JPRS ID: 10076 TRANSLATION ECONOMIC FORECASTING FOR THE DEVELOPMENT OF LARGE TECHNICAL SYSTEMS BY S.A. SARKISYAN, ET AL.
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27 October 1981
Translation
ECONOMIC FORECASTING FOR THE DEVELOPMEIVT
OF LARGE TECHNiCAL SYSTEAAS
By
S.A. Sarkisyan, et al.
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JPRS L/10076
27 (7ctober 1981
ECONOMIC FORECASTING FOR THE DEVELOPMENT
OF LARGE TECHNICAL SYSTEMS
~ Moscow EKONOMICHESKOYE PRQGNOZIROVANIYE RAZVITIYA BOL'SHIKH TEKHNICIiES
 KIKH SYSTEM in Russian 1977 (signed to press 10 Jun 77) pp 1318
l3ook by S.A. .�.,,Ykisyan, D.E. Starik, P.L. Akopav, E.S. Minayev and
V.I. Kaspin, written under the editorial review of Doctor of Economic
Sciences V.A. Lisichkin, Izdatel'stvo Mashinostroyeniye, 3,800 copies,
318 pages]
COlVTENTS
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
CHAPTER 1: SCIENTIFIC AND TECHNICAL PROGRESS ANID THE DEVELOPMENT OF LARGE
TECHNICAL SYSTEMS
 1.1. The Scientific and Technical Revolution and Large Technical
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2. General Principles of Research and Analysis of the BTS 8
1.3. Particular Features in the Development of Large Technical
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 16
� 1.4. The Life Cycle and Scheme of zhe Analysis Process of a BTS . 22
CHAPTER 2: FOItECASTING THE DEVELOPMENT OF LARGE TECHNICAL SYST~i4S
2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
P'unctions and Tasks of Forecasting . . . , . . , . _ . , , . 29
A Classification of Forecasting Methods . . . . . . . . . . 37
Expert Forecasting Methods . . . . . . . . . . . . . . . . . 44
Forecasting on the Basis of the Extrapolation and Interpolation of Trends . . . . . . . . . . . . . . . . . . 55
Probability and Statisttcal Methods in Forecast Research 95
Composite Forecasting Methods . . . . . . . . . . . . . . . 124
CIiAPTER 3: CRITERIA FOR ESTIMATING EFFECTIVENESS OF LARGE TECHNICAL SYSTEMS
3.1. Principles for Formulating the Effectiveness Cr;~eria of BTS 134
3.2. Economic Effectiveness Criteria and Types of Economic
~ Effects of BTS . . . . . . . . . . . . . . . . . . . . . . . 145
 a  jII  USSR  FOUO]
[III  USSR  3 FOUO]
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_
Page
1 CHAPTER 4: FORECASTING THE COST ESTIMATES OF LARGE TECHNICAL SYSTEMS
4.1.
Forming the BTS Cost Estimates . . . . . . . . . . . . . . .
167
175
4.2.
Basic Princip les and Methods for Forecasting Cost Estimstes
3
 4
c Patterns in the Formation and the Forecasting Methods
Basi
.
.
.
for the Costs of NIR and OKR of Large Technical Systems
214
4
4.
Basic Formative Patterns and Methods for Forecasting the
.

Costs of Serial Production of the BTS and L'heir F�tnctional
' '
225
4
5
Elements . � � � � ' ' ' '
Basic Formative�Patterns and Forecasting Methods for BTS
.
.
Operating Expenditures . . . . . . . . . . . . . . . . . . .
246
CHAPTER 5: METHODS QF DETERMINING THE ECONOMIC EFFECTIVENESS OF LARGE
TECHNICAL SYSTEMS
5.1.
General Characteristics of Methods . . . . . . . . . . . . .
253
5,2.
The Dynamic Method for veterndining National Economic
254
Effectiveness of the BTS . � � � � � � � � � � ' '
3.
5
Appror_imate Method for Determining Natianal Economic Effect
.
(From the Example of the ATS) . � � � � � � � � � '
2~2
277
5.4.
Dynamic Method for Determining CostAccounting Effect of BTS
5.5.
Approximate Method for Determining Cost Accounting Effect
(From the Example of an ATS) . � � � � � � � � � �
280
6
5
of
Determining the Limit Price of anAircraft in theStage
.
.
�
281
_
 5.7.
.
Its Development � � ' .
Estimating EconomicEffectiveness . of BTS Functional Elements
282
. . . . . .
289
Bibliography . .
. . . . . . . . . . . . . . . . . . . . . . . . .
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[Annotation] The book examines the basic principles and methods of economic fore
casting, as well as the specific featurec of their application to the class of large
technicai systems (BTS). Tha development patterns o� technology are shown under the
conditions of the presentday scientific and technical revolution, the reasons for
the occurrence of 1,3rge technical systems and their distinguishing features and
classification. Also taken up are the basic concepts dealing with systems, their
life cycle and the principles of technical and economic analysis. The methods are
given for forecasting the development of the BTS, t`::e criteria and methods for
assessing the ecoiiomic effectiveness of the sys.*,e^.s, the models and methods of fore
casting their values.
The book is designed for scientific workers and engineers whose sphere of profes
sional interests includes the questions of forecasting and an economic assessment
of scientific and technical progress. It can also be usef ul for instructors and
students in machine building VUZes.
17 tables, 50 illustrations, and a bibliography of 50 titles.
LIST OF STANDARD TRA,.'VSLATIONS
1. takt3.kotekhnicheskiye trebovaniyatacticaltechnical. requirements
2. ogytnokonstruktorskaya razrabatkaprototype design work
3. opytnaya sistemaprototype system
4. ekspluatatsiyaoperation
5. seriynoye proizvodstvoserial production
6. sebestoimost'production cost
7. tekushchiye zatratycurrent expenses
8. kapital'nyye vlozheniyacapital investments
9. stroiLel'nomontazhnyye rabotyconstructioninstallation work
10. nauchnoissledovatel'skiye rabotyscientific research
11. avanproyektpreliminary project, design
12. tekhnicheskoye zadaniyetechnical requiremer.t, sgecification
13. tekhnicheskiye predlozheniyatechnical proposals
14. eskiznyy prvyektdraft design
15. prorabotkastudy
16. mametirovaniyemockup construction
17. tekhnologiyatechnology, production method
18. opytnyy obrazetsprototype
19. raboc'hiye chertezhiworking drawings
c
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FOREWORD
' A most important condition for increasing the efficiency of sociai prcduction and
improving product quality, as was pointed out by tiie 25th CPSU Congress, is an ac
celerated pace of scientific and technical progress based upon comprehensive pro
grams [1]. The comprehensive programs and the longrange develapment plans compiled
on their basis for the interrelated national Economic sectors linked together by
the common end result of research, design and production activities, are an effec
tive means for ensuring the planned and timecoordinated development of science,
technology and production. The developing of the comprehensive scientific and tech
nical development programs and longterm plans involves the necessity of surmounting
a whole series of ambiguiries.
Among them one would mention the ambiguities relative to: the development goals;
the means and methods of achieving the goals; the resources ensuring development;
the total development effectiveness; the comparative eftectiveness of possible de
velopment areas under the conditions of future resource constraints. The firgt two
types of ambituities can be overcome by special methods based, as a rule, on non
formal (heuristic) and formal (extrapolati.on) forecast assessments.
The theoretical and practical aspects of forecasting can be found in a large number
of articles and monographs published in the USSR and abroad. Th,.~se studies take up
 a large range of questions related to the gnoseology and methodology of forecasting,
the restilts of the practical implementation of individual methods are given, and the
questions of organizing forecast activities are examined. However, it must be
pointe.d out that many questions in the theory and practice of forecasting still re
main debated.
At the same time the existing publications virtually do not deal with the method
ological aspects of forecasting the resources which ensure scientific and technical
development as we11 as the questions related to assessing the economic coriseque.nces
of scientific and technical progress. Active control of scientif ic and technical
progress becomes effective only under conditions where, along with assessing its
results, consideration is given to the entire spectrum uf resources essential for
carrying out one or another direction in scientific and technical development.
The problems of f orecasting and assessing economic efiectiveness from scientific
and technical progress assume particular acuteness in line with the nece4sity uf
managing the development processes of large technical systems (BTS).
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The BTS are a direct consequence of the presentday scientific and technical revolu
tion which made a start to the age of the conquering of space, the use of atomic
and nuclear energy, computers, the automation of groduction processes and so forth.
In being marked by great complexity, the BTS require signif icant resource outlays
on their creation and series production. At the same ti.me the use of the BTS for
their specific purpose creates an opportunity of obtaining an ecanomic effect in
various spheres of human activity. From this arises the problem of correlating the
resources consumed in the various stages of the life cycle of the systeins with the
effect obtained during the period of their operation. The solving of this problem
should correlate two aspects of scientific and technical development for the BTS,
namely the technical and economic, and this is essential for drawing up the long
range pians.
The book presented for the reader's consideration attempts to systematize an examin
ation of the methods of choosing alternatives for the development of the BTS from
the viewpoint of their integrated technical and economic eva:Luation. In cansider
ing that the BTS properties which determine the specific methods of their economic
forecasting are most inherent to aircraft systems, a majority af the applied ques
tions is examined in terms of aircraft systems of various classes and purposes.
The book has been written using materials from the theoretical research by the
authors and from an analysis and generalizatton of Soviet and foreign literature on
the questions raised. The leader of the au+,hor collective is Doctor of Economic
Sciences, Prof S. A. Sarkisyan. Chapter 1 was written by S. A. Sarkisyan; Points
2.1, 2.3 and 2.6 of Chapter 2 were written by Candidate of Economic Sciences, Docent
E. S. Minayev; Points 2.2 and 2.4 by Candidate of Technical Sciences, Docent V. I.
Ka.spin; Points 2.4.5 and 2.5 jointly by V. I. Kaspin and Candidate of Economic Sci
ences, Docent P. L. Akopov; Chapter 3 by S. A. Sarkisyan and Doctor of Economic Sci
ences, Prof D. E. Starik; Chapter 4 by S. A. Sarkisyan and P. L. Akopov; Chapter 5
by D. E. Starik.
The authors would like to thank Senior Science Associate Ye. V. Tabachnaya, Engrs
Yu. A. Teplov and A. S. Chernaya for the help given in p'reparing the manuscript for
publication.
The book does not claim to be an exhaustive exposition of all the aspects of the
 posed problem, and for this reason the authors would be grateful for critical com
ments and proposal.s which should be sent to the following address: Izdatel.'stvo
 Mashinostroyeniye, No 3 First Basmannyy Lane, B78, Moscow, 107885.
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CHAPTER 1:.: SCIENTIFIC AND TECHNICAL PROGRESS AND THE DEVELOPMENT OF LARGE TECHNICAL
; SYSTEMS
1.1. T''`;e Scientific and Technical Revolution and Largp Technical Systems
The deuF:lopment of technology in recent decades has shown a transition from techni
cal deGices to technical systems and this to a significant degree determines the
essenc.e of the presentday 5cientific and technical revolution.
The advances made in in3ividual scientific and technical sectors, in nuclear physics
and power, electronics and computers, aircraft and missile construction could be at
tained only by the creation of systems.
.
If ne bears in mind modern science as a whole, in it one could scarcely find a con
ceFL capable of rivaling the word "system" in terms of breadth of use. Biologists
and physicists, cyberneticians and psychologists, cosmologists and economists ana
lyze and model a system.
The same thing can be said about modern technology. Not so long ago specialists af
a corresponding specialty designed means of communications or transport and then,
depending upon the specific technical parameters of this equipment, develoPed
auxiliary facilities which would ensure thei.r successful use. The present develop
ment stage of technology is characterized by the designing not of individual pieces
of equipment but rather technical systems which incorporate all the elements essen
tfal for carrying out a certain complex function.
A modern aviation or missile complex, a production control system, a telephone net
work serving millions of subscribers or a large power system could be created only
by considering the complex interaction of the entire system of oparations and dif=
ferent types of equipment. All this equipment must be designed simultaneously, in
a strict relationship subordinate to carrying out the basic function, and an omis
sion in any of the elements can tell decisively on the entire system.
Large technical systems are the result of the action of fully automating the system
functioning processes and the development of computers. The increased scale of aL
tivities performed by equipment, the complexity of the problems solved and at the
 same time zhe necessity of a more rapid pace of decision taking have led to a situ
ation where the historically formed systems of control and data processixig have been
unable to promptly produce optimum solutions. In particular this is characteristic
for systems with great operating speeds where a delay in taking an essential deci
sion can lead to catastrophic consequences. Aircraft are a vivid example of such
 systems.
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The complexity and diversity of the problems solved and the specific conditions of
realizing them have led to a sharp rise in the number of specifications of the
flight and surrounding medium which intluenced the course of carrying out the set
task. As a result over the quarter obperformiaircraftnconu
ment.s has risen by more than has been reduced by 6 7
trol operations, due to the sharply increased flight speed,
fold. At the same time human revious speed qipment even with intense
training has remained on the p
As a consequence of the simultaneous execution of a range of involved Onboardncom
the necessity has arisen of automating the aircraft controT process.
puter equipment has appeared on the aircraf t.
The process of automating the various functions performed by aLrcryft iaSeaecrued
_ particularly intensely in aviation where the speed factaX has alwa s p yed
cial role. Here is a characteristic example from the develapment history of combat
aircraft demonstrating the need to develop complex technical systems. The combat
aircraft employed in World War II had anachine gun and canon weapons and aiming was
done visually. Due to the speed differences of a bomber and a fighter, the latter
had an opportunity to execute several combat turns attackiiigdeclinedthe
with an increase in speed the number of p
_ Analogous trends can also be 2raced in civil aviation. Thus, in an air traff ic con
_ trol system, with an increaseinhsafetynby traditionalnmethods becamefimpossible.
in the air, the ensuring of flght
For example, for solving the problems involved in figuring the optimum aircraft
routes for all the centrally trips
so sforthg itn ould be necessaryttosex
as the capacity of the routes, altj.tudes
 amine more than 10100 variati.ons and choose the optimum one.
auto
Tlie latter can be carried out oI~ymu tabeaemphasy,zedethat computmosters
 mated flight control systems
functions in such systems and namely assessing developingacceleratedsituation
taking, as before are carried out directly by p ple
transforming and processing the increasing
optimum solutions using conputers
are the particular features which put large technical systems in a special c1ass o
systems? They are:
1) The complexity of the structure and behavior of Che system, that is, the pres
ging
relationships
ence of such complex intertwined overlapping
parameters of the system whereby a chanSe in one
others; the presence of complex and overlapping ties between the elements in the
system;
2) The irregularity of effectsoleads tottheanecessity~oftdecision takinglunder
 of the very system's conduct which
conditions of ambiguity and sometimes active counteraction;
~ed~e having their
3) The presence of subsystems of an hierarchical
own particular goals from which the overall goal
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4) A high degree of automation making it possible by utilizing the computers to
create flexible control, encompass complex dyr.amic processes with an enormous num
ber of parameters and optimize the decisions being taken.
With all the diversity and complexity of the problems solved by the BTS, it is also
possible to isolate features for their systematization and classification.
1. The specific features or the degree of purposefulness and specificnQSS of the
system; the degree of clarity and certainty in the formulation of the goal; the
degree of its formalization, the range of goals (that is, the number and diversity
of goals) and the hierarchy of goals. It is possible to isolate singlegoal sys
tems, that is, systems designed to solve one single task, multigosl systems for
~ solving multiple tasks and functional systems which solve an individual aspect or
facet of a general task.
2. The degree of integrity, that is, the degree of the permanent dependence of the
component parts, elements and processes of the examined systems or stages and the
y directions of the tasks being carried out. Integrity is characterized by the number
and diversity of harmonious links, component parts and elements of the system, by
the degree of determinism in their reciprocal conduct and functioning and by other
features.
3. Complexity or the degree of objective complexity; this is determined by the
 total number nf elements and links between them, from the diversity of elements and
links, from the number of hierarchical levels, from the number of functional sub
systems and from other features. Depending upon the number of elements, the charac
~ ter ot the links and the conduct it is possible to isolate the following systems:
,I a) Simple or smallsystems with a limited number of elements (10104), the links
li and conduct of which are a determined nature;
i b) Complex or largesystems with a large number of elements (104107) with a mass
~ variable number of links; the beh.avior of such systems represents a random process
~ which moves toward 3 certain limit, and for this reason such systems are of a prob
' ability sort; characteristic for them is a high degree of automation for the control
' processes; in particular, modern aerorocket, spacemissile and other aircraft sys
tems belong in such systems;
c) The ultracomplex or selfdevelopingsystems with a number of eleznents up to
I 1030 in which successful adaptation to randomness will be carried out by the random
~ ness of the internal structure.
