OF THE
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MASSACHUSETTS
OF TECHNOLOGY
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Research Program on the
Management of Science and Technology
THE TECHNOLOGICAL PROGRESS FUNCTIONr
A NEW TECHNIQUE FOR FORECASTINGAlan R. Fusfeld
'
#438-70
January 1970
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS
02139
U{^v:i0-^^
Research Program on the
Management of Science and Technology
THE TECfiNOLOGlCAL PROGRESS FUNCTION?
A NEW TECHNIQUE FOR FORECASTING*
Alan R. Fusfeld
January 1970
"
#438-70
-=Based on a paper read at the annual meeting of TIMS in Atlanta,
October 3, 1969. This paper is being published in Technological
Forec asting Volume 1, Issue 3, Winter 1970.
,
The author
is a special student at MIT's Sloan School of Management,
E52-530, 50 Memorial Drive. Cambridge, Mass. 02139, while on
leave of absence from his senior year at The Johns Hopkins University,
Baltimore, Maryland.
/rv
I
,-1
r
•
yjf-
^^^^ or.
ICTO
76
t.
THE TECHNOLOGICAL PROGRESS FUNCTION:
A NEW TECHNIQUE FOR FORECASTING
1.
BASIC ASSUMPTIONS
:
The core of technological trend forecasting has most
often been based upon
against time.
a
plot of
a
technical parameter
This study concerns the implications for
forecasting, when these technical parameters are plotted
against cumulative production.
For the examples covered in this study, the relationships between technical characteristics and cumulative pro-
duction all have the mathematical form:
(1)
Ti = a(i)^
where
and
value of parameter at the i^h unit,
Tj^
=
i
= cumulative production,
a,b = constants.
This dependence of
a
technical value upon production is defined
here as the technological progress function
.
This paper relates
the development of that function, the implications of the math-
4.
ematical form just mentioned for trend forecasting, and
information compiled from
variety of case studies.*
a
It is of interest to note that the form of the technological
progress function is similar to that of the common industrial
"learning curve", as well as the learning curves of psychologists,
The industrial "learning curve" relates cost to production
(1,
2,
3)
.
Yi -
where
Yid)"^
unit
y-
= cost of the i
i
- cumulative production
y-^
b
= cost of
1
constant
=
st
(2)
unit
.
and the psychological learning curve relates the efficiency
of performing
a
task to the number of repetitions
Ejj
where
= k(N)
b
E„ = efficiency of performing N
N
(4,
5,
6,
7)
(3)
th
task
= cumulative task number
k,b = constants
*
Studies completed included:
civil aircraft-speed
automobiles- horsepower
military aircraft- speed
electric lamps- lumens
turbojet engines- specific weight
computer programs- figure
turbojet engines- specific fuel
consumption
of merit
hovercraft- figure of merit
5.
They are similar to the technological progress function,
T^ = a(i)
,
because all three show the measure of some
characteristic, for which improvement is desired, as an
exponential function of the cumulative number of repetitions
or production.
The environment in which technology develops is clearly
of additional interest and appears to affect the rate of
learning through discrete changes in the learning constant,
b.
It must be specified here,
that environment refers to
the external economic factors which surround the production
process.
An example of this effect was the change in the
government investment and market potential relating to the
which
automotive industry in the late twenties
(8,
multiplied the rate of progress, b, by
factor of 6.2.
a
9)
,
2.
INTRODUCTION AND BACKGROUND INFORMATION
:
The development of the technological progress function,
T^,
from
characteristic of technological improvement, evolved
a
consideration of:
(1)
problems and background factors inherent in
industrial progress relationships,
(2)
indications from psychology that general phenomena
of "learning" were present,
and
2.1
(3)
difficulties in existing techniques of forecasting.
PROGRESS FUNCTIONS
Briefly,
:
industrial progress relationships are functions,
such as production costs/unit, maintenance costs/unit, and
manufacturing costs/unit, which can be written as
of costs associated with unit number one,
number, and a learning constant.
functions because
efficiency.
a
a
function
the cumulative unit
They are termed progress
reduction in costs indicates
a
gain in
These functions, as previously mentioned, have
been found to have the precise form
yi
-
(1,
10,
2,
a(i)-b
11):
(1)
When plotted on log-log paper, this gives
a
straight line,
whose slope, -b, is the rate of progress, that is, the rate
at which efficiency is improving.
