mechanics pf the Learning Curve. Northrop
Aircraft, Inc., Hawthorne, CalifOrnia, March, 1952.

 

 

Russell, J.H. "Predicting Progress Function Deviations",
IBM TR 22.4, 1967.

Rutan, E.A. Theory of Learning Curves. Chance Vought Air-
craft, Inc., Dallas, Texas, October, 1948.

Schreiner, D.A. "The Manufacturing Progress Function, Its
Application to Operations at IBM, Endicott", Paper pre-
sented at 12th Annual ASQC Convention, Boston, Mass.,
May, 1958.

Smith, RJW., Lansing, W.C., and Horton, H.G. Improvement
Curve Handbook. Boeing Airplane Company, Seattle,
Washington, 1956.

 

 

Source Book pf World War I; Basic Data, Airframe Industry,
Vol. ITUAir Materiel Command, Dayton, Ohio, 1947.

 

 

 

Stanford Research Institute. An Improved Rational and
Mathematical Explanation 9f Ehe Pro ress Curve ip Airframe
Production. Stanford: SR1, August I9, I949.

V

 

 

 

 

. Relationships for Determining

the Optimum Expansibility pf the Elements Q: a Peacetime

Affcraft Procurement Program. Stanford: SRI, December 31,
prepared for Air Materiel Command, USAF).

 

 

 

 

 

Zieke, R.P. "Progress Curve Analysis in the Aerospace Indus-
try". An Engineer's thesis presented to the faculty of
the Department of Industrial Engineering, Stanford
University, June, 1962. (Published by the Boeing Company,
Seattle,Washington, June, 1962.)

344

UNPUBLISHED MATERIALS

Baloff, N. "Manufacturing Startup: A Model". Unpublished
Ph.D. dissertation, Stanford University, 1963.

 

, and Becker, 8. "Group Learning Curves: A Power
Function?" Working Paper, Graduate School of Business,
University of Chicago.

Billon, S.A. "Industrial Time Reduction Curves as Tools for
Forecasting". Unpublished Ph.D. dissertation, Michigan
State University, 1960.

Conway, R{W. and Schultz, A., Jr. "The Manufacturing Progress
Function". Department of Industrial and Engineering Admin-
istration, Cornell University, (n.d.).

Reguero, M.A. "An Economic Study of the Airframe Industry”.
Unpublished Ph.D. dissertation, New York University,l957.

Stupar, M. "Forecasting of Airplane Man-Hours". Unpublished
manuscript, Headquarters Air Materiel Command, 1942.

GENERAL REFERENCES

Ezekiel, M. and Fox, K.A. Methods pf Correlation and Regres-
sion Analysis. New York: John W11ey and Sons, 1959.

 

 

Hoel, P.G. Introduction to Mathematical Statistics. New'Yaflx:
John Wiley andSons,l965.

 

 

Reis, D.R.A. _"Extensao da Funcao de Progresgo da Producao a
Algumas Industgias Brasileiras". Dissertacao de Mestrado
em Administracao de Empresas, EAESP-FGV, S.Paulo, 1973.

Tintner, G. Econometrics. New York: John Wiley and Sons,Inc.,
1952.

 

 

. Mathematiques et Statistiques pour 1es
Economistes. Paris: DunodT 1962.