4. Controllab:tlitythe degree of automation of control over the functions carried
out. According to this feature it is possible to establish three basic classes of
the BTS: I. Information retrieval systems (IPS). II. Automated control systems.
III. Automated natianalscale control systems.
Automatic telephone systems would be put in the systems of class I. Close to them
in terms of the problems solved are the information retrieval systems which use an
Plectronic computer to retrieve scientific and technical information. Here the com
puter is the central element of such systems dnd provides the link between the con
sumers (subscribers) and the information sources. Approximately the same principle
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 has been used to develop and operate at present systems which lacate free hotel
rooms, se11 air tickets (the Sirena automated ticket sales and reservation system
serves 250 cities) andt lexity tofodata transmissionsand processingideclass
_ of large systems in which the. comP
pends essentially upon the number of stibscribers (users).
Table 1.1
F ~teposi'tary of Information
Subscribers Name of System Sources
Scientific workers
Visitor
Leadership, adminl.stration
of sector or ministry
Aeroflot reservation clerk
Ret.rieval of scientific
and technical information
Hotel reservation
Obtaining information on
course of carrying out
production plans,
materialtechnical svpply
and so forth
Air reservation
Libraries, repositories
of scientific and tech
nical papers, microfilm
holdings and so forth
Hotels
Enterprises under the
ministry
Aeroflot ticket service
whtch has tickets and
monthly flight schedules
A common feature of the class II systems is traffic control where a person acts as
_ an operator controlling a proi~e s diagnostician
control loop both for the ent ystem
_ systems one could put:
a) An automated air traffic, takeoff and landing control system which ensures
safety in carrying out training ofithesrun~aayncathe pacityits of a
given airport and simultaneously maximum
b) The control system for a large aircraft or space device;
' c) An automated control system for production processes (production processes in
petrochemistry, the cement industry, the extraction of inetals from ores, the rolling
of inetals and so f orth;
_ d) The control system for energy or transport systems and so forth.
 A basic feature of the class III systems is the use of classII systems have
f unctionsl purposz within a single system unif ied b} a comanon go P
ing upn the degree of detailing for the component elements and functions bothhthe
class II systems, by a class III system one can understand, for example,
entire automated air traffic control system encompassing the territory of an entire
' countr;? as well as an individual subsystem concerned solely with the questions of
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dependable communications between the airfrelds comprising the air traffic service.
The following basic features of the class III systems can be established:
a) Control of the processes is significantly more complicated than in the class II
systems; entire complexes and associations of the class I and II systems Which
_ carry out different functional tasks can serve as objects of controa.;
b) The hierarchy of the system's structure and the higher the level of hierarchy
the less contact with the specific functions performed by the lower rank systems;
c) The presence of class I infomation retrieval systems as data sources;
d) Control of the major operations using various automatic data processing and dis
plzy devices.
 Thus, the basic feature af complex systems is information procesf;es linking the in
dividual elements into a single whole for ensuring optimum control. For precisely
this reason a system is not a simple combination of its own subsystems but rather
possesses particular properties which none of its individual parts has.
Cybernetics, information theory and algorithm theory are concerned with the ques
tions of controlling the BTS. However, in the process of developing the BTS, a
multiplicity of important problems arises going beyond cybernetics and the other
.M abovementioned sciences. One of them consists in creating an.economically optimum
system in terms of its set functions. The development of science and technology
provides an opportunity to create a great diversity of technical devices or elements
of the BTS capable of carrying out qualitatively uniform :unctions. Due to the dif
�ereaces in ttl2 physi:,a1 processes :rY!icYe e*!sure the realization of a certain func
= tion, these devices possess different functional characteristics and a design or
= technological appearance, they consume different types of energy and so forth. The
= listed features determine, on the one hand, the operational efficiency of the tech
nical devices and, on the other, the cost of their creation, production and opera
tion.
c The diversity of functionally equivalent elements for the BTS gives rise to an even
greater diversity in the variations of constructing it as each of these is capable
of realizing the set behavtor. The variations of the BTS synthesized in a certain
initial range af devicas or elerents wtiich are indistinguishahle in terms of func
tional features wi11 possess their own characteristics of cost and effectiveness.
The latter gives rise to the problem of selecting a preferential alternative out of
the multiplicity of systems which realize the set behavior. The choice of alter
 natives can be made abjectively only under the condition that this is done on the
basis of analyzing the economic consequences of developing the BTS and the entire
spectrum of expenditures on their crear_ion, production and operation.
The prablem of selecting an economically optimum system is solved by the methods of
general systems theory and systems analysis, the theory of the economic effective
ness of capital investments and new technology and scientific and technical fore
casting.
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1,2, Ceneral Principles of Reaearch and Analysis of the BTS
The principles for the research and anayobslandethishcomprisestthe foundation
upon the methodology of studying complex ]ects
of socalled s}�stems theory. General SYVOn Bertalanffy atcanphilosophicaloseminar
the 1930's by the Austrian biologi3t L. ro osing
at the University of Chicago. He developed an "organismic"niturninconsisted of
to view living organisms not as aggregates of cells which, i
colloids and organic molecules but as an organized unified system.
An examination of objects representing a certain aggregate of interrelated and inter
complex
dependent elements as a unified (unifi~oach. siTheesywhole stemsf approachigained universal
function has been termed a systems app
recognition and was fruitfully employe due studyitsanddialecticalanalysis mThisia
objects and processes of their development
approach represents the extension of wztlWhich henomenaoareainterdeterminedsandlch
view nature as a single related whole incybernetic, technical and other systems
interdependent to eco 3~~SCOfbthe~materialworld.
which are component p
roach is the unifying principle which makes it
s
The methodology of the systems app
possible to extend its principles to diverse scientific areas. The method~ingi le
based upon the principles of the integrity of the studied object and the p P
the
of isomorphism. The principles of ingtaathe complexityrofctheostudied
system's structure and make it possible to rePresent
system in a broken down form. The principle of isomorphisml is used for an analy
sis of the laws which explain the inner similarity of objects and structures of dif
ferent nature and purpose.
In the sphere of technolyhas already
t1iat modern technology, as
technology as:
a) Its basic object is various types
tion control systems, coMnunications
approach has been effective due to the~~s
been pointed out, in Y.ts essence is sys
of systems (aviation, missile, space, produc
systems and so forth);
b) Tha process of creating technical systems is itself
ried out the coordinated work of numerous prototyp r resent independent sys
production and operating organizations which, in turn, P
tems;
c) The process of producing the technical systems,or their construction is carried
out in a certain system and, as a rule, this process involves numerous enterprises
which are elements of a complex production system;
~een betweensobjecttructuralselementsfrom siew
1By isomozphism one understands a unif similarity
point r~f their structure, the relation ip
well as between the objects and the external environment.
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d) The process of the functioning of technical systems is a system in which there
is an interaction not only between the elements of the technical system but also be
tween the different specialized systems.
Since the main distinguishing feature of a large system is the close link of all its
elements and parts, a systems approach to the .analysis of a BTS means a considera
tion of these relationships, a study of the individual objects as structural parts
of more complex systems and the ascertaining of the role of each of them in the
overall process of the functioning of the BTS.
Let us examine certain concepts of systems theory which are most often used in the
text bzlow. For this let us formulate again the concept of a"system" and isolate
its basic properties which distinguish a system from any other aggregate of ele
ments.
~ In general systems theory, by a system one understands an aggregate of objects or
elements which possess certain properties and are interconnected and by these inter
connections the system is unified into a single whole. The system possesses a cer
 tain structure which allows a breaking down of the hierarchy of elements. It inter
acts with an external environment and can be viewed as an element of a broader sys
_ tem that is superior to it. The structure of a system is such that its elements
possess the properties of a subsystem in relation to it. The system is designed to
perform a certain activity which can be broken up into a number of interrelated op
erations. From the def inition. of a system it f ollows that the most important con
cepts in general systems theory are the element, operation, external environment,
; structure and hierarchy.
An element of a system is what lies at the basis of the hierarchy in breaking down
the system and cannot be broken down further.
In accord with the role that the system's elements play in the process of achieving
the set result, ttze socalled system central e1ement2 is isolated among them (the
elements) and by this one understands the entity (the aggregate of interrelated
el.ements) capable of performing an elementary operation.
An operation is an aggregate of actions aimed at achieving a certain goal. In the
process of performing an operation, the central elerPnt will be linked to other
parts of the system aad the interaction with them carries out the operation. How
ever, the characteristics of the central element have a determining impact on the
functional properties of the entire system.
Any system operates in a certain environment. The environment is the aggregate of
all elements where a change in the properties of these influences the system as well
as those objects the properties of which are altered as a result of the system's
2In certain instances, the term "central subsystem" is used. For example, the air
craft is the central element (subsystem) of a system of aircraf t which would in
 clude the aircraft and the ground facilities such as the airfield, controls, com
munications and so forth.
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behavior. For this reason, both the system as a whole as well as each element in
the system have inputs which characterize the actions of the environment on the syG
tem and its elements and outputs which characterize their effects on the environ
ment.
The interaction of the system with the environment as well as that of the system's
elements with one another can be represented by structural models or f unctional
_ models.
A structural model, depending upon the aim of analyzing the system, can be of three
types: an external model in which the system is represented in a canonical form and
all its links with the environment are expressed by inputs and ouzputs; a hierarchi
cal model in which the system is broken down by levels according to the principle of
the subordination of inferior levels to superior ones; an internal model in which
the composition and relationship between the system's elements ate shown.
The f unctioning of the system can be represented by the following: by a model of
the system's life cycle characterizing the processes involved in the system's exist
ence from the genesis of the idea of its creation to its "death" (the ceasing of
functioning); by a model of the system's operation representing the aggregate of
processes involved in the system's functioning for its basic purpose.
All these models characterize the system's method of action (the method of existence
and functioning) in space and time.
~eenvironrnent
i i
system yr
yi
_ y
I
L
9e
Fig. 1.1. Canonical model of a
_ technical system with inputs (outputs):
xl(yl)information; x2(Y2)energy;
Let us examine the particular f eatures of
canonical models of systems. As is shown
in Fig. 1.1, the basic input vector compo
nents are: xlthe information input which
controls the activity of the subsystem or
is subject to processing by the system;
x2the energy input which ensures the de
velopment of the system or maintains it at
the set level of productivity; x3the
material or object input which represents
the flow of materiel to be processed by the
system (the material means for the opera
tions performed by the system); x4the
personnei input which provides the system
with personnel prepared for participating
in the functioning processes.
X3(Y3)object; x4(Y4)~ersonnel; ~e designated inputs represent organized
xg(yB)disturbances; filter. inputs and their presence is ensured by the
purposeful activities of people. In addi
tion to the organized inputs there are also unorganized ones which, as a rule, im
pede the system's activities or these might be callsd the disturbance inputs xB com
ing from the environment (interference, noise, constraints and so forth). Thus,
the input o.f a BTS is a vector
x=(X1, X2, X3, x4, X$).
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Each input will have several components so
xi =(xij), i= 1, 2, n; j = 1, 2, m;
Xij (Xijg)> g= 1, 2, p,
where ithe type of input; jnomenclature of input; gsource of input.
The result of the system's activit'Les, the output vector y, can be characterized by
analogous components:
y=(yi, y2, y39 ya, yB),
where ylthe information output characterizing the result of the system's informa
tion activity;
Y2the energy output c.haracterizing the transfer of energy from the system to
the environment and zhe loss of the system's elements (the exhaustion of
their life, the nonconformity to demands or flaws) as well as production
wsstes;
Y3the object output characterizing the result of the purposeful action of
the system (what the system produced);
Y4the personnel output characterizing the movement of personneZ;
ygthe output disturbance characterizing the system's ancillary actions on
the environment.
Obviously, as is the case for the inputs, the oiitput vector components can be rep
resented in the form _
Yi =(Yij), i= 1, 2, k; j= 1, 2, Z;
_ yij =(Yijg), g= 1, 2, s,
where itype of output; jnomenclature of output; gpurpose of output.
The characteristic inputs and eutputs of a passenger airplane as a system are shown
in Table 1.2. An analogous approach to systems analvsis using canonical models can
also be applied to production systems. The characteristic inputs and outputs of a
system in terms of the production of large technical systems are shown in Table 1.3.
From Tables 1.2 and 1.3 it follows that, regardless of the difference in the struc
ture and the functions of the designated systems, their inputs and outputs keep
fully within the given input and output classification. The latter, in particular,
means that the compared systems, regardless of their differing nature, are isomor
phic from the viewpoint of the external structure.
A stucly of a canonical model in terms of a specific system makes it possible to
disclose the relationships of the systems. The inputs and outputs here can be ex
pressed by parameters which comprise the system's functional model (the model of the
operation).
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13
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In the process of systems analysis and forecasting it is important to know now only
their link with the environment but also their structure. In considering that the
different natured systems are isomorphic in terms of their inner structure, let us
= examine the principle of constructing hierarchical models for the inner structure of
systems using the example of aircraft systema.
In the general instance an aircraft system is divided into subsystems of a certain
rank (Fig. 1.2, a). In the diagram So is a certain supersystem (for example, the
transport system) in which the aircraft system is included as a subsystem (a system
of the first rank).

The system of aircraft S2 consists of several aircraft subsystems S21, S2m and
several suppart subsystems for their functioning 52(m+1), S2.(m+r4), which, in
turn, will be the subsystems of the second rank. For example, in an air transport
 system the aircraft systems of the second rank can be the air transport systems in
the economic regions of the nation while the support systems for their functioning
are the air traff ic control system, the system for the development and overhaul of
aircraft, the material and technical supply system and so forth.
m The secondrank aircra`_t systems, in turn, can be broken down into the thirdrank ral edi
 systems which bss 1for several betid ntical
ate support su ystems
with the support systems of a higher rank.
. Fir_ally, each thirdrank system bSuchnadb eaking downrcanaleadato
systems and support systems of the
ensuring the identicalness of the inferior rank systems.
If the systems S2mi1, S2mi., are identical in terms of the composition of the
aircraft and their functions, tien the inner structure of each of these identical
systems can be represented by a single scheme (Fig. 1.2, b). An identical aircraft
system includes the aircraft (aircrafts), the takeoff system (airf ield), the con
trol system and the repair and support system. The aircraft can be divided into
expendable and reusable subsystems and so forth.
An analysis and assessment the systems for the purposes of forecasting their de
velopment are the basis for the scientific choice and disclosure of the relation
ship between the goals of the system, the means of achieving them and the resources.
The basic goal of systems analysis is the taking of a decision on the ways to im
prove the system or process. A decision describes the difference between two states
and determines the method for moving the system to a new state. The implementation
of the decision is the process of moving the system to a new state.
In t:erms of the contents of the anal.ysis problem, systems can be divided into four
_ types: the problems of optimizing the designed parameters af the sys.tem; the prob
lems of selecting a preferential alternative (the selection of a preferential sys
tem); the problems of allocating the assigned resources in the stage of making up
the complex systems (in forming a"mix") under the conditions of an ambiguous situ
ation; the problems of allocating the resources available to the systems (for
thenlastTis thesproblem of
achieving the heal~oblemseofPdeveloping specific
problems are t p
using them.
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V
V
1) ~
C
0
V
o�
c
b
a
�
E
C
~
F
U
~
V
R
OZ
It V ~ V
7
5 6
b)
   ZOnnt
C4CIlIrMAI
Q
8
1
Fig. 1.2. Tree of system's hierarchical structure for aircraft
Key: 1Aircraft systems of rank n; 2System rank; 3Support
system of rank n; 4Aircraft; SReusable subsystem;
6Disposable subsystem; 7Takeoff system (air.field);
8Control system; 9Repair and support system
The process of analyzing large technical systems includes the following areas af
research.
1. Determining the ultimate goals of the system.
2. The working out of alternative methods and means for achieving the set goals
and variations of systems from which the most preferential must be selected.
3. Ascertainiiig the required resources to implement the designated alternatives
and the constraints on them.
 4, An analyGiG of the interaction of the goals, alternatives and resources, in
cluding interrelated events such as: the selection and formation of the evaluation
criterion and the constraints whic'rt define the area of possible decisions; a compar
ison of alternatives by a criterion, including an opt3mization of the decision with
an analytical form of a criterion; defining the ambiguities and an analysis of their
influence on the calculation results; judgments complimenting the analytical anaty
sis; taking a decision on the choice of the preferential varia2ion of the system
considering additional information on possible situations, interacting systems,
 available resources and so forth. If the results are unsatisfactory, then a deci
sion is made to carry out a new cycle of analysis with a revision of the set goals
and the elaboration of new alternatives and resource constraints.