It should be noted here
that,
as previously mentioned,
the technological progress
function has the same form and expression, the difference
being
a
positive rather than
a
negative slope.
Various studies of industrial progress functions have
shown that there are
a
number of causes or factors which
are common to most of the different types
(1,
2,
11,
12,
13).
The more direct factors are:
1.
engineering or desing improvements;
2.
servicing technician progress;
3.
job familiarization by workmen;
4.
job familiarization by shop personnel and engineering
liason;
and
5.
development of
a
more efficient parts upply system;
6.
development of
a
more efficient method of manufacture.
The innate factors are:
1.
the inherent susceptibility of an operation to improve-
ment
and
2.
;
the degree to which this can be exploited.
These functions can be called "micro-economic progress
functions"
and
(MEPF) because:
1.
they are similar in mathematical form,
2.
each relates to decisions of the "firm".
However,
their applicability to development decisions is not
8.
as simple as it would seem initially since the functions
contain two areas of problems.
Both areas are relevant
to the subject of the technological progress function be-
cause the first helped to prompt its development and the
second helps to explain its behavior.
The first area involves certain MEPF characteristics
which present problems of decision that have been unsolvable
because of their link with technological innovation.
The
problems begin when one attempts to determine the component
costs of the first unit of production and finds that one must
arbitrarily set
a
limit upon the background development
A mechanism by which background and development
expenses.
costs would be linked to all of the generations of
a
product
might help to alleviate this difficulty of first unit costs;
of course,
this would first require
a
model for technological
change.
Further and more fundamental complications become apparent
when the development of new generations of equipment is considered,
as opposed to the difficulties of measuring cost for
any one generation.
Technical changes and design modifica-
tions are part of the inherent forces that cause the MEPF
to behave as it does, but there are no provisions for setting a
limit on these changes; the engineering change procedure of
firms being arbitrary in nature.
That is to say that, after
a
certain number of modifications in production procedure
or design,
the product in reality will be a "new" product
and will be so labeled by sales and management.
This is often
the case with developments resulting from both conscious and
unconscious defensive research, which is designed to improve
a
product's competibility
The locating of an "old-new" product line or determining
where
a
major change begins are questions posed by users of
the MEPF and seem to beg
a
progress function which cuts
The technological pro-
across individual generation lines.
gress function is such
a
relationship and contains the same
factors of engineering learning and motivation of the MEPF,
which not only cause problems in its use but also provide
a
major stimulus for technological advancement.
The other problem area relates to deviations from normal
(line a-rity on log-log paper) MEPF behavior.
of these differences are important,
The explanations
since they help to clarify
the effects of external factors of the environment on tech-
nological progress.
Some
of
these
deviations
are
due
to changes in the psychological motivation of the personnel
involved,
as would be the case where a product line is sud-
denly scheduled for discontinuance or where the success of
new development is announced to the employees
(1,
2,
14).
a
10,
others would be due to changes in design or the introduction of new people to the job.
Finally,
if the product
or system is being constructed on parallel assembly lines
the addition or subtraction of new lines, perhaps
or shifts,
with changes in demand, will cause anomalies in the MEPF
(14).
All of these deviations cause a change in the
slope of the progress function.
Thus,
it is implied
that changes in the rate of progress (i.e., the slope of
the progress function) are not random but are a function
of critical parameters forming part of the external environ-
ment.
2.2
PSYCHOLOGY
:
In addition to the information supplied by analysis of
progress functions, further insight has been gained from
psychological learning theory.
Since there is considerable
dispute over exact definitions of what learning is or is not,
let me point out that, here, psychological learning refers to
perceptible gains in performing given tasks.
These tasks in
academic studies have ranged from solving puzzles and going
through mazes to simple studies of response time for given
stimuli.