The obtained decisions are the basis for elaborating the specific, operational pro
gram and economic forecasts. The integrity of the compiled forecasts wil.l be
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~1 r, s
~
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largely determined by the objectivity of the criteria used as the basis for select
 ing the systems and the considered conEtraints, by the correctness of the formal
methods, by the depth of analysis and so forth. However, along with this of impor
tant significance will be how complete and thorough analysis is given to the pat
terns inherent to the development processes of the BTS and the relationship of thpse
processes to the overall development trends of science and technalogy in related
spheres of activity employing different systems as well as in the sectors of materi
al production.
1.3. Particular Features in the Development of Large Technical Systems
Large technical systems are developing systems. In studying the BTS it is essential
to bear in mind two aspects of systens development: genetic, that is, the study of
a system in its development, and functional or the study of the actual actions of a
system and its functioning. From the viewpoint of the methodology in economic fore
casting of interest is the genetic analysis, that is, the examination of the origin
and partieular features of a system's development.
There are two approaches to explaining the nature of the processes in scientific and
technical development: ontological and teleological. The sense of the f ormer ap
proach is that the development processes are viewed as a manifestation of a self
developing synamic process or the result of activities by a selfdeveloping system.
In other words, scientific and technical progress is viewed as a response to the
opportunities and problems confronting science and technology. Here is assumed the
presence of factors which are internally inherent to science and technology and
 cause the process of scientific and technical development.
The supporters of this view refer to the fact that the inventions which have caused
major consequences are accidental and not determi.ned by external causes, or, in any
 event, are determined by certain concealed factors which are outside the sphere of
action of the main driving forces of history (the discovery of the antib iotic prop
erties of penicillin, the discovery of radioactive decay, the invention of the laser
and so forth).
The teleological viewpoi.nt holds that scientific and technical progress is con
sidered as the result of an objective process determined by social need or a great
economic demand. The primacy of the external (social) effects on scientif ic and
tecnnical progress assumes that the rate and direction of the latter can be pre
 dicted only to the degree that scientific and technical progress itself is the con
sequence (that is, the reaction) of changing needs or demands externalYy superiin
posed on the system of research and development. In other words, if a social need
is recognized, then the technical means for satisfying it can be provided.
These two approaches are diametrically opposite viewpoints and while the former
fully excludes the possibility of controlling the development processes the la,tcer
assumes that these processes are fully controllable.
How do large technical systems develop? From the viewpoint of establishing the pat
terns in the scientific arrl technical development of the systems of greatest inter
est is an examination of the class of competing BTS, the examples of which wauld be
air.craft systems.
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The schemes for the functioning of a s}stem is represented by its canoni.cal model
which depicts the aggregate of factors characterizing the process of its function
ing through an external structure, the system's inputs and outputs. The latter are
determined by the relationships between the designated system and the e:nvironment.
Characteristic of technical systems are three groups of inputs and outpu:ts, the in
formation, energy and material. Their content depends upon the presence of a com
peting s}stem and the relations between Competing systems.
Characteristic of the relationship between competing aircraft systems are two peri
ods of life: the period of their nanconflicting competition and the conflict period.
The systems will have different inguts and outputs in accord with these periods.
 The canonical model of a competing systeyn can be most fully represented for the con
flict period. In this instance all the inputs can be divided into three groups
(Table 1.4): those deppnding upon the researcher X1, X2, X3, X4, those depending
upon the competing cystem X5, X6 and those depending upon nature (if nature does not
operate as a competing system) X7. The results of the transformation of the inputs
by the system will describe the system's outputs.
_ Let us describe in somewhat greater detail the significance of the inputs and out
puts for a certain aircraft system S.
For each such system the basic object of effect is a cerCain aggregate of goals (the
 system gflal) which will be the basic content af input X5. The results of the
system's effect on the system goal will be described by the output Y5. In turn, the
effect ef the system goal on the designated system will be described by the input
X6 and the change in the state of the Zatter as a result of this effect by the out
put Y6. .
Z'hus, the ef�iciency level of a competing system depends, on the one hand, on the
conformity of the controllable inputs X1 and X2 to Che needs of the system, and on
the other, upon the state of the inputs regulated by the competing system, X5 and
X6. For this reason the development of aircraft systems has a competitive nature.
Each of the competing sides endeavors to increase the efficiency level of its system
and thereby reduce the efficiency of the competing side's system. Under these con
ditions, even during the nonconflicting period, relative efficiency of the competing
system shows a wavelike nature. Af ter one of the sides has improved its system for
the purpose of raising its efficiency, the opposite side endeavors either to mini
mize the gain in efficiency achieved by the competitor by countermeasures or to make
its system as advanced as the competitor.
Consequently, it can be stated that the development of competing systems has a dual
nature. On the one hand, this development is a response to a change in the state of
the system goal, that is, the competing system in order to prevent a decline in the
effi.ciency level of one's system by elnploying countermeasures. At.the same time,
a change in the efficiency level of the competing system can occur and often does
occur as a result of spontan2ous discoveries and inventions (for example, the de
_ velopment of more advanced equipment, semiconductor electronics and so forth).
Thus, the motivating force in the development of a BTS is simultaneously the social
needs and the inner possibilities of scientific and technical progress which open
~ up new, previously unknown prospects for their improvement and application.
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Table 1.4
Inpu ts
Outputs
X1information,
determining program of system's work
X2anergy,
(esources ensuring development,
safekeeping, functioning and repair
of system)
X3conditions and constraints,
imposed by interacting systems
X4conditions and constraints,
imposed by national ecoriumic
interests
X5object,
or the system goal (the objects of
_ the system's effect)
X6competing,
X7conditions and constraints,
imposed on system by naturE
Y1information,
describing resul.ts of system's in
formation activities
Y2eriergy not depending upon competing
system
(loss of system's elements, con
sumption of resources as a resul.t
of system's functioning)
Y3conditions and constraints,
imposed by the rPsults of system's
functioning on interacting systems
Y4effect of system on national
economy
YSgoal or specific
(the results of the system's func
tioning for its basic purpose and
the change in the state of Lhe
system goal)
Y6energy dependent upon competing
system
(change in system's state, loss of
system's elements as a result of
counteraction and resources on re
pair of system)
Y7effect of system on nature
What has been said prFdetermines the strategy of analysis and building of systems
 whereby the choice of the optimum directions of systems development is catried out
 proceeding from the set goals of their functioning but consi3ering the means (pos
sibilities) which are provided by scientific and technical progress.
The development of new technology can have an abrupt or evoliitianary nature. From
this viewpoint scientific and technical progress consists of definite stages (mark
ers) which differ qualitatively from one another. These stages are not absolute
and their relativity consists primarily in the fact that each new stage is a dialec
tical negation which includes an aspect of succession, maturation and development
and the synthesizing of certain elements from pervious stages [28]. The transition
to a new stage is not a single act or a boundary point of development as the techni
cal and scientj.fic revolutions or their stages can be superimposed one on the other.
The rPlativity of the stages and the revolutionary periods of science and technology,
the links between them and their possible superimpositionall of this does not show
that technical and scientific progress is a continuous chain of revolutionary
changes. The pace of scientific and technical progress periodically alters. Scten
tific and technical development always includes not only the abrupt shifts and
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revolutions but also periods of evolutionary movement. Abrupt davelopment occurs
in the transition to qualitatively new physical phenomena and materials and is ex
 pressed in the appearance of new classes of systems.
The change from cycles of evolutionary technical development to abrupt shif ts in ite
 functional properties can be easily traced from the example of the change in the
speed of ineans of transport. The iunctional parameters evolve within the limits of
 a certain class of systems which are unified by common principles in the function
ing of the major subsystems [airpianes with piston engines (PD), aircraft with gas
turbine engines (GDT) and missilesJ.
In the general case, the evolutionary process of functional characteris'Lics in sys
 tems undergoes a number of sequential phases: the phase of embodiment, the initial
phase, the phase of intensive development (maturity) and the phase of obsolescence
(Fig. 1.3, curve 1).
i
limit of functional
properties
v
u
G
~
~
0
w
x
v I
~
I inttial maturity a eing time
~ phase phase p~ase
i
I Fig. 1.3. Dynamics of mo,st important BTS performance:
mlFunctional properties; 2Cost; 3Efficiency of
' system
The embodiment phase which precedes the appearance of the prototype inc].udes re
search on the physicochemical principles of the system's functioning, the methods
of creating a useful effect based on the results of the theoretical and experimental
research and the possible spheres of the new syszem's application.
 The initial phase, or as it can be called the incubation period, coincides in time
with the beginning of materializing the scientific and technical ideas. During this
stage the firsC models appear of the functional subsystems which employ new physical
and physicochemical processes which fundamentally distinguish these subsystems from
their predecessors.
During the incubatian period the basic efforts of the research and development
organizations are aimed at ensurin.g the stability of the occurring processes as well
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as studying new phenomena which appear in testing the BTS which employ such sub
systems. As a rule, the prototypes of the new BTS do not go into industrial pro
duction or operation. More advanced articles which employ the same functioning
principles are developed on their basis and undergo experimental testing. The
growth rate of the functional parameters during this period is still slight but con
tinuously increases.
The intensive develupment phase can be termed the period of the system's maturity.
The maturity period encompasses the time from the appearance of the first industrial
models to the moment when the potential provided by the nature of the occurring
processes is virtuaYly exhausted. Characteristic of this period is the highest
growth rate of the funcr_ional parameters. It is essential to note in particular
that a series of modifications and modernizations occur in this period. The func
tional parameters increase during this period by a slight amount in comparison with
the base article but here the spheres of use of the BTS are substantially widened.
Over time the increase rate of the parameters gradually declines and the moment
comes beyond which the increase rate in the parameters begins to drop continuously.
This is caused by the influence of impeding factors for the given type of equipment
(for example, the piston engine restricted the possibility of develaping supersonic
aircraft). ,
_ The last phase in the development of the BTS is the equipment obsolescence phase
. when the possib ilities of further improving the equipment are exhausted in terms of
the old fundamental_ bases and the growth rate of the f unctional possibilities de
cline sharply. During this period, as a rule, there begins the materialization of
new scientific and technical ideas aimed at broadening the theoretical limits of
functional characteristics which restrict a further rise in operating efficiency and
a broadening of the sphere of use of the BTS. The latter is accompanied by a quali
tative shift in the functional performance of the BTS subsystems and properties
 (Fig. 1.3, curve 1').
A combined examination of the change patterns in the functional parameters of tech
nical systems and their cost estimates3 makes it possible to spot the most inportant
feature in the systeAn's changed efficiency which can have a decisive impact on its
development.
Numerous research has shown that the cost estimates of systems respond regularly to
a change in their functional properties and parameters (this question will be ex
in being
amined in detail in Chapter 4). The evolution of functional properties,
accompanied, as we have seen, by a greater complexity of the systems, leads to an
intensive rise in their cost estimates (Fig. 1.3, curve 2). Here, while the g,rowth
 of the functional properties is restricted by the system's nature and as a conse
quence of which its development rate moves to zero, the system's cost rises expo
nentially.
3By cost estimates here and below we under.stand expenditures on development (re
search and development), industrial production and operation of the systems.
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sOR oFFlcini. UsF oNtX
The efficiency of a system, as a result of the interaction of the functional prop
erties and cost of the system, with the growth of the functional parameters ini
tially increases and then, upon reaching a certain maximum value, begins to dimin
ish sharply (Fig. 1.3, curve 3). Consequently, it is possible to speak about a
certain area of an economically optimum existence of the system beyond which it is
necessary to use fundamentally new systems for the same purposes. The locating of
these areas is one of the most important tasks in economic forecasting as it opens
up an opportunity to effectively control the scientif ic and technical developmant
processes of the BTS.
The development processes of a certain type of modern technical systems cannot be
viewed in isolation from other systems as well as outside of the scientific and
technical development processes in the sectors involved in their creation and pro
duction.
 Thus, in selecting the development strategies for a certain class of systems it is
essential to consider not only the direct result of this development in the form of
the greater eff iciency of the system. It is also important to take into account the
side effect which can appear as a consequence of implementing the results of the
given system's development in systems of a different class or purpose.
Scientific a::d technical progress is expressed not only in a change in the proper
ties of the r,fS and in the use of the results from this development in other areas
of human activity. The ensuring of the set functional properties of the systems
often requires the employment of fundamentally new means and methods of their crea
tion and production. This leads to a situation where in the process of scientific
and technical development the material and technical base of the sectors producing
the new systems undergoes profound changes. Fundamentally new equipment and produc
tion processes are introduced and these provide high precision in the working of the
parts and joints as well as high purity and uniformity of structures both in working
traditional and fundamentally new materials.
 In parallel with this an improvement occurs in the processes of creating systems for
the purpose of raising labor productivity, reducing labor intensiveness, shortening
the cycles and increasing thQ efficiency of control. The enterprises which produce
the BTS elements automate the processes involved in controlling conditions in the
 heattreating and plating shops, the processes of milling part contours using
hydraulic and electric tracking systems and machine tools with program control are
 evermore widely used. Machining is replaced by cold upsetting, cold extrusion,
Qlectroupsetting and rotary working. Ultrasound and photoelectronic, magnetic pow
der and capillary methods are employed for quality control of the initial materials,
castings, forged pieces, finished articles, joinCS and ussemblies with a high degree
of precision and reliability. The organizational management structures are being
improved and automated production control systems (ASUP) and automated development
control systems (ASUR) are being introduced.
Thus, the process
the technical lev,
as in the related
ment processes as
developmznt rat.es
should be applied
of developing the BTS is a multiaspect one involving a rise in
:1 in the sectors involved in their creation and production as well
sectors. All of this shows the necessity of viewing the develop
a whole and considering the relationship and intercausality of the
in the individual areas. In other words, a systems approacn
to analyzing the BTS development processes.
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1.4. The Life Cycle and Scheme of the Analysis Process of a BTS
The large technical systems exist in space and in time. The time period from the
plan to create a system until it is taken out of operation is termed the life cycle
of a system. It includes several stages, each of which consists of a number of
events and levels. The duration of a system's life cycle depends upon its purpose
and technical potential. The basic stages of a BTS life cycle (Fig. 1.4) are: sci
entific research; prototype design work; series production; operation. The begin
ning of a system's life cycle is preceded by the phase of social, economic and
scientif ictechnical forecasting. This includes the range of work on shaping the
tasks of the systems and assessing the possibilities of science and technology over
the long run.
~41 plan
, 2  ~
~ ~ ve~voeon
~
0
0 3
:
y
a
~
~Q
74
1
~
V
~
17 hAemm..,ai
7
 Fig. 1.4. A System's Life Cycle
Key: 1Scientific research; 2Prototype design work; 3Series production;
4Operation; SSystems research; 6Designing; 7Creation of proto
 type (head) systems; 8Elaboration of subsy5tems; 9Testing of sub
_ systems; 10Assembly of system; 11Testing of system; 12Modification
of system; 13Series production of systems and subsystem; 14Series
production of modified system; 15Operation of system; 16Operation of
modified systems; 17Taking out of operati.on.
Scientific research starts with ths plan of the BTS (the phase of shaping the con
cept). The genesis of the plan starts with an awareness on the part of the organi
zatlon in charge of utilizing the system for its basic purpose of a need to develop
or replace tne existing systems because of a widening or change in the nature of the
tasks or the development of a f.undamentally new system caused by the appearance of
new tasks< Thus, an awareness of the new tasks and new conditions is the starting
point of the plan.
22
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The initial prerequisites for the genesis of the plan are fundamental changes in
the nature of operations which shape the hasic principles (the doctrine) in the
sphere of the systems' �unctioning. Here the doctrine operates as the organizing
principal. In turn, the successes in developing the new systems and the appearance
oF fundamentally new types of systems def initely influence the content the doc
trine. Thus, in the process of creating the systems there is a constant interaction
between theoxy and practice, as follows: new tasksdoctrinethe plan for the
systemnew systemdoctrine.
The forming of the plan includes a series of events the basic ones being: research
carried out by the client for the purposes of analyzing the new tasks and elucidat
ing the demands to be made on the systems designed to solve them; the shaping of
the initial tacticaltechnical requirements (TTT) for the new systems in considering
the nature of the new tasks and the scientific and technical possibilities fore
casted for the immediate period (it is important that the demands reflect as fully
as possible the goals which the new system seeks to attain an3 nrovide the designers
and researchers with room for searching for rational ways to solve the new prob
lems); research conducted by scienti.fic and industrial organizations in the aim of
seeking out new scientific and technical principles and ways for solving new prob
lems; the elaboration of several variations for the initial design of the system,
that is the preliminary project (predesign project) for the purpose of formulating
the system's appearance, the basic relationships, the ways for solving the basic
technical problems and the required resources for the creation and functioning of
such a system; research on the efficiency and optimization of its parameters for the
purposes of choosing the preferred variation.
The end result of the plan stage is proposals or recommendations on solving the
problem and these would include the content of the plan in the form of the descrip
tion of the system, the volume and sources of resources required for its creation
and functioning and an estimate of the development and production times.
For choosing an optimum systen it is essential to work out not only several alter
native systems within one plan (several alternative subsystems within one system)
but also several alternative plans. The alternative plans would include fundamen
tally different systems.the cocimionness of which consists only in the commonness of
the pursued goals.
The second stage in the life cycle of a BTS is the prototype design work (OKR) which
includes the designing, manufacturing of prototypss (prototyre production) and the
testing of systems. As a rule, by the start of designing the less preferential ver
sions of the systems have been weeded out and designing is carried out with a
smaller number of variations.
The system's designing starts with an adjustment of the tactical and technical re
quirements made on the system. In working out the projects for several variations
of systems there is an alternating of the process which follows the scheme of
synthesisanalysissysthesisanalysis and the discovery of new possibilities is
not to be excluded. For this reason the client's requirements here must `ue con
sidered as a guidE> for the areas of the search although they basicully should al
ready govern the developers. It must be pointed out that designing also presupposes
the continuation of research on new problems discovered in the process of drawing
up the plan and in designing.
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Designing ends with the elaboration of the working drawings. The system's analysis
carried aut at this stage has specif ic features. In the �irst place, thp analysis
of the system in the designing is carried out on a theoretical basis wi*_hout test
ing it out in a fullscale experi.ment. The Lesting provide5 an apportunity to
check out the conformity of a whole series of calculated initial data and conclu
sions to the experi.ment. Substantiation of the conformity of the individual calcu
lated parameters to the experiment provides great certainty of the analysis' cor
rectness.
Secondly, in selecting the preferred system in the research and design stage it is
very difficult to assess the expenditure of resources on the prototype and series
production and operation of the systems with sufficient precision and reliability.
This can lead to the taking of incorrect decisions. This can be done with much
;reater accuracy and reliability from the results of the actual expenditure of re
sources on the development of the prototype system. For this reason at the given
stage a specific analysis of the system should be run in taking the decision about
the series production program.
 In the stages of the series production and operation of the systems there will be:
the production of the subsystems, the assembly and installation of the systems as a
whole, the functioning of the systems and the maintaining of them in a state of
technical working order and functional readiness as well as the repair of the sys
tems. The operation of the systems makes it possible to finally assess the theo
retical research carried out in the process of creating the system as well as to
improve the algorithm and methods of system analysis. The life cycle of a system
ends with its taking out of operation as a consequence of obsolescence. A system,
 as a rule, is modernized by replacing some of its elements and by developing others.
As can be seen from the description of the basic stages in a system's life cycle,
 the analysis and assessment of systems are carried out in all stages starting with
t:ie formation of the plan and ending with the decision to take it out of operation.
The analysis and assessment of aircraft systems in the interests of forecasting
their development in the early research and design stages are carried out under the
conditions of an ambiguity of the situation and initial data and the presence of re
source constraints. These conditions in the selection of the technology have led to
the rise of a new scientific discipline, systems analysis, as a methodolo.gy for
selecting systems under conditions of ambiguity and resource constraints. Systems
analysis, in
Systems analysis, in being based on systems theory and using the ma*hematics of op
erations research, compliments them in its logical methods of decision preparation
under the conditions of the ambiguities developed by decision theory. Systems
analysis is the basis for a scientific choice and tlie elucidation of relationships
between goals, the means for achieving them and the resources.
In comparison with operations research which provides a quantitative assessment of
the results of systems use zn a specific operation by using strict mathematical
methods, systems analysis recognizes such an assessment as insufficient for select
ing the preferred variation as a result of the presence of a number of ambiguities.
For this reason the solution to the selction problem is supplemented by other meth
ods, namely: by judgment methods based on logic and on formal experience and by
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engineering methods. In the latter the prime role is assiFned to the art of recog
_ nizing common interrelated development patterns of aystems and situations.
lhe process of systems analysis includes the fo1lowing research areas:
1) Determining the ultimate goals of the system;
2) The elabozation of alternative metho3s and means for achieving the set goals
and the variations of systems from among which the most preferential must be
 selected;
3) The elucidation of the required resources for implementing the designated alter
natives and constraints in them;
4) An analysis of the interaction of the goals, alternatives and resources includ
= ing interrelated events, such as: the choice and shaping of the evaluation criter
ion and the constraints which define the area of acceptable decisions; a comparison
of alternatives using the criterion, including the optimization of the decision with
an analytical form of the criterion; elucidation of the ambituities aiid an analysis
of their influence on the calculation results; judgments or logical analysis compli
menting the analyt;.cal analysis; the taking of a decision on selecting the prefer
red variation of the system considering the additional information on possible situ
ations, interacting systems, available resources and so forth; if the results of the
analysis are unsatisfactory, then the decision is taken to carry out a new cycle of
analysis with a revision of the set goals and the elaboration of new alternatives
and resource constraints.
The multiplicity of states in which a system is found during its life cycle also
determines the necessity of a continuous systems analysis process. As a result of
the increase in the amount of information and the degree of its reliability, one
can speak of a multistep (iterative) process of systems assessment. As a first
step one might point to preliminary analysis based on judgments and simple analyti
cal models in the course of which the required informatian on the possible goals
 and areas for searching for alternatives, on operations models and so forth will be
more fully disclosed.
The basic components in the systems analysis process are: the goal, operational
and design research, an analysis and selection of the criterion and the constraints,
 modeling of resources, the criterion function and constraints and the selection of
alternatives or the optimization af the system.
The goaloriented research consists in elaborating the alternative goals and choos
ing the preferential alternative. The selection of the tasks (goals) the fulf ill
ment of which should be ensured by the system is either the result of a systematic
analysis of the dynamics af the tasks which arise as the situations change or the
_ result of the generalizing of the experience and views existing on a superior
management level.
The process of defining the goals is subordinate to certain rules of which we would
point to the two most important. In the first place, the tasks of the interior
level systems should be compatible with the tasks of the superior level systems and,
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_ conversely, the tasks of the superiorlevel systems should be synthesized from the
tasks of the inferiorlevel ones and stem not only from the needs but also from the
possibilities of the systems. Here there should be a hierarchy o� tasks correspond
ing to the hierarchq of systems, that is, the superior Ieve1 systen shauia r.a:�e a
structure of ineans (their tasks) so that the aggregate of the inferiorlevel systems
comprising it (the aggregate of their tasks) ensurea the achieving of the system's
goals as a whole. Secondly, the attainability of the goal depends upon the expendi
ture of resources on fulfilling it. A goal can be chosen where the available re
sources do not make it possible to create the system ensuring its attainment. For
" this reason, the final determination of the goals can be given only in the process
of analysis as the setting of the task will change both depending upon the resource
constraints which also can be revised from the results of the analysis and upon the
technical possibilities disclosed in the course of the analysis (for example, the
possibility of creating a multigoal system capable of carrying out a broader range
 of tasks instead of a specialized system).
In the process of operations research, on the basis of an analysis of the probable
conditions for carrying out the operations and the system parameters in the desig
nated period, a logical description is given and the mathematical models are formu
lated for the possible variations of standard operations (methods of attaining the
goal).
Depending upon the specific purpose, the operations performed by aircraft systems
can be divided into information (inspection, communications and so forth) and trans
port operations.
Depending upon the workfront, the scope of activity and the degree of involving tech
nology and human resources, operations can be divided into volume (supervolume) and
local (sublocal).
Tl:us, in the process of operations research there is a choice of goals and the
methods of attaining thezn. This makes it possible to formulate the operational
links and constraints which are part of the model of the criterion to assess
(select) the alternative systems.
In parallel the possible technolagy is studied and elaborated for achieving the
selected goals in the form of variations of design decisions (in the instance of the
discrete positing of a problem, a comparison of a finite number of variations) or
the acceptable ranges for the change in the system's parameters (in the instance of
the continuous positing of the problem, the systematized sorting out of an infinite�
number of variations in the designated range of parameCer changes).
Project or design research, like the entire process of systems analysis, is an
iterative process. In designing on the basis of the tactical and technical re
quirements worked out by the client from the results of operations research, the
 system's structure is formed, the base subsystems are selected and modif ied, new
ones are designed and the system is synthesized and analyzed.
In the event of an unsatisfactory solution, the design process is repeated. The
iterations are carried out until a satisfactory solution has been ob,tained.
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The use of base (standardizcd) subsystems in the system being designed is of a
contradictory nature. On the one hand there is a reduction in expenditures on the
designing and series production of the system, and in addition, the development time
:,t the systam is shortened, while en the QLhary rhP I1sP of such subrystems can lower
the level of the technical advancement and operational efficiency of the system.
In the conflict period, the superiority of one of the competing systems over another
to a significant :iegree is determined by the degree of the system's degredation and
reconstruction rate. This, in turn, is largely dependent upon the mass production
of reserve means used for the reconstruetion as well as upon the labor intensiveness
of the reconstruction processes. It is not to be excluded that these considerations
may be crucial in examining the designated contradiction.
The genesis of the idea of creating new means is related, as was already pointed out,
to two sources: the rise of new tasks and the achievements in technical progress.
In this regard the genesis of new plans and alternative design variations for the
systems must be expected in organizations entrusted with the solving of new prob
lems (the client) and the organizations directly developing the technology (the
developer). Obviously only the combined activities of these organizations ensure
the elaboration of alternatives which conform to the demands of the problems being
solved and of scientific and technical progress.
As was already pointed out, in the BTS, as a rule, the central element of the sys
tem is the most revolutionary link. Within the system, as a rule, there are two
or three generations uf central elements (for example, of aircraf t) having identical
purpose but different efficiency levels. The system's average efficiency level as
a whole will depend upon the degree of heterogeneity of the systems of the competing
sides, that is, upon the proportional amount of differentgeneration central ele
ments within the systems of both sides. In this regar3 a study of the replacement
rate in the generations of central elements within the systems of a competing side
should be one of the objects of systems analysis.
The choice of an alternative for tYae next generation of central elements should be
made proceeding from the view that the incorporation of new types of central ele
ments in the system makes the system an optimum one. A system which has been opti
mized under the supposition that it will include only new elements can be nonoptimal
under the conditions where the system possesses central elements of several genera
tions.
This conclusion follows directly from the socalled Bellman optimality principle.
According to this principle, an optimum sequence of decisions possesses the property
where, regardless of the initial state and the decision taken at the first moment,
the following decisions should be optimal relative to the state arising out of the
initial decision [34].
The heterogeneity of the BTS places a number of demands on the other subsystems.
The basic demand is compatibility or the reserving of possibilities for compatibil
ity with the central elements of several generations.
Thus, in the process of project or design research, alternatives are worked out for
the design variations or the acceptable ranges for the changes in the system's
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 parameters as well as the demands incorporated in the group of design links and con
straints in a system of disciplining conditions in solving the problems of an eco
nomic assessment of systems.
In the process of carrying out the goaloriented, operational and project research
extensive use is made of forecasting methods and these make it possible to reduce
the degree of ambituity ir. the notion of the future goals, tasks and possible pa~hs
in the scientific and technical devel.opment of BTS f unctional elements.
In the process of criterion research, on the basis of analyzing the goal orienta
tion of the standard operations, the possible criteria are determined for evaluating
 the system and the mathematical model of the criterion function (the goal) and the
overall appearance of the disciplining conditions (the matrix of conditions and vec
tor of constraints) are formulated.
The carrying out of resource research entails the necessity of setting numerical
perameters for the crit?rion function, the maLrix of conditions and constraint vec
tors determining the group of economic ties. In the process of this research the
following are determined: the resources required to implement the alternative pro
grams of the "resourcessystem parameters" link as needed for an analytical descrip
tion of the criterion function and disciplining conditions as well as the con
straints imposed on the amount of the resources.
Resource analysis is carried out according to the types (material, labor, �inancial
and so forth) as well as in terms of the stages of the system's life cycle and ele
ments. In the process of this research, the necessary and sufficient degree of ag
gregating the forecast estimates is determined. In accord with the scope of the
initial information, a choice is made of the forecasting method and the composition
of the essential factors and variables dEtermining the effectiveness of the system
and cY!aracterizing the state of the stages of their life cycle is defined. The
mass of statistical information is formed, it is analqzed and processed, forecast
resource models are constructed and the accuracy and reliability of the forecast
calculations are assessed.
The choice of alternatives for the optimization of the system is made using the
selected criteria considering the formulated constraints which are determined by the
operational, design and economic links.
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CHAPTER 2: FORECASTING THE DEVELOPMENT OF LARGE TECHNICAL SYSTEMS
2.1. Functions and Tasks of Forecasting
The enormous impact of scientific and technical progress on the development level of
productive forces necessitates the presence of constant controlling actions on the
nature of scientific and technical development and the introduction of their results
into the industrial production sphere. The management of scientific and technical
development is an important element in production management.
Production management has gained the widest development in a socialist society.
Under the conditions of a socialist economy, national economic management is an ob
jective necessity. Public ownership of the means of production makes it possible to
have control on a scale of the entire national economy. Production management is a
most important function of the socialist state.
Management includes three basic elements: planning, the organization and management
itself (or control) of production. These management elements are interrelated and
interdetermined and represent a single process, a single management system. The
primary element is planning which determines the production development goals. The
organizational structures and procedures are forraulated for the established goals.
Within the set structures and procedures production is controlled under the inter
ests of attaining the goal.
In keeping with the development of the productive forces and the accelerated base of
scientific and technical progress, the role of management has grown and the manage
ment system has become more complex and advanced. Pr.oduction planning and primarily
longrange planning and forecasting have assumed particular significance.
Planning as an element of management is an lnformation process. A particular fea
ture of this process is the presence of a time shxft of the information output in
relation to the information input. In planning the information flows on the past
(retrospective information) are the input while the flows of information on the
future (prospective information) are the output (Fig. 2.1).
Along with retrospective information, in adopting plan decisions, information is
also used on the state of the planning objECt and the environment (background) at
the moment the plan is worked out; this is information on the present. In relation
to the planned period, this information is also information on the past and for
this reason it can be conditionally classified with the retrospective information
(conditionally retrospective information).
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retroapection lperspective
ess
Ien tn of 4"' 1~n~t~ of
ret~`ospecti 0 p a ng
1
F
~ ~ 1 z t 2 ~~_z
retrospaction i lann~n
~ fi3rizon I to norizon
~ 
past present future time
_ Fig. 2.1. Characteristics of information inputs and
outputs of planning process
Key: 1Object; 2Background
The amount of the time lag of the information nutput and inpuothehtimeointerval
pends upon the length(the lead time) of planning, that is, upon
in the future for which the plan is worked out. The greater the amount of the lead
the more needed the length of retrospectionl and, consequently, the greatfr the time
lag between the information input an3 output of the planning process. Depending
upon the lead time, four stages are distinguished in natlonal economic planning.
1. Operation calendar planning (with a lead time from an hour to a month).
2. Current technical and economic planning (up to 1 year).
3. Perspective and longrange planning (up to 15 years).
4. Forecasting.
The planning stages are oriented not only in time but also in the planes of the
functional and territorial articulation of the planning object. The scope of the
functional and territorial levels of the hierarchy by the planning stages (the space
of their functianing) varies.
Forecasting and longrange and perspective planning encompass the superior levels of
the functional and territorial hierarchy: from the national economy down to the
enterprise. Operationalcalendar planning encompasses the inferior levels of the
hierarchy on the planes of the functic^al and territorial articulation of the
1The ].ength of retrospection is the time interval of the object's functioning in the
past (from the retrospection horizon to the present) for which the necessary and
sufficient retrospective information is available. The retrospection horizon is
the name given to the most distant point in the past on a time scale at which
there is the necessary and sufficient information.
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planning object, from the work area to the enterprise. Current technical and eco
nomic planning holds an intermediate position between them.
The designated scheme for the scope of the hierarchy levels in terms of planning
stages is continuously being transformed. Under the impact of the extensive use of
computers in planning, the planning stages with a short lead time (operational
calendar and current technicaleconomic planning) are encompassing everhigher
 levels of the hierarchy. At the same time, under the impact of the accelerated
pace of scientific and technical development, the stages with a longer lead time
(perspective and longrange planr.ing and forecasting) are encompassing the ever
lower levels of the hierarchy and the planning horizon2 is being widened.
Planning can be divided into two stages: forecasting and planning per se including
the first three stages and termed the plan elaboratian stage. Direct links between
these stages occurat the boundary of longterm planning and forecasting. They have
a common sphere.of application in the functional and territorial planes and an iden
tical scheme of information flows.