It has been noted in a variety of projects concerning
both animals and people, that the efficiency of performing
a
11.
given tasks increases with the cumulative number of repetitions
(4,
7,
5,
14,
It should also be pointed out
16).
15,
that none of the studies differentiate between the fre-
quency of the repetitions as long as the frequency was within
The studies found that,
the maximum retention interval.
Ejj
where
Such
a
efficiency in performing the N^^ task,
Ejj
=
N
- the cumulative task number,
K, a =
and
(2)
= K(N)^
constants.
relation was particularly apparent in
with children by Melcher
by Bush and Hosteller
(4)
(7)
.
a
1934 study
and in a 1955 study with dogs
A similar but different relation-
ship regarding repetitious activity can be derived from work
presented by Frank Logan in his book. Incentive (16).
noticed of his subjects, that they,
"...behave in such
Logan
a
way
as to maximize reward while at the same time minimizing effort."
Such behavior is not only identical with the aims of man's
economic endeavors, but is also
progress
a
.
a
causal factor of technological
Since the progress functions discussed previously show
similar dependency on repetition of tasks,
it would appear
that the same type of learning discussed above is involved.
2. 3
FORECASTING
:
The third area that has contributed to the present develop-
ment of the technological progress function is technological
12,
forecasting.
as Delphi,
The primary methods already available,
trend
extrapolation,
such
trend correlation, and
growth analogy, do not allow for the easy or precise handling of environmental factors
incentive to find
a
(17),
This provided further
progress function that would be a6
petent as other techniques under constant forces and yet
allow for environmental change.
com-
13.
3.
TECHNIQUES USED
IN ANALYSIS;
The work involved
in investigating
logical progress function
lines of reasoning.
note
and defining the techno-
was completed through two separate
Both paths were developed while taking
of:
(1)
an observation, through analysis of the
the independent
MEPF,
that
variable should be the cumulative production
number;
(2)
an implication in learning theory that the form of the function
should be similar to that of the
MEPF
and the general structure
characteristic of improvement functions; and
(3) a
need of forecasting techniques for a term through which
environmental change could be introduced.
The paths chosen for development are common
endeavors.
They were those
to
most
of theoretical derivation
scientific
from known
relationships and empirical model building from real world data.
14.
The theoretical side bases itself on the eauivalence, for
at least certain areas,
of the rate of patent output with
technological /growth or prc^-ress.
Through relationships
derived or shown by Schmookler (l8) and Villers (19), the
rate of patent output was related to investment and expected
profit functions.
Prom this point, substitutions were made
from investment/profit functions, denoting technical advance.
The result was a rela.tionship that equated the level of
technological improvement to a constant multiplied by an
increasing Quantity dependent function raised to a positive
exponent
,
i.e.,
T.
where
T.
= the level of
= K(
f(i)
)^
technology at the
(3)
i
th
unit of
production,
f(i) « a function which varies with the
number, i,
and
K,c = constants.
production
15.
Although it seemed to substantiate earlier hypotheses, it
could only be taken as
a
further indication that this might
be the correct approach, since it was not precise enough in
nature to stand by itself.
The results of the theoretical derivation cleared away
doubts from the proposed directions of the empirical phase
of the study.
At this stage, it was clear that the work
should attempt to correlate technological improvement with
the cumulative quantity of production.
In addition, thoughts
of incorporating the results with other progress functions
motivated the gathering of data designed for more extensive
correlations.
Real data were then sought to provide information on technical parameters, production, and costs with background infor-
mation to be provided where possible.
One could then observe
the behavior of technological parameters with respect to pro-
duction under the sets of conditions forming the background
environment.
However, before describing the case studies and conclusions
some of the difficulties involved in this work should be pointed
out.
The most difficult problem was that of obtaining accurate
data,
particularly with regard to cost information.
This was
16.
solved by combining data from several sources and v/here
possible having the material validated by someone fa .iliar
with the field.
Other nroblems arose concerning the choice of technical parameters.
In this care decisions were made from
background information and observation of changes in the
parameter with resoect to cost changes.
4.