The fundamental distinction between planning per se and forecasting is the nature of
the output information, that is, the directive nature of planning information (plan
directive) and the orientation or guideline nature of forecast information (forecast
orientation). These dif�erences are caused by the significant reduction in the
accuracy and reliability of the information produced on the future with an increase
in the depth or length of planning.
b)~
v=J(rl
time t
d)
Fig. 2.2. Dynamics of the confidenr_e interval for
assessing parameters of a planning object
_ 2The planning or forecasti.ng horizon is the most distant point in the future on the
time scalp at which the state of the planning object is assessed.
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Fig. 2.2 gives a diagram for the change over time in the confidence interval for
assessing the state of a planning object for one parameter x. If the state �ivene.
..i1t,P curves for x(T) g
object is described by the set oz parameters 1~~ � thAn r
(n + 1) order, where nthe number
in the graphs are transformed inte of athe object.
of parameters describing the stat
In constructing the graphs (Fig. 2.2, a) it is conditionally assumed that the retro
 spective information on the object is~u~he reliable. this ina
mistakes arising out of interference i
stance are not considered. .
However, here it must be pointed out that the value of older retrospective informa
J tion is reduced, its predictive force is lowered, that is, there is a discounting
of retrospective information. older is, the fewer
the germs of the future and the greater
and
The perspective information worked oulf the confidence i terval.nat~unrincrease
has a cer t a i n r e l i a b i l i t y w i t h i n t h e limits o
in the length of planning leads, with a constant confidence pro ba b i l i t y ( F i g. 2� 2, the b) p= cons t., t o a w i d e n i n g o f t h e confidence interval ofWit h estimate
t a ti v a u Fig.
o f
2.2 a, it has the form bounded by the diverging curves).
the confidence interval (the confidence interval is bounded by the equidistant
curves, see Fig. 2.2, c), the confad accuracy reducedproduced(see
2,2, d). Thus, the reliability an that
planning is substantially reduced with an increase in the length of planning,
is, there is a discounting of the perspecti.ve information.
A directive nature cannot be ascribed to perspective unreliable informdeline for
this information indicatss With probable
shorteralengthtof planningnd Forecasts even with
future planning decision sible to reduce the uncertainty
relatively small degree of reliability make it Pos lower the risk of the present
of our knowledge about the future and, consequently,
planning decisions and the harm from their nonoptimality which can arise beyond the
_ planned period.
As we see, the time factor is primary in delimiting the concepts of forecasting and
planning per se (the elaboration of a plan). It determines the limits of the proy
esses of planning per se and forecasting. The length of forecasting theoret~J.r_�11
is no t limited. In practical terms
reliabilityso�ltheoestimates forithe
proceeding from the necessary
state of the planning obJ'ect in the future. Thus, in terms of the lead time, ore
casting holds the superior Ievel in the hierarchy and then comes the elaboration o
plans.
Let us give the basic concepts of theory:
prognostics [18]. A forecast is a p
ccrtain object (process or phenomenon) at a certain moment of time in the future
and (or) the alternative ways of them. its
formulating the development foreca
development trends. Prognostics is the science studying the patterns of the fore
casting process.
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In certain sources concepts are encountered which replace the concept of forecast:
ing: prediction and foresighC. Prediction is a reliable judgment based on a logi
cal sequence concerning the state of a certain object (process or phenomenon) in
the future. Prediction is the advance reflection of reality based on a knowledge of
the development laws of an object (process or phPnomenon).
Prediction and forecasting differ in terms of the reliability of the future assess
ments and foresight is a broader generic concept which includes both of the previ
 ous ones. Thus, the logical formulas for the different types of processes of pro
ducing information on the f uture (foresight) can be written thus: forecasting
"it probably will be," prediction"it will be" and planning"it should be."
The concept of futurology as a_.science dealing with the future has become wide
spread in foreign terminology. Being in a certain sense the equivalent of the term
"prognostics," the concept "futurology" significantly and unjustifiably broadens
the subject of the science, making it allencumpassing and including all aspects of
the problem of the future.
_ There are also other viewpoints on the question of the relationship between the
concepts of forecasting and planning. At times an opposition is ascribed between
forecasting as the foreseeing of spontaneous uncontrollable socioeconomic processes
characteristic of capitalism and planning as the defiecing of development trends in
the future for controllable processes in society and the national economy under
socialism. Such an approach to the forecasting of national economic development is
invalid, as forecasting and planning (the stage of plan elaboration) have the same
informational and socioeconomic nature.
In other instances the nature of the output information is considered to be the pri
mary factor in delimiting the concepts of planning and forecasting, that is, the
directive nature of the plan and the noncompulsoriness of a forecast. This dis
tinction between plans and forecasts is secondary and is caused, as was already
pointed out, by the time factor and the related greater level of forecast ambiguity.
The supporters of this view feel that the plan and the forecast of national economic
development can be compiled for the same period. Obviously the presence of two sets
f or the same futcre perioda directive uniform indication and a noncompulsory multi
variant guidelineare merely capable of misleading production and depriving a pro
duction collective of a unity of goal.
_ There is the viewpoint that forecasting is the preplanning studies, that is, the
process preceding planning. The unity of the tasks of planning and forecasting and
the commonness of their principles and methods make it illadvised to have a funda
mental division and opposition between these concepts. There is a single produc
tion planning system as a system of producing information on the future and this
includes the forecasting stage and the plan elaboration stage.
Thus, production planning is a unified system for generating information on the
future and this system does not have a formal limitation in time and consists of
two stagesforPCasting and plan elaboration. With the specific features of these
planning stages, they are united primarily by the commonness of the goals (produc
ing information on thE f uture) and the tasks.
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For gaining information on the future it is essential to study the national eco
nomic laws and to determine the causes and dtiving forces of this development.
This i.s the basic task of planning and forecasting. Social requirementss techni
cal possibilities and economic advisability are the basic driving forces in the
development of production. In accord with this it is possible to point to three
basic tasks for planning and forecasting: the setiing of the national econamic de
velopment goals, the seeking out of the optimum ways and means for achieving them
 and determining the required resources for attaining the set goals. ,
The choice of goals is the result of analyzing the sociopolitical tasks which must
be carried out in a society and which reflect the objective action of the economic
laws of socialism.
The selection of goals is preceded by the elaboration of alternative goals, by the
constructing of an hierarchical system or "tree of goals," by the ranking of the
goals and the choice of the leading links. The initial prerequisites for goal se
lection a're, on the one hand, the real possibility of solving the given alterna
tive 'and, on the other, its optimality in terms of the efficiency criterion.
The next task of planning is to study the possible ways and means of attaining the
set goals. The ways and means of attaining the goals are determining on the basis
of analyzing the development c+f the aational economy and scientifictechnical prog
ress. Here in the forecasting process there is a restricting of the area of alter
, native ways and neans for achieving the set goals, that is, the area of optimum de
cisions is defined. The sole alternative criterion which is optimal in terms of
the accepted vector is determined in the process of working out the plan.
 It must be pointed out that, depending upon what task is carried out first, two
types of forecasting are recognized: research (or exploratory) and narmative.
The research or exploratory forecasting is the name given to the drawing up of
forecasts for objectively existing development trends on the basis of an analysis
of historical trends. This type of forecasting is based on the use of the princi
ple of development inertia where the forecast is oriented in time "from the present
to the future." A research forecast is the picture of the forecast object's state
at a certain moment in the future as obtained as a result of examining the develop
ment process as movement by inertia from the present to the forecast horizon.
The forecasting of the development trends of the forecast ob,ject where these trends
should attain certain sociopolitical and economic goals at a set moment in the fu
ture is termed normative. In this instance the time orientation of the forecast is
"from the future to the present."
The discrepancy of the normative and research estimates of a forecast object at a
given moment of future time is the consequence of the "needpossibility" contra
diction. A composite forecast is based on the research and normative forecasts.
The choice of the goals and means for attaining them without fail should be com
bined with setting the resource requirements. In setting the resources it is essen
tial to view the planning and forecast resource matrices (financial, labor, material
and enPrgy) as well as the production capacity and time resource matrices. Also
assessed are the required resources and the probable constraints on their amount
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within the range of the lead time of the plan or forecast. The forecast's resource
matrices are the most important initial data in drawing up the national economic
balances in longrange planning.
 The driving forces of development do not operate in isolation, they are interrelated
" and interdetermined and can be shown in the form of a connected triangular graph
(Fig. 2.3). The vertexes of this "causal triangle" identify the driving forces of
production development and its edges are the twoway ties hetween them. For this
reason the tasks of planning and forecasting cannat be viewed separately. In the
process of forecasting and plan elaboration without fail there is an analysis of
the interaction of the goals, the methods and technical means of attaining them and
the required resources for realizing them and using the accepted efficiency criteria
the optimum national economic developmznt paths are determined.
Social needs
Economic Technical
advisability possibilities
Fig. 2.3. Triangular graph of driving forces
_ of development
Regardless of the commonness of the tasks, their positing in forecasting and plan
ning differ. In planning there is the following scheme: goaldirective, the ways
and means of achieving them are determined while resources ar2 limited. In fore
_ casting the scheme is different: the goals are theoretically attainable, the ways
and means of attaining them are possible while the resources are probable.
' As is seen, the plan will contain only one (optimum) solution while the forecast
will have a range of alternatives. This particular feature is also a consequence
of the time factor as the large ttme lead causes a high degree of ambiguity in the
, information on the future and, consequently, a widening of the confidence interval
of the f orecast estimates (the probability nature of the estimates). The tasks of
planning and forecasting also differ in terms of the breadth of coverage. While
the tasks of forecasting are global ones, the tasks of the other stages of planning
_ are tasks of a lower rank. Thus, the global goal of forecasting national economic
development in the USSRthe creation of the material and technical base of commu
nismis transformed in the Tenth FiveYear Plan as a more coucrete goal of a lower
rank. "The main task of the TenLh FiveYear Plan," as is pointed out in the Basic
Birections for USSR National Economic Development in 19761980, "is to consistently
carry out the communist party's course of raising the material and cultural standard
of living of the people on the basis of the dynamic and proportional development of
social production, a rise in its efficiency, the acceleration of scientific and
technical nrogress, the growth of labor productivity and the gre3test possible im
provement in the quality of work in all the national economic units" [1]. The goals
of the current plans are defined in accord with the main task of the fiveyear plan.
The aim of each inferior planning level is to ensure the achieving of the goal by
the superior level, that is, a compatibility of goals among the different planning
 levels should be achieved in planning.
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National economic planning is carried out on a basis of the conscious use of the
law of planned, proportional national economic development and is correlated with
the basic economic law of socialism. Marxi.stLeninist economic science is the
scientific basis of planning theory.
In our nation national economic planning has been carried out from the �irst years
of the founding of the Soviet state. In 1920, under the direct leadership of V. I.
Lenin the GOELRO [State Commission for the Electrification of Russia] Plan was
worked out. This first forecast plan for the socialist reorganization of the Soviet
republic's national economy through largescale machine industry and eZectrif ication
was designed for 15 years.
In the following ,years a number of other forecasts was worked out. in lyLU, under
the leadership of the Soviet scientist G. S. Strumilin, a demographic forecast was
worked out and this was a forecast for the size of our nation's population for
19201941. Prior to the Great Patriotic War, under the leadership of T. S. Khacha
turov, a forecast was drawn up for the development of transportation for 1015
years. In 1945,1946, the USSR Gosplan drew up a national economic development
forecast, in 1948, a plan for the transf.ormation of nature, in 19591960 a general
perspective of national economic development for a 15year period (up to 1975) and
then f or 20 years, up to 1980. In 19671969, a plan was elaborated for the develop
ment and location of the productive forces up to 1980.
The longrange fiveyear plans compiled considering these forecasts played an im
portant role in the development of the socialist national economy. At present the
initial projections of national economic developnent are being worked out for a 15
year period (19761990) and the forecasts up to the year 2000.
Starting in the 1950's, in a number of the capitalist nations, and primarily the
United States, a great deal of attention has been devoted to forecasting and its
Gcience and attempts have been made to compile development plans (programs).
However, the political and economic structure of a capitalist society which is de
termined by the private ownership of the means of production and by capitalist pro
 duction relationships creates an objective impossibility of effective management,
plannitg and forecasting of production development. Characteristic of a capitalist
economy are longterm studies only for individual, relatively stable sectors (chief
ly for the military sectors) while unsuccessful attempts have been made to unify
their results in the socalled macreeconomic structure.
The deviating of the forecasts and plans from actual reality in the capitalist
world is caused by the discrepancies between the particular patterns characteris
tic of the individual industrial complexes and the social conditions of their mani
festation.
The financial crisis which has engul.fed many capitalist nations in recent years has
again convincingly demonstrated the impossibility of efficient economic management
in the capitalist nations, including its highest form, state monopolistic capital
ism.
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2.2. A Classification of Forecasting Methods
In the modern Iiterature on forecasting a good deal of attention is devoted to the
questions of classifying the forecasting methods. At present one could count more
than a score classification systems of vari,ous authors. However, up to the prescnC
there has been no unified classification of forecasting methods which is useful,
sufficiently complete and open (in the sense of a possibility of broadening).
Probably, prognostics as a science has not yet reached a development level where it
would be possible to create a unified classification, and for this reason it is not
the aim of the given section to conpensate for this shortcoming as a whole. Here an
attempt has been made to formulate the goals of a classification; rn PX?inina rt,o
possible ways of attaining them and to review certain examples from the past.
What are the aims in classifying forecasting methods? Two basic aims could be men
tioned. In the first place, there is classification for the purpose of studying
and analyzing the methods and, secondly, classification for the purpose of selecting
a method in working out the forecasts of the object.
As is known, there are two basic types of classificatior: successive and parallel.
A successive classification presupposes the separating out of particular groups
from more general ones. This is a process which is identical to the dividing of a
_ generic concept into specific concepts. Here the following basic rules should be
, observed; the basis of the division (the feature) should remain the same in the
formation of any specific concept; the groups of the specif ic concepts should ex
; clude one another (the demand of the absence of overlapping classes); the groups
of specific concepts should exhaust the group of the generic concept (the demand of
the full coverage of all objects of classif ication).
The parallel type of classification presupposes a complex basis of classification
consisting not of one but rather of a whole series of features. The basic principle
of such a classif.ication is the independence of the selected features each of which
is essential, all of them togQther are simultaneously inherent to the subject and
only their aggregate provides an exhaustive idea of each class.
, A successive classification can be given a visual interpretation in the form of a
certain geneological tree and for this reason makes it possible to encompass the
entire area of classification as a whole and determine the place and relationships
of each class in the general system. It is more accessible for the purposes of
study and makes it possible procedurally to represent the classified area of know
 ledge in a more orderly manner. In the parallel system of classification, each class can be interpreted as a certain
area in the ndimensional space of classification features. This interpretation,
naturally, is less visual and procedurally is not as convenient for preserLting and
studying the classes. However, the classification possibilities o� such a system
 are greater than the successive approach, since the complex specific features make
it possible to provide a more detailed classification with no overlapping of the
classes. For this reason, in practice, for example, in the process of selecting a
class of inethods for an object characterized by the given set of parameters, this
_ classification is more effective than the successive one.
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Examples of both types of classifications are known among the classifications of
forecasting methods. Certain authors prefer to .give a classification of forecasts
and others a classification of the forecast objects. Let us give certain examples
of classifications and briefly describe them.
The classification of G.*M. Dobrov [13] gives extrapolation, expert estimates and
modeling as the basic classes for forecasting methods. Then follows a level of 8
types an3 below that 19 generalized names of inethods. .
In examining this classification one can note a violation of the principles of an
ideal classification in it. On each level there is not a unified classification
feature and the demand of an absence of overlapping types is not observed (the types
of the pnodeling class can be partially put among the expert methods whiie types of
the extrapolation class are partially among the mathematical models). On the in
ferior level such narrow specific methods as an interview are given simultaneously
along with the almost allencompassing groups (incidentally also overlapping) such
as mathematical economics models and probability statistics models.
4 The classification of E. Ianch [50] on the upper level cannot be reduced to a single
feature: 1) intuitive methods, 2) methods of exploratory forecasting, 3) methods of
normative forecasting, 4) methods with feedback. As we can see, the second and third ~
classes are determined by the aim of forecasting while the first and fourth are de
termined by its appara*.us. The overlapping of the types is also apparent. Thus,
~ the Delphi method from the first class uses the feedback principle with an expert,
that is= it could be put in the fourth class, the writing of a senario (the second
 class) usually preceeds the constructing of the tree of goals, that is, it is in
 cluded in a normative forecast (third class) and so forth.
V. A. Lisichkin [49] gives a system of features for forecast classification. This
system is a parallel one for 18 features, one of which is a method used for the
forecasting.
Thus, this is not a pure classification of inethods but rather a mixed classification
in terms of the types of objects, goals, the tasks of the forecast and the methods
of carrying it out.
Here are the classification features: the nature of the forecast object; the scale
of the forecast object; the number of forecasted objects; the nature of the link of
the forecasted object with other objects; the nature of the change process in the
object; the lead time of the forecasted event; the degree of localizing the forecast
on the scale of probable situations; the method used for forecasting; the number of
methods used for forecasting; the nature of the process of compiling the forecast;
the relationship of the predictor to the forecast object; the system of knowledge
underlying the forecast; the form of expressing the forecast's results; the goal of
the forecast; the purpose of the forecast; the degree of understanding and soundness
of the forecast; the method of testing the reliability of the forecast; the area of
science the object of which is being forecast.
The given mixed system of classification features provides an opportunity to put
each of the forecasts in a relatively narrow class and describe it from various as
pects, however the utility of such a classification is not very clear from the view
point of the two possible aims formulated above.
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In the work [23], this system of features has been developed and presented in the
form of an hierarchical structure. In this form it is more complete and finished.
Here four basic aspects of classification aze examined; the process of makiiig the
forecasts, the object of forecasts, the predictor and the forecasting method.
In the classif ication by the aspect of the forecasting method, the forecasts are
divideci into two groups: those based on unsystematized kn4wledge (everyday) and
those employing a system of scientific knowledge (scientific). The latter are
divided into hypothetical, theoretical and empirical. Each of these classes is
divided in terms of the employed type of inethods into general scientific, inter
scientific (the method uses the apparatus of a specific science) and special scien
 tific (methods used only in a narrow area of science). The classification is then
made in terms of the number of inethods employed in making the forecast. If there
is one then it is a simplex forecast, if there are two methods it is a duplex, and
if three or more then it is a compound forecast. Then follows the feature of the
time lead by which forecasts are classified into longterm, mediumterm and short
" term. Finally, there is the feature of forecast accuracy by various scales: by
the scale of probabilities, by the scale of parameters and by the semantic scale.
Without going into a detailed analysis of this classification, we would point out
that the presenting of it in the form of a polyhierarchical tree has advantages
from the viewpoint of the procedure for expounding the entire range of forecasting
problems. At the same time, in essence, it remains a paralleltype classification,
however it does not solve and does not ease the problems of the choiee of forecast
ing methods.
An attempt to approach the problem of selecting the forecasting method on the basis
of a classification of information data on the forecast object has been undertaken
in [48]. In accord with the data defined by the classifier, the initial information
is assigned a certain eightdigit code which is then compared with a table of the
known f orecasting methods. Appropriate methods are selected in the process of this
comparison. The data classification features by categories are given below.
Quantitative Data
RandomNonrandom
SingularMass
DiscreteContinuous
PeriodicNonperiodj.c
StationaryNonstationary
ReliableUnreliable
RepresentativeNonrepresentative
Qualitative Data
SinglefactorMultifactor
HomogeneousHeterogeneous
ScalableUnscalable
CyclicalTrajectory
StationaryNonstationary
ReliableUnreliable
RepresentativeNonrepresentative
Each place in the code can contain a 0, 1 or 2 and the 2 is used in the instance
that the given place is indifferent to the forecasting object or me.thod. The table
of inethods gives 38 names and their corresponding information codes.
It should be pointed out that the very idea of selecting a method according to the
information possessed about the object is extremely enticing, however the realiza
tion of this idea cannot be considered satisfactory. In the first place, the given
classifier has uot been sufficiEa~ly worked out as the concepts of reliability,
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representativeness, cyclicalness, trajectoriness and statianariness for the quanti
tative data are unclear. Secondly, in order to identify the method it is not
enough to consider just the initial information but rather it is also essential to
poL3ess information on the nature of the object, the goal and the tasks of the fore
cast, as well as the demands which are made upon the forecast's quality.
Even in the instance that all the listed data are present, the process of selecting
the method remains a creative, unformalized process which should not and cannot be
_ replaced by the mechanical substituting of numbers at least at the present develop
ment level of prognostics.
From the examination of the abovegiven classifications of forecasting methods it is
apparent that at present this problem cannot Le considered satisfactorily resolved.
In our view, at present it is impossible to present a unif ie3 c].assif ication which
would satisfy both aims f ormulated at the start of the given section. For this
reason we propose two classifications: the f irst of the successive type for the pur
poses of visual presentation and an analysis of the methods and a second of the
parallel type for the purposes of facilitating the choice of the method for the spe
cific forecasting object.
Fig. 2.4 shows the first of the designated classifications [35]. On the first level,
the classification feature (according to the inf ormation basis) divides the methods
 into two classes: factographic and expert. The factographic methods are forecasting
methods which use as the information base real facts that occurred in the past.
These facts can be recorded on any infortaation carrier and have both a quantitative
and qualitative nature. In opposition to the factographic methods, the expert meth
ods are based upon the processing of opinions and judgments by spet.ialists or ex
_ perts in one or another area of knowledge and these are obtained in the process of
various specialized procedures for their collection.
The classification feature of the second level has been formulated as the method of
employing the information about the object. In the class of factographic methods,
three types have been established for this feature.
The first type is the aggrebate of extrapolation and interpolation methods. Charac
teristic for this type of inethods is the use of initial information for constructing
fitting functioeS and Tthe obtainedevaluehofstheefunctionaforlthzisoughttforecastnishe
found dependenc
set.
The second type of factographic methods ~.s based upon a study of the relationships
between two or more variables in the forecast object with the subsequent determining
of the future values of some variables using the values of others which are known or
have been determined by other methods. lhe apparatus of multidimensional mathemati
cal statistics underlies such methods. In this type of inethods a specific place is
held by the socalled lead methods which are based on a study of relationships be
tween the scientifictechnical information and the scientifictechnical progress.
As a rule, for all the methods of this type, ne t or anothertstatisticalamodelhithebe
tween tle variablPS and the constructing of o
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ForecaSting methods.
Factographic
Expert
Extrapolation & II Statistical II Analogies I I Direct estimates
interpolation
w
0
N
~
~
N
0
N
N
~
O
~
vi
N
Gl
c
0
4!
.u
'i7
cd
O
eI
k~
~
J.l
fO
w
p
O
c0 C!
~
U
N
9
41
c~d
U
p
~
Ob
~
~
V
4)
a.
i
c23
4
i
'r
~
_rl
�rl
~
�rl
1.~
c0
>
9
~
co
R f O
O
44
0
O
~
~
~
N
rl
N
rl _e~l
'rl
s~
0
~
,
~
.d
0
O
~J
o
a
a
o
o cn
P. tn
m
m
o
p
o
cu
~
1~,+
~
r+
ip
.
~
~k
cd
v
Cd
~
N
o
U
P
v
y
~
W
W
W H
a
a
+
f=
x
~
H
l
,
,
�
.
�
_
~
~
J
J
Fig. 2.4. Classification of forecasting methods
With feedback
~
d
W
~
~
~
d
00
W
54
q
14 $4 ~
. . �
actual forecast is obtained by the extrapolation or interpolation of the dependent
variables in relation to the independent variables.
The following type of inethods is based upon a study of the future development nf
certain objects following the development patterns of their analogous objects. Here
it is possible to use both quantitative and qualitative information. :ioreover,
within this type one can isolate the research on analogies between objects of the
same nature but having a certain historical brpak in the development level and anal
ogies between objects which differ in .*.heir nature. In the first instance, for ex
ample, this could be two countries standing at different levels of social and tech
nical development, and in the second, analogies known from foreign sources of lit
erature between the development of a biological system and the spread of a new pro
duction method and so forth. It is essential to point out that in the practice of
scientif ictechnical and economic forecasting, particularly in our nation, this
type of inethod is applied in practice comFaratively rarely.
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In terms of the method of using the information obtained from expert specialists,
there are two types of methods: direct estimates and those with feedback. Here
the difference is that in the �irst instance the expert information obtained is
processed and given directly as a result. In the second type of inethods, the result
is obtained in a process, as a rule, of several approximations and at each step the
results of processing the previous one influenced the experts, that is, there is
feedback with the experts.
The inferior level of our classification is comprised of comparatively narrow groups
of inethods which are close in their essence.
It is essential to point out that the presented classification encompasses only sim
ple (singular) methods3 and does not include the composite ones. As a rule,
composite methods of varying complexity are employed in practice. For example,
the PATTERN method [25] includes as component elements the following: the writing
of a scenario, morphological analysis, t;ze constructing oi a Lree of goals and col
lective expert estimates. The matrix forecasting method includes the construcLion
of a model graph of the object and collective expert estimates. The patent methods
ordinarily employ the construction of a certain classification tree for assessing
the importance of patents, the methods of statistical analysis for examining the sta
tistics of patenting and its internal and external relationships, the methods of ex
trapolation and analogies of patenting dynamics and introduction dynamics.
Obviously with the development of the apparatus of prognostics, the number of singu
lar methods will grow and at the same time the complexity of the comriehensive
methods will rise with an increase in the demands made upon the quality of the fore
casts. In this regard, as was pointed out above, at the present Zevel in the de
velopment of prognostics it is hard to propose a classification of the paralleltype
forecasting methods which would make it possible to select uniformly a method for
forecasting a certain range of parameters in an object. It is only possible to list
g:oups of preferential methods which are employed under various conditions and for
different objects.
In selecting a forecasting method, it is essential to consider the following basic
~ factors: the type of forecast to be worked out; the volume and type of initial in
formation about the object; the ratio of the base time To and the Iead time 'ry of the
forecast.
In terms of the first factor, it is possible to isolate exploratory and normative
forecasts. Let us designate the first by the digit 0 and the latter by the digit 1.
For the second factor, the information oae, it is possible to isolate four forecast
subgroups: 0there is a very limited amount of information about the object; 1
there is qualitative information; 2there is statistical information about the ob
ject or estimates of the basic probability characteristic; 3there is determined
quantitative information.
3By singular methods one understands methods which are not broken down in a certain
sense into other, simpler ones based, as a rule, on a certain type of information
and employing a specific area of mathematical procedures.
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, a\ MH~ ,y01u~
a~A~r�roptrayu~' o6o6~CKm~
pd e6etnm!
0
'
f
' /
(
h ~
n\
/
tiqwa
~
o
Fig. 2.5. Space of forecast classification
<
~
T
features
b~ena ~ o r
v T nfo~u.~t..r.  3bo~~t Qbj_Cr', bTvna
,.ey: a=m~~..~.,, r
orNOaa ~
�
r9
of fore.cast; cExploratory; dNorm
NM~I Maqrra .
a),ao~eenmc
ative; eDetermined quantitative
information; fStatistical informa
p.0 s
tion; gQualitative information;
e)
"
hLimited amount of information
b )
f ~ i
d
g )
h) ~'4
,O�
~ ~
(b
Q
For the third factor which cannot be examined for all the number combinations of
the first two, it is possibl to isolate the following forecast subgroups: 0with
a ratio of TO/Ty 3; 1with a ratio of T0 /Ty 3. The figure 3 has been taken as a
result of generalizing the opinions of a number of authors on the minimum acceptable
ratio of the base time to the lead time for a forecast. In their ma.jority these re
late to the extrapolation methods of exploratory forecasting.
For normative forecasting, it would be correct to formulate the gradations for the
third factor depending simply upon the lead time: shortterm, mediumterm and long
term forecasts. For maintaining the commonness of the classification, we will keep
the two gradations for the normative forecasts as well using the digit 0 for the
short and mediumterm forecasts and the digit 1 for longterm ones.
Tnus, we have a threedimensional space of features in which there are 16 areas cor
responding to the various values of their codes (Fig. 2.5). For each of these areas
it would be possible to name several methods from those listed above (see Fig. 2.4)
which would preferentially be used under conditions corresponding to the value of
its coordinates. The simple methods ordinarily are incorporated in the comprehen
sive forecasting methods for this area. It is pessible to give the following tenta
_ tive division of inethods, using the numeration of the methods shown in Fig. 2.4 for
the following areas:
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Area Area
Code Number of Preferential Methods Code Number of Preferential Methods
 000
8,
9,
10,
13
100
101
9,
9
10, 14
10, 13, 14
OOZ
010
9,
7
10,
8
13
9
10, 13
110
,
9,
10, 14, 15
O11
,
9
,
10,
,
13
111
9,
10, 12, 13, 14
020
,
3,
4,
5,
6
120
121
4,
4
5, 6
6, 10, 13
5
 021
4,
5,
6,
9, 10, 13
130
,
10
,
, 11, 12, 14, 15
030
031
1,
59
2
7,
8,
10 13
9, ,
131
~1
, 12, 15
The given parallel classification can be used as a guideline in the problems of
selecting suitable methods. The solution to this problem as a whole remains a com
plex unformalized process as was already pointed out in the given paragraph.
2.3. Expert Forecasting Methods
The area of employi.ng expert forecasting methods is the scientifictechnical objects
and problems an analysis of which either completely or partially cannot be submitted
to a mathematical formalization. These methods provide an opportunity to construct
an adequate model of future scientifictechnical development on the basis of opin
ions of persons working in science and technology (experts).
The use of expert opinions as sources of information about the forecasted object is
_ based upon the hypothesis that they possess ideas about the ways to solve particular
or global problems, apriori judgments about the importance of different decisions
and intuitive guesses about alternative and possible variations for the development
 of the studied object.
Let us examine the most widely found methods of expert evaluation which are employed
in scientific and technical forecasting.
2.3.1. Individual Expert Estimates
Individual expert methods are based on the use of the opinions of exgerts who are
specialists in the appropriate specialty, independently of one another. The most
frequently employed are two methods of making a forecast: on the basis of a conver
sation between the forecaster dtheebasis expert
gram (the itzterview methods) ; on
work by the expert on posed questions (the analytical estimate method).
The interview method is the simplest method of expert evaluation and here the spe
cialist's opinion is elucidated by an expert. The analytical estimate method, on
 the contrary, provides an opportunity for the expert to use all the information
needed by him about the forecast object. The expert draws up his ideas in the form
of a report.
P The basic advanthe designated
experts 1However~stheseimethodsaareg
maximum use of t
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little suitable for forecasting the most general strategies due to the limited know
ledge of one expert specialist about the development uf related areas of science.
2.3.2. Collective Expert Estimates
The method of collective expert estimates is based on the principles of elucidating
the col.lective opinion of axperts on the development prospects of the forecasting
object.
The use of these methods is based on the hypothesis of the expert's ability with a
sufficient degree of reliability to assess the importance and significance of a sci
entific and technical problem, the prospectiveness of developing a certain area of
research, the time for completirig one or another event, the advisability o� select
ing one of the alternative development paths for the forecast object and so forth.
" The advantage of these methods is the possibility of exchanging opinions between the
expert specialists, the orientation of the ideas toward the strategic goals and the
use of internal and ex,ternal feedback in the heuristic process. The basic shortcom
ings of these methods are the possibility of an influsnce of the authorities and the
opinion of the majority, the difficulty of a public abandoning of one's viewpoint
and so forth.
At present, expert methods based on the use of special commissions have become wide
spread and here the groups of experts at a"roundtable" discuss one or another prob
lem in the aim of coordinating their opinions and working out a uniform opinion.
This method has a drawback in the fact that the expert group in its judgments is
basically guided by the logic of compromise.
In contrast to the commission method, in the Delphi method, instead of a collective
discussion of one or another problem, there is an individual questioning of the ex
perts ordinarily in the form of a questionnaire for elucidating the relative impor
tance and dates for the occurrence of hypothetical events [50]. Then the question
naires are statistically processed, the collective opinion of the group is formed,
the arguments in favor of various judgments are generalized and all the information
is provided to the experts. The participants of the expert evaluation are requested
to review the estimates and explain the reasons for their disagreement with the col
lective judgment. This procedure is repeated several times (three or four times).
As a result, the range of estimates is narrowed. The drawback of this method is
the impossibility of eliminating the influence which the organizers of the question
naires have on the experts in drawing up the questionnaires.
As a rule, the basic questions in drawing up the forecast using an expert collective
include: the formation of a representative expert group; the preparations for and
carrying out of the expert evaluation; statistical processing of the obtained re
sults. The basic rules for solving these questions will be examined below.
Forming the representative group vf experts. In forming the expert group, the basic
questions are to determine the quantitative and qualitative membership of the group.
The selection of experts starts by determining the areas of scientific, technical,
economic and administrative questions which are involved in solving the given prob
lem; then lists of persons competent in these areas are drawn up.