F.IJSENTATION OF DATA;
The empirical studies discussed here verified both the
preliminary hyootheses and the theoretical derivation and
were based upon ana.lyses of data regarding the aircraft
industry, the electric lamp industry, and computer pror^rammmin"
4.1 AIRCRAFT DATA
.
:
The data from the aircraft industry concerned turbo-jet
engine development over a period of nearly twenty years and
was synthesized from two Air Force Institute masters theses
(20,21), sup'oorting information supplied by the Pratt &
Whitnoy Division of United Aircraft (22), and Aviation I'acts
and Figures 1958 (23).
From these sources, production cost/
unit, production by year, ciAmulative production, and
trie
tech-
nological parameters were observed.
The results have shown, in Figure 1, that technological
progress, as represented by s-ecific weignt
(
dry engine weiglit/
Figure
TURBO-JET ENGINE CHARACTERISTICS
o
f
OO
o
OT O
CM ci CM CM oj
o o o o o
<j-
•
i
1—
r-~
.-I
c-J
•
•
•
•
in
<0 00
Q
in o o o o
I
r—l
E^
1-)
I
m
CO CO
U-t
[-<
t^
y-i
C7>
H
H ^
[M
-
<J-
09
4J
to
CO
•r-1
3
o
<u
Xs.
o
•H
IH
•f-i
l-l
u
(U
a.
CO
•r-l
—o
_
c;
cr>
1
17.
pounds of thrust) and specific fuel consumption (pounds of
fuel/hr ./pound of thrust)
is log- linear to a logarithmic
quantity axis, where quantity indicates the cumulative pro-
duction of engines within the turbo-jet family.
It also
appears that an arithmetic time axis may be used in place of
a
logarithmic quantity axis on many occasions for the same
accuracy, whenever production undergoes constant percentage
increases with respect to time*,
4.2
ELECTRIC LAMP DATA
;
The Electric lamp industry data was obtained by combing
information from James Bright
ment
,
(24)
Industry
,
'
and Arthur A. Bright
(25)
book Automation and Manage -
s
'
s
book. The Electric Lamp
for 60 watt lamps.
These sources enabled the
study to be concerned with technological progress as re-
presented by the output of the lamp in lumens, production
by year, and cumulative production.
The many development
changes made it unnecessary to examine each individual generation
of lamp.
The results confirm the existence of a technological
progress function behaving in accordance with the generalized
MEPF principles.
*
Figure
2,
where this is illustrated, actually
The bracketed points at the end of the "specific weight" curve were
obtained from current advertising information of Pratt & Whitney
presented in Aviation Week during March,
1969.
pif^ure 2
ELECTRIC
LMP
LUMEN OUTPUT
O
O
o
o
U5
d
o
•r-l
•r-l
H
o
01
U
•r-l
C
P
D
o
>
kJ
•H
o
ta
r-l
3
o
O
o
o
w
(0
6
k4
18.
indicates the existence of two rates of progress.
One ex-
tends from 1912 until 1920 and the second from 1920 until
1940.
This would imply that the external environment
affecting the lamp industry underwent
a
production which had been increasing at
shift.
a
Since the
steady 9.39% per
year throughout the first period concurrently suddenly
shifted downward to a rate of 2.74% growth at the same transition point (Figure 3),
it may be concluded that one or
more changes in the environment increased the value of the
exponent "b" in the technological progress equation, while
at the same time decreasing the rate of growth of production.
fUe
It IS interesting to note comparison between the progress
function for lamps as shown in Figure
forecasting method shown in Figure 4.
a
2
with the more usual
This data substantiates
possible claim that forecasting results with an arithmetic
time axis are as good as they have been because of the substi-
tution effect.
That is,
an arithmetic time axis may be sub-
stituted for a logarithmic quantity axis, when production
quantities increase in reasonably constant percentages with
time,
4.3
as in Figure 3.
COMPUTER PROGRAMMING DA TA:
An additional example has been drawn from a specific
experience in programming.