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. For obtaining a qualitative forecast, a number of demands are made on tha partici
" pants in the expert evaluation, the main ones being: a high level of general erudi
tion; profound special knowledge in the arQa to be evaluated; the ability to ade
= quately depict the development trends of the studied object; the presence or a psy
chological set for the future; the presence of an academic scientific interest in the
auestion being studied with the lack of any practical selfinterest as a specialist
in this area; the presence of production and (or) research experience in the desig
nated area.
A questionnaire is used to determine to what degree the potential expert meets the
listed requirements. The method of the expert's selfevaluation of his competence
, ~ ~ rL.o ovnarr cjeCer
is frequently employed in addition to thls. in a se~~assessL.~.^_,. r
mines the degree of his knowledge on the question being studied also using a ques
tionnaire. The processing of the questionnaire data provides an opportunity to ob
tain a quantitative assessment of the potential expert's competence using the fol
lowing formula:
M~
Z Y1
= K  0,5 + (2.1)
ui p ,
~j Y/ max
I~
where Yjthe weight of the gradation given by the expert for j(j = 1, m) char
acteristics in the questionnaire, number of points;
Yj'maX the maximum wexght (the scale limit) for j characteristicss number of
points;
mthe total number of competence characteristics in the questionnaire;
_ athe weight of the group marked by the expert in the selfevaluation scale,
number of points;
pthe limit of the expert selfevaluation scale, number af points.
It is rarher diff icult to set an optimum size for an expert group. However, at
present a number of formalized approaches to this question has been worked out. One
of them is based upon setting the maximum and minimum size limits of the group.
Here they proceed from two conditions: 1) the high average competence of the expert
groups; 2) the stabilization of the average assessment of the ferecasted character
istics.
The fiist condition is used to determine the maximum size of the expert group nmaX:
~
r
cK,,,.x K
 ,
nmat
1
where cconstant;
Kmax the maximum possible competence �or the emplpyed Competence scale;
Kithe competence of expert i.
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This conditions assumes that if there is a group of experts whose competence is max
imal, then the average value of their estimates can be considered "true." Voting is
_ used to determine the constant, that is, the group is considered elected if two
thirds of those present have voted for it. Proceeding from this it is accepted that
c= 2/3. Thus, the maximum size of the expert group is set on the basis of the in
equality
n
~j Kl
1
ri
3
Kmax
(2.3)
Then the minimum size of the expert group nmin is determined. This is done by using
the condition of stabilization for the average estimate of the forecasted character
istic. This condition is formulated in the following manner: the inclusion or ex
clusion of the expert in the group has an insignificant influence on the average es
timate of the forecasted amount
BB' < E, (2.4)
Bnax
where Bthe average estimate of the forecasted amount in points as given by the ex
 pert group;
B'the average estimate given by the expert group from which one expert has
been excluded (or included therein);
Brtax the maximum possible estimate of the forecasted value in the adopted esti
mate scale;
ethe set value for the change in the average estimate in including or ex
cluding the expert.
The amount of the average estimate is most sensitive to the estimate of an expert
who possesses the greatest competence and who has set the greatest number of points
with B< RmaX/2 and minimal for B> BmaX/2 and for this reason for testing the reali
zation of the condition in (2.4) it is proposed that the given expert be excluded
from the group.
In the literature the rule is given of calculating the minimum number of experts in
a group depending upon the set (acceptable) amount of the change in the average
estimate of .
nmin = 0.5 ~e + 5J.
(2.5)
Thus, the rules of (2.3), (2.4) and (2.5) provide an opportunity to obtain estimate
values for the maximum and minimum number of experts in a group.
The final size of the expert group is formed on the basis of the sequential exclu
sion of the littlecompetent experts and here one uses the condition (KmaX  Ki) s rl,
where rtthe set amount ot the limit for tne acceptable deviation of competence for
expert i from the maximal. Simultaneously new experts can be included in the group.
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The group size is set within the limits nmaX Ne n4 nmin�
In addition to the abovedescribed procedures, in the methods of collective expert
estimates they also employ a detailed statistical analysis o� the expert conclusions
an3 as a result of this qualitative characteristics of the expert proup are deter
mined. In accord with thes2 characteristics in the process of carrying out the ex
pert evaluation the quantitative and qualitative composition of the expert group
can be adjusted. The methods of determining these characteristics are examined be
low.
Prepcxration and car.ryiny out of expert evaZ2iaf'.2072. The preparations for questioning
the exparts includes the elaboration of questioanaires which raould contain the range
of questions on the forecast object. The structural and organizational set of ques
tions in the q,uestionnaire should be logically linked to the central task of the ex
pert evaluation.
Although the f orm and content of the questions are set by the specific nature of the
forecasting object, it is possible to set general demands for them: the questions
should be formulated in generally accepted terms, their formulating should exclude
 any semantic ambiguities and all the questions should logically correspond to the
structure of the forecast object and ensure a uniform interpretatior..
By form the questions can be open and closed, direct and indirect. A question is
called open if the answer to it is not regulated. Questions are considered closed
if their formulation contains alternative answers and the expert should choose one
(or several) of them. Indirect questions are used in those instances when the aim
of the expert evaluation must be concealed. Such questions are resorted to when
 there can be no confidence that the expert, in giving information, will be totally
sincere c. frae of outside influences which would distort the objectiveness of the
answer. Let us examine the basic groups of questions used in carrying out a collec
tive expert estimate.
1. Questions presupposing answers in the form of a quantitative estimate, that is:
on the time of the occurrence of events, on the probability of the occurrence of
events or for an estimate of the relative influence of factors. It is advisable to
use an uneven scale in determining the scale of values for quantitative characteris
tics. The choice of the specific uneven scale depends upon the nature of the de
pendence of the forecast error upon the lead time.
2. Questions requiring an informative reply in a concise form: disjunctive, con
junctive or implicative.
3. Questions requiring an informative answer in a complete form: with an answer
 in the form of a list of information about the object; with the answer in the form
of a list of arguments a�firming or rejecting the thesis contained in the question.
These questions are formed in two stages. In the first stage ths experts are asked
to formulate the most promising and least elaborated problems. In the second stage
from the designated problems they choose those that are fundamentally solvable and
have direct bearing on the forecast object.
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The procedure of carrying out an expert estimate can vary, however here also it is
possible to establish three basic stages: 1) in the first stage the experts are
used to clarify the formalized model of the forecast object, to foruulate the ques
tions for the questionnaires and adjust the membership of the group; 2) in the
second stage the experts work directly on the questions in the questionnaires;
3) after the preliminary processing of the forecast results the experts are used
for consultation on the lacking information needed for the final formulating of the
forecast.
StatisticaZ processing of expert estimate resuZts. In processing quantitative data
contained in the questionnairev, statistical estimates are determined for the fore
casted characteristics and their confidence limits as well as statistical estimates
IO'C Ltle agreemeni. c)l `Liie CXYCL L vpiiiiOnS.
The average value of the forecasted amount is determined using the formula
n
B Bi/n,
 i=1
where Bi the value of the forecasted amount given by expert i;
nthe number of experts in the group.
Moreover, the variance is determined D= Ij (Bi  B)21/ n1 and the approximate
L
value of the confidence interval J_ t~ D
n1'
where tthe parameter determined from the Student tables for the set level of con
fidence probability [the number of degrees of fieedom k=(n2)]. ,
The confidence limits for the values of the forecasted amount are figLred according
to the formulas: for the upper limit Ag = B+ J; for the lower limit AH = BJ.
The coeffi.cient of variation for the estimates given by the experts is determined
from the following dependence V= a/B, where Qthe standard deviation.
In processing the results of the expert estimates for the relative importance of the
_ scientific areas, the mean value, the variance and the coefficient of variation 1re
figured for each assessed area. Moreover, a concordance coefficient is figured and
this shows the degree of agreement among the expert opinions on the importance of
each of the assessed areas and paired rank correlation coefficients which determine
the degree to which the experts agree with one another.
For this the importance estimates given by the experts are ranked. Each estimate
given by expert i is expressed by the number of the natural rank in such a manner
that the number 1 is given to the maximum estimate and the number n to the minimum.
If all the n of the estimates differ then the corresponding numbers of the natural
series are the estimate ranks of expert i. If there are the same estimates among
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those given by expert i, these estimates are given the same rank equal to the arith
metic average of the corresponding numbers of the natural series.
= 1, m; mthe
The total of the ranks Sj assigned by the experts to area j(j
number of studied areas) is detexmined by the formula
n
' Sj Rij ~
i=1
where Ri~the rank of the estimate given by expert i to area j. m
The mean value of the total of estimate ranks for all the areas equals S= I Sj/m�
j=1
The deviation of the total ranks obtained by area 3 from the mean value of the total
ranks is def ined as dj = Sj~. Then the concordance r,oefficient calculated for the
aggregate of all the areas approximated for the estimate is
12 n '
n~ (m�m)nj Ti
tt
a
The amount Ti tete is calculated with the presence of equal ranks (athe
e=1
number of equal rank groups, te the number of equal ranks in the group).
The concordance coeff icient assumes values within the limits from 0 to 1: W= 1
means the tui.l agreement of the expert opinions and W= 0full disagreement.
The concordance coeff icient indicates the degree of agreement in the entire expert
group. A low value f or this coefficient can be obtained if a commsSoinionsipns
is lacking among all experts as well as due to the presence of opposin$ OP
among expert subgroups although agreement can be high within a subgroup.
For ascertaining the degree to which the expert opinions agree with one another, a
paired rank correlation coefficient is used
m
lg~
~ (rn'  nr)  ~Z (T'1 T1.1)
where *�the difference (for the modulusRlintheamounts of the expert ranks for
 area jiet by experts i and (i+l) ; ~yj _ ~ jRi+l'� j(�
The paired rank correlation coefficient can assume the values 1 < p< 1. The value
p= 1 corresponds to a full agreement in the opinions of two experts. The value
p=1 shows that the opinion of one expert is the opposite of the other's opinion.
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For determining the significance level of the values of the coefficients W and
pi, i+l it is possible to use the Xsquare criterion. For this one calculates the
m
_ 12 2: d2
value X' _ jx1 � (the number of degrees of freedom k= m1) and from
mn (rn + I) T,
the appropriate tables the significance level of the obtained values is determined.
2.3.3. Brainstorm Methods
Among the intuitive forecasting methods, a significant place is held by brainstorm
methods. The given methods are based upon involving all the experts in an active
creative process. For using these methods in forecast studies there is an opportun
ity to obtain productive results over a short interval of time in a situation of the
creative generating of ideas with the direct contact of experts. The following
brainstorm methods can be named:
a) The direct brainstorm method the aim of which is to generate as many possible
new ideas for solving a problem situation;
b) The method of a destructured relative estimate;
c) The confidence group method the aim of which is to establish the agreement among
a small group of participants in the method;
d) The method of inducing mental and intellectual activity the aim of which is to
tind the rational choice of one or another solution to a problem situation without
the establishing of quantitative estimates;
e) The method of the controlled generation of ideas the aim of which is to disclose
promising and original ideas to resolve a problem situation;
f) The method of stimulated observation the aim of which is to find logical solu
tions to the discussed problem situation with the formulated constraints;
g) The operational creativity method the aim of which is to find the sole solution
 for the discussed problem situation.
The basic rules of brainstorming consist in the following: 1) The statements by the
parti.cipants should be terse and clear and detailed reasoning is not required; fu11
statements can reduce the pace and impede the development of the essential and fruit
ful state of emotional stress; 2) skeptical comments and criticism of previous state
ments are categorically prohibited; 3) each of the participants has the right to
speak as many times as he wishes but not sequentially; 4) the floor is given first
 to those who wish to comment on the previous statement; 5) it is not permitted to
read without interruption a list of proposals which coiyId be prepared by a partici
pant ahead of time.
Additional proposals to resolve a problem situation can occur among the participants
later and for this reason all proposals which occur after the brainstorming can be
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written down and turned over to the organizers of the session. For brainstorming it
is most productive to have a group consisting of 1015 specialists in a session oi
from 20 minutes to 1 hour.
The explanation for this established fact is probably that in the process of brain
storming a majority of ideas are voiced by association with the previous idea.
Along with encouraging such associative thinking it is essential to provide rapid
questioning of all those who desire to be heard on the previous idea. For the oc
currence of associative thinking it is essential to haxe a certain minimum "thresh
old" group size which generates an aggregate of different quality viewpoints on the
discussed problem situation. The answer to ths question of the makeup of the group
presupposes a proper selection of participants:
1) From persons of approximately the same position (degree or title) if the partic
ipants know each other (the presence of superiors intimidates subordinates);
2) From persons of different positions (degrees or titles) if the participants are
not acquainted; in this instance it is essential to make each of the participants
equal by giving him a number in cal?.ing on the participant in turn by number, since
the group could include candidates of sciences, for example, along with academicians.
The essential condition of the participants specialization in the area of a prob].em
situation is not required for all group members. Moreover, it is very desirable
that the group include specialists from other areas of knowledge who possess a high
level of general erudition and understand the sense of the problem situation. A con
structive approach to f ulfilling the formulated condition consists in the coordinat
ing of the goals of the brainstorming, the forms of informing the participant of the
~ initial information and the competence or informativeness of the participants in the
area discussed (by inf ormativeness one understands the level of special knowledge
for the participant in the discussed area).
A description of the problem situation includes: the reasons for the occurrence of
the problem situation; an analysis oF the causes and possible consequences of the
arising problem situation (it is advisable to overstate the consequences in order to
more acutely feel the need for resolving the contradictions); an anal.ysis of world
_ experience in solving a similar problem situation (if this exists); a classif ication
(systematization) of the existing ways of resolving the problem situation; the for
mulating of the problem situation in the form of a central question with an hier
archy of subquestions (a question should be sufficiently simple in its internal
 structure since the narrowing of the problem enceuragas the efficiency of brain
storming).
The leadership of the brainstorming should be assigned to forecasters who are ex
 perienced in leading scientific discussions and problem posing and who know the pro
cedural questions and methods. If the discussed problem situation is complicated
and has anarrow specialized nature, then a specialist on the discussed question
should be called in as a cochairman for directing the brainstorming. Moreover, the
group should include: methodologists who are specialists in the area of prognostics
and who have experience in holding sessions and processing the results; initiators
who are specialists in the area of the studied problem; analysts who are highly
skilled specialists in the area of the studied problem and who are capable of sum
marizing the past, assessing the present state of the object and the research trends
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on the problem; amplifiers who are specialists in the designated problem with de
veloped deductive thinking.
All the participants ir. the brainstorming should possess developed associative think
ing. The brainstorming process should be carried out under conditions which help as
much as po5sible in establishing a creative mood with a maxi.mum concentrating of at
tention among the participants on the discussed problem. Since an idea advanced at
a given moment could previously "mentally beZong" to another participant waiting for
_ the floor, the result of the brainstorming is considered to be the fruit of the col
lective labor of the entire group. The most valuable are the ideas directly tied to
previously voiced ideas or arising as a result of the uniting of two or several ideas
ideas into one.
It is desirable that a problem situation be presented in writing beforehand and here
specific questians should be raised, such as: the goal of the brainstorming; useful
ideas for solving the problem; a list of factors relating to the discussed problem
situation which anticipate new approaches to solving it; a list of different view
points on the discussed problem; a list of questions which must be answered in order
to resolve the problem more rapidly and effectively; a plan for resolving the dis
cussed situation.
The leader of a brainstorming session should organize his opening speech in such a
manner as to arouse the mental susceptibility of the participants and force the par
ticipants to feel the need of doing what the leader is asking. The process of ad
vancing new ideas occurs in the following manner. An idea voiced by one partici
pant in the discussion gives rise to a reaction which, because of the ban on criti
cism, is formulated as an accompanying idea.
As a rule, at the outset of the session it is essential to have required questioning
as of ten the lea3er must stir up the participants for 510 minutes and create an at
mosphere of a free exchange of opinions. In conducting the brainstorming the leader
follows the abovelisted rules and in addition he should:
1) Focus the participants' attention on the problem situation, setting its limits
by the specific requirements of the problem situation and the terminological strict
ness of the ideas voiced;
2) He should not declare any idea faults, he should not discuss and not interrupt
the examination of any idea; he should examine any idea regardless of its seeming
inapplicability or unfeasibility;
3) Welcome the improving or combining of ideas. He should give the floor first to
those who wish to make a statement relating to the previous statement;
4) He should support and encourage participants as this is so essential to elimin
ate their reticence;
5) He should create an unrestrained atmosphere thereby helping to increase the ac
tivity of the participants.
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The active work of the leader is presupposed only at the outset of the brainstorm
ing. After the participants have been sufficiently aroused the processaf taeFaS�
rocess the leader plyS
 moting of new ideas occurs spontaneously. Tn this p
sive role in controliing the participants according to the rules for conductiTeg~er the brainstorming session. It remembered
verageiofrtheaexamined ques
the number of statements, the
tion and the greater the probability that new ideas will occur.