Data for this example was made
pi^re
ELECTRIC LAMP PRODUCTION VS
3
TllIB
O
n
C^
H
O
CO
>
O
M
H
O
Q
O
in
CM
:=>
Pi
1^
o
as
T
H
—TT—
I
\
1
Figure
ELECTRIC LA]\1P LUMEN OUTPUT
TREND FORECASTING
u
o
M
o
f3
o
M
01
4
19.
available through the help of John Cleckner, a student at The
Johns Hopkins University.
The example represents the rate of
progress in the development of a single computer program
developed solely by him under special arrangement with an area
firm.
A figure of merit was determined which would best represent his ow
aims while developing the program.
This figure of merit equals the
lines of output divided by the running time.
The computer pro=
gram dealt with an estimation of bakery goods to be sold according
to the day of the week,
store, and particular item.
In an analogy to the development of products,
it was
decided to regard the last best figure of merit (Figure
5)
particular run as the figure of merit to be processed.
T|iis
for a
may
or may not be the true figure of merit for that run, but it
would be an accurate representation of the figure of merit of the
best program available at any given point in its total development.
The results illustrate several areas of interest.
The
first is the applicability of the technological progress function
to the area of programming.
There is also the observance of
different slopes corresponding to different development phases.
Finally,
since the work was done by one person,
the analysis indicates
the possible role of psychological "learning" theory in techno-
logical progress.
ADDITIONAL EXAMPLES:
Other cases studied
demonstrated similar technological
Figure
A
COMPUTER PROGRAM DEVELOPMENT
L.
>.
H
O
w
>
o
a
o
ex!
H
o
o
5
20,
progress functions, that is, linearity on log-log paper
when the technical progress parameter is plotted against production.
Those cases not discussed here are civil aircraft-
speed, military aircraft-speed, automobile -horsepower
hovercraft-figure of merit, and an office machine developmentfigure of merit.
One could get the impression, from the data just discussed,
that the trends observed were basically limited to items of
hardware, but this was not found to be the case, as evidenced
by studies of the totally different area of agricultural
efficiency.
From information pertaining to the pounds of rice
produced per acre in Japan over
a
period of twelve hundred years,
three log linear trends were noticeable (26,27).
The two major
lines were a "long term" or ancient trend line and a modern
trend line of much steeper slope.
By calculating backwards,
it was found that they intersected at a period of great external
change-- the "opening of Japan".
The third line was a moderate
decrease in slope, which prevailed during the period of World
War
II.
21,
THE TECHNOLOGICAL PROGRESS FUNCTION
:
if we assume that Initial studies are correct
Thus,
,
technological progress functions exist and are of the same
general nature as other micro-economic progress functions
It is then further implied,
denoted by
a
.
that technological progress, as
positively increasing function is logarithmically
linear with respect to a logarithmic quantity axis under con-
stant external forces.
Over periods of time during which
external forces vary, the slope of the line, that is, the rate
of progress, may undergo discrete shifts.
Such
a
function is
denoted as,
T. =
1
where
and
a(i)^,
(4)
T^ = the value of the technological parameter,
i
= the cumulative production number,
a
= a constant associated
with unit number one,
b
= the rate of progress,
a variable,
of the external environment.
which is
a
function
22
5.1
MATHEMATICS
:
A brief glance at the mathematics shows that where b is
a
constant:
log
= b log
(Tj_)
(i)
+
log
and regarding the differential of
dT-L
(5)
(5)
b di
=
T^
Equation
(a)
i
(6)
(6)
indicates that, b, being constant, the percent-
age change in a technical parameter is a linear function of
the percentage change in cumulative production.
it should be noted that:
In addition,
if
i
log
and
=
(constant)
= Kt + log
(i)
=
di
Kdt
e
(7)
,
(constant)
(8)
(9)
.
i
Equation
a
(9)
represents a percentage change in production as
linear function of time, that is, what has been referred to
as a "substitution effect".
Where
=
di
Kdt,
i
d T^
= bKdt,
which is the traditional trend forecasting relationship
5.2
RATE OF PROGRESS
(10)
(28)
;
From analysis of background information involved in the
work discussed above and current studies, the rate of pro-
grass, b, has been designated as a function of:
L/ the effective sii:e of the technical labor force;
o(,
an intelligence factor-average educational level;
n,
a
,
pre-learning factor-average experience level;
I,
the level of investraent;
I,
the rate of change of the level of investment;
d,
a r.\aturation
na,
the anticipated raue of change of raarket derriand-slope
factor-durability of item-corripatability;
of sales curve;
and
c,
tha corrjTiunica'cion or diffusion rate.
Currently, work is being done to define the exact nature of
the function just described within the inherent constraints
that b is greater than, or equal to,
that if L,c^, 6,
6.
SUMMARY
I,
7ero and never infinity and
or m go to ^ero,
then b also goes to ?ero
:
This study has been concerned with the technological pro-
gress function.
The study proposes that the technological pro-
gress function may be based on the cumulative production unit
number on
a
logarithmic scale, as opposed to the arithmetic time
series base used for most forecasts.
In this form,
the techno—-
logical progress function allows for environmental changes to
act upon the rate of progress in
a
precise manner, that is, dis-
crete changes in the slope of the function.
be noted,
However,
it should
that no less care must be taken when employing this
24,
technique in addition to or instead of existing methods,
particularly with regard to physical constraints imposed
upon the system.
7.
FUTURE WORK
-
The technological progress function is a tool to be used
with other techniques by the foresighted corporate and military
development planner, who can no longer overlook the effect of
technological change upon his proposals for future development.
Although, it is doubtful that
a
true "Newton's Law" for pre-
dicting technological advancement will ever be made, it is
believed that the development of the technological progress
function is
a
step towards more effective predictive methods.
Considerable work remains to be done in exploring the
nature and anomalies of the technological progress function
itself.
There is, of course, the need for a precise definition
of the rate of progress function.
In addition,
there are
avenues of research concerning macro-economic implications
and ways in which the function might be of maximum value to
technologically oriented firms.
25.
REFERENCES
(1)
H.
Asher, Cost-Quantity Relationships in the Airframe
Industry
(2)
.
Project Rand-291, July, 1956.
W, B. Hirschmann,
"Profit Prom the Learning Curve",
Harvard Business Review
(3)
J.
L.
,
Jan-Feb, 1964.
Kottler, "The Learning Curve- A Case History In
Its Application", The Journal of Industrial Engineers
,
AIIE, July-Aug, 1964.
(4)
R. R. Bush, F. Llosteller,
Stochastic Models of Learning
,
John V/iley and Sons, Inc., New York, 1955.
(5)
J.
Deese, Psychology of Learning
,
McGraw-Hill, New York,
New York, 1967.
(6)
C.
I.
Hovland, I. L. Javis, H. H. Kelley, Communication and
Persuasion
,
Yale University Press, New Haven, Conn.,
1953.
(7)
H. T. Melcher, Children's Motor Learning
Vision
,
Vv'ith
and V/ithout
The Johns Hopkins University Ph.D. Dissertations,
1934, p. 333.
(8)
W.
Owen, Automotive Transportation , The Brookings Institutim,
Washington, D.C., 1949.
(9)
F.
Stuart, ed.. Factors Affecting Determination of Market
Shares in the American Auromobile Industry
University Yearbook of Business, Series
New York, October, 1965.
2,
,
Hofstra
vol.
3,
26.
(10)
J.
Abramowitz, G. A. Shattuck, Jr., "The Learning Curve",
G.
IBM Report No. 31.101, 1966.
(11)
J.
Kneip, "The Maintenance Progress Function", The
G.
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ACKNOWLEDGEMENTS
The author offers special thanks to John Cleckner, Class of 1970,
The Johns Hopkins University, for his suggestions and help in preparing the data on computer programming.
The author is also indebted
to Lisa Geiser, Class of 1972, Goucher College and Nancy Smith,
Class of 1972, Goucher College for their help in developing additional
data.
In addition, Norman Turkowitz of Pratt and Whitney must be
noted for supplying the supporting information for the aircraft
engine study, for which the author is also very grateful.
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