The stated ideas should be taped in order not to miss a single idea and to be able
to systematize them for the following stage.
After the holding of the brainstorming the ideas raised inthe esituationsanalysis
systematized. The systematization is carried out by the problem
group in the following stages: 1) A nomenclature list is compiled of all stated
ideas; 2) each of the ideas is formulated in generally accepted terms; 3) p
� ing and complementary ideas are determinaT~aestablishedeby then whichrtheaideasncanebeorm
of comprehensive ideas; 4) the features iven
unified; 5) a list of ideas is drawn up by groups; in each group the id,eas are g
in the order or their comtr.onness: from the more general to the particular which com
plement or develop the more general ideas.
In forecasting use is also made ef the eofotwofprocessescofran ordinary brainstorm
(the D00 method) which is an int gration
and the destructuring of advanced~inteefor dpracti al feasibility in thedpro S
a specialized procedure for evalua g ideas
cess of brainsrorming, when each of the in thetdestructuring stcrit ageiissto y
;e ~.articigants in the session. TYe basic
examine each of the systematized solely the brainsto in;4g3vecargumentse
_ path to achieving it, that is, the particiPants s a counter idea can be
which reiect the discussed idea. In the destructuring proces
proposed which would contain an assertion about the impossibilityofsallonithetpos
id~;a, it would formulate the existing constraints and advance a prpo
sibility of eliminating these constraints. The structure of the
thisritdis,essen
rule, is as follows: This cannot be because.... tn carrying met tial to use...." Thus, the result of carrying out the second Sofgideastor ea~Ch ofh
od is the drawing up of a list of critical comments on a group
the ideas as well as a list of counterideas.
 The leadership of a brainstorming session in the destructuring of ideas is provided and by a leader who: discloses the contethe nt stormingdand
scribes the groups of systematized ideas; recall
concentrates the attention of the participants on the need for a thoroughecformuism
of the ideas proposed for discussion and the advancing of counterideas;
lates the most general idea of the first group and invites the members to have their
say.
rocess in the destructuring of ideas is governed by the same xules
The brainstorming p
 as in the idea generatior stage. However, the leader gives basic attention to pre
venting the voicing of arguments which substantiate the discussed idea as well as
_ encouraging proposed counterideas.
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After all those who so desired have voiced their critical comments on the discussed
idea, the leader proposes the next idea for examination. The leader determines
whether the next idea will be a particular idea o� the discussed group of ideas or
will belong to a new group of ideas. In taking this decision he is guided by the
considerations of maximum criticism for all ideas belonging to the discussed group.
 The leader can propose that the participants criticize a group of ideas all at once
but this is allowable only in the instance when the number of ideas in the group
does not exceed fiveseven and they are simple in terms of their inner structure.
The destructuring process is repeated until each of the systematized ideas from the
list has been criticized. The advanced criticisms and counterideas are tape re
corded.
The third stage of the D00 method is to assess the critical comments obtained in
the destructuring process in order to draw up a final list of practically applicable
ideas aimed at resolving the problem situation. The evaluation of the criticisms is
as important .as t'!e destructuring of the ideas since in the destructuring stage all
possible constraints impeding the practical implementation of the idea are formu
lated while in the generation stage the lcnowledge of concrete conditions under which
the ideas should be realized is voiced.
The processing and analysis of the destructuring results are carried out by the
problem situation analysis group. The group can be supplemented by those special
ists who are empowered to take decisions on carrying out the ideas (this is partic
ularly important in those instances when decisions must be taken quickly on a multi
plicity of problems all at once).
The D00 method makes it possible to find a group solution for an arising problem
situation, excluding the path of compromises. A solution in the form of a single
opinion is the result of the dispassionate and successive analysis of the problem.
However the method does not offer the ranking of ideas in significance or the find
 ing of an optimum way to achieve the set goal and f.or this reason should be comple
mented by a collective expert evaluation with the subsequent statistical processing
of the evaluation results.
2.4. Forecasting on the Basis of the Extrapolation and Interpolation of Trends
In examining the singular forecasting methods we have not set the task of fully rep
 resenting the entire list of them either in terms of composition or in terms of the
depth of examining individual methods. The following considerations have underlain
 the choice of the compo~;.i.tion of examined methods and the degree of detailed examin
ation of each of them. In the f irst place, the practical feasibility and utility of
the meChods in the tasks of forecasting BTS development, secondly, how well the
given method has already been described in the literature, and thirdly, the newness
and promise of the method in developing the theory and practice of prognostics.
Thus, let us turn to the probl.em of using the mathematical methods of extrapolation
and interpolation in forecast research as this is the rost fully elaborated and
widely used type of factographic forecasting methods.
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2.4.1. The Formal and Forecast Extrapolation
For persons who are little acquainted with forecast problems, as a rule, the ques
tion arises of why extrapolation is examined in prognostics at all as it has beer_
presented exhaustively and in the greatest detail in mathematical literature. It is
merely a question of taking the formulas, substituting the initial information 1nd
obtaining the results. kThat results can be obtained with such an approach is clear
ly demonstrated by R. Ayres [46]. For example, in formally extrapolating the growth
trends for the speeds of the various types of transport over the last two centuries,
by the year 1990 we will obtain speeds which significantly surpass the speed of
light. In an analogous manner the curve for hwnan life expectancy af ter the year
2000 rushes toward infinity. The extrapolated growth trend for the explosive power
of ineans of dPStruction cr.eated by man rises virtually without restriction even
after 1981. One could give a number of other results of formal extrapolation which
run counter to couanon sense.
R. Ayres defines such forecasts as excessively exalted. On the other hand, he gives
examples of blinded farecasts which do not permit the researcher to predict the pos
sible consequences of future events, the prospects of a trend, the influence of the
 environment and so forth. "A simple extrapolation of trends does not ptesuppose the
understanding of factors underlying any phenomenon and it is usually eno1lgh that
these (concealed) factors remain unchanged over time" [46].
Ttie problem of formal and forecast extrapolation ha.s been examined more closely by
G. Haustein [43]. He says that it is possible to have extrapolation on the basis of
the patterns inherent to a system proceeding from the e%.isting development trends.
Mathematically the optimum f itting of results to the initial data using a polynomial
to a certain degree corresponds to this.
With a different variety of extrapolation, the amounts characterizing actual data
are correlated to hypottieses about the dynaraics of the process over the long run.
The elaboration of the hypotheses is not carried out just on the basis of past de
velopment. The closeness of the theoretical and actual data is not turned into the
sole criterion for the choice of the function.
Below we will distinguish between a mathematical or formal extrapolation and fore
cast extrapolation. By the former we understand those methods of extrapolation (and
interpolation) of dynamic series whereby no use at all is made of information about
the physical or logical essence of the examined process, nonformal procedures for
selecting the functions are not employed while the results are not varified by any
hypotheses concerning future development of the object.
The most suitable example of mathematical extrapolation is the calculating af coef
ficients for the polynomial breakdown of a function for a set r.ange of values.
According to the Weierstrass theorem, a function which is stable within the inter
val of values (a, b) can be represented in it with any degree of accuracy in the
form of a polynomial. For any value of x on (a, b), there is the valid inequality
ly_f(X)I < e, where eany small positive number.
In increasing the number of terms of the polynomial in the breakdown, it is possible
to increase accur~actice othiscseeming advantageSendseupowithdanloss ofeaccurve curacyith
all pei.nts. In p
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As a rule, the economic processes and the processes af scientif ic and technical de
velopment arise out of a certain constant, steady tendency which in prognostics is
usually termed a trend, and a certain random component expressed in the fluctuation
of indicators around the trend. These random components can appear as the results
of random fluctuations in the external or internal variables of the object as well
as random measurenent results. In borrowing terms from information theory, the
abovedescribed increased accuracy of approximation leads to the emphasizing of
noise (the random components) and does not filter the signal (the trend) out of
the noise.
In contrast to a formal extrapolation, a forecast extrapolation is aimed at using
various methods to seek out the simplest type of function which provides a maximum
approximation to the trend of the process, considering its particular features and
_ constraints and conforming to the hypotheses about its future development.
The pror_edure of selecting the type of approximating function in this instance, as
a rule, includes a series of nonformal aspects such as assessing the conformity of
the function to the points of a dynamic series. The very forecast research here
consists of several stages usually of the following composition: the primary proc
essing and reprocessing of the initial series; the cho3ce of the type of extrapola
tion function; determining the parameters of the extrapolatioz function; the extrap
olation itself and assessing the accuracy of the obtained results.
Let us point out that a formal extrapolation can be incorporated as a stage of re
search in a f_orecast extrapolation. On thP other hand forecasts are frequently
worked out on the basis af just formal extrapolation. In this regard let us examine
it in more detail.
The mathematical basis for extrapolation aad interpolation methods can be found in a
section of fuaction approximation in the theory of numerical analytical methods.
The approximation problem is posed in the general case in the following manner [12]:
a given functionf(x) must be approximately replaced by a generalized polyri.)mial
Q (x) = CoTo (X) + C~mt (x) + � . . + C,"Tm (x). (2.6)
so that the deviation, in a certain sense, of the function f(x) from Q(x) in the
given set X={x} is the least. If the set X consists of a finite number of points
xo, xl, xn, then the approximation is termed point, and if X is an interval
a< x< b, then the approximation is termed integral. Let us examine the first in
stance.
The most important for practice are the degree polynomials of the type
Q (Y)  ao + alx +nsXI + . . . + QmX'n.
(2.7)
ItI terms of them it is possible to formulate the approximation problem in the f ollow
ing manner: for the given function f(x) to find the polynomial Q(x) af the lowest
possible degree m assuming at the set points xi(i = 1, 2, n, xi # xj with i# j)
of the same value as Che function f(x), r_hat is, one where Q(xi) = f(xi)(i = 1, 2,
n). The given system of points xl, x2, xn is termed the basic point of
interpolation while the polynomial Q(x) is the interpolation polynomial.
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Let us point out that for extrapolati.on purposes it is possible to use the same in
terpolation polynomials in which are substituted the x values lying beyond the in
terval (a, b) or the given set of points X={x}. Hawever, mathematics recommends
that extrapolation be very cautiously used. FQr example, in assigning the initial
points in the form of a sequence of equidistant values with an interval h, it is
reconunended that orthogonal polynomes be used for an extrapolation only for a value
Ax = h/2 or if the designated function has rather smooth ends.
On this level the forecaster must make much bolder conclusions, for example, in
setting the initial series of values for a variable over a year to extrapolate this
for 5 or even 10 years to come. Because of this there is the obvious necessity of
involving certain additional information not found in the dynamic series.
However, let us return to the forma.l interpolation. If n< m, then it is possible
to assume m= n and determine the factorization coefficients for ai from the system
of equations:
ao +njxo � . . +anXO _yo; 1
na _ f., alxt + . . . + p.Cj  lfli (2.8)
no + aix� + . . . + Q�xn  yn.
The determinant of this system is a Vandermond determinant A04p4q, xn. Formulas (2.26) and (2.2$) can
be used for extrapolation of the y values. For x4 xp it is advisable to use (2.26),
and t=(xxp)/h < 0. With x> xk and x close to xk, it is convenient to use (2.28)
and t = (xxp)/h > 0.
For calculating one value of the function f(x) using any of these formulas, there
must be n divisions (n2+3n
, l 2  1 multiplications and (n2+1) additions and altogether
. (1.65n2 + 4.85n  C.3) conditiona arithmetic operations. In knowing the speed of a
computer it is pu55ible to assess the machine time expenditures for calculating one
point on the standard interpolation (extrapolation) program.
2.4.3. The Methods of Preliminary Processing and Presenting Initial Data in a
Forecast Extrapolation
In contrast to a formal extrapolation, a forecast extrapolati.on does not come down
merely to calculating dependence factorization coefficients for apriori selected
polynomials or to calculating the parameters of a predetermined function. In a
forecast extrapolation, in the process of analyzing the set sAries of values as well
as the essence of the described process, hypotheses are advanced on the nature of
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its further development and on the basis of them the type of extrapolating function
is chosen. '
~ Ae experience of working out such forecast models shows, very essential here is the
_ method of presenting the initial data and the procedure of their preliminary proc
essing. Let us examine certain possibilities for realizing these procedures.
With a large number of empirical points, the choice of the extrapolation function
can be facilitated by smoothing the initial series. As was said above, for a fore
cast extrapolation the essential thing was to eliminate the random deviations
 (noise) from the experimental sequences.
Smoothing is carried out by polynomials which approxi.mate the groups of experimental
points using the least square method. The best smoothing is obtained for the mid
points of the group and for this reason it is desirable to 'choose an uneven number
ef points in the group to be smoothed. The very groups of points are taken by com
position as moving through the entire Cable. For example, for the first five points
yl, y2, y3, Y4, y5 the average y3 is smoothed and then for the following five y 2, y3,
y4, y5, y6 the y4 is smoothed and so forth.
The remdining end points are smoothed using special formulas. The most widespread
f orm of smoottxing is linear, that is, using a firstdegree polynomial. For smooth
ing for .*.hree points, the formulas are as follows:
3 + !/s +
~ ~
(5y i = �g J1 2Jo  yi); (2.29
 1
y+, _ V y_1 I 2y0 I 5y,),
where yp, ypvalues of the initial and smoothed function at the midpoint;
Yl, Y1values of initial and smoothed function to the left of the midpoint;
Y+l, Y+1vaYues of the initial and smoothed function to the right of the mid
point.
The formulas for y_1 and y+l are used, as a rule, only for the ends of the interval.
~ Analogous formulas exist for smoothing series for five points:
~ t
o = b (y_2+f1+y0+y+1+y+2).
~
yt  10 (4yy + 3y_j I 2JoIyj;
~
y.t � To (y' + 2yn I 3y' + 4y:) (2.30)
y_,  b (3y_1 2y_1 + !/o  y9);
t
J{2� 6 !/jj+Jo +2y: +39s)�
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A block diagram for an algorithm to smooth a sequence of points according to the
formulas of (2.30) for a group of f ive points can be represerted in the form given
in Fig. 2.7.
Fig. 2.7. Block diagram of algorithm for
smoothing a sequen,e of n points
Key: [only Russian terms translated in
appropriate boxes]
1Start; 2Input; 4Calculation...
of end points from formuias (2.30);
SStart of cycle from i =3 to n= 2;
6Calculation; 7End of cycle for i;
10Start of cycle from i= 1 to n;
11Calculation; 12End of cycle
for i; 13Print; 14End
Smoothing, even in a simple linear version, is in many instances a very effective
method for disclosing a trend in superimposing random interference and measurement
errors on an empirical numerical series. In the block diagram shown in Fig. 2.7 a
possibility has been provided fAr carrying out repeated smoothing of the initial
numerical series. The number of sequential smoothing cycles is set by the value K.
The chuice of the amount of K should be made depending upon the type of initial
series, upon the degree of its assumed distortion by noise and upon zhe goal pur
sued by the smoothing. Here it is essential to bear in mind that the effectiveness
of this procedure declines rapidly (in a majority of instances) so, as experience
shows, it is advisable to repeat it from one to three times.
As a certain objective criterion from which it is possible to judge the inadvisa�
bility of a repeated smoothing, one can use the expression: max {Iyi  Yj.}I S e,
where Ea positive number chosen from considerations of the accuracy of data pre
sentation ard the accuracy of the subsequent processing algorithms.
y
An example of the processing of a numerical series using the smoothing method is il
lustrated in Fig. 2.8. The broken line shows the initial dynamics 'Lor passengex
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1000 persons
f foa
900
700
501
304
201
tol
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~
~
?
t
1
in tial
0

i
i
/
D
.
departures over the years for one of the
nation's airfields. Curves 1 and 2 cor
respond to the smoothed series for three
and five points.
In addition to the linear smoothing, a
signif icant number of formulas are known
for nonlinear smoothing by higherdegree
polynomials. However, in practice they
are employed comparatively rarely (at
least above the third degree) for the
reason that for a satisfactory realiza
tion they require only largesized table,
in addition, the edges af the table are
not sufficiently well smoathed a.nd the
formulas themselves become cumbersome.
Nevertheless, in the case of large ini
tial f iles, complex types of curves and
the use of computers, their application
is fully justified.
While smoothing is a procedure ai.med at
the primary processing of a numerical
series for the purpose of excluding ran
dom fluctuations and elucidating a trend,
fitting is used for the more convenient
presentation of the initial series, in
leaving the numerical values as before.
Fitting is the name given to the reductian
of the empirical formula y= f(x, a, b) to
the type
Y = a1X+b1. (2.31)
1957 1960 1YDJ Tyiu
. years
The formula (2.31) examines s two
parameter initial function. ii,is is due
Fig. 2.8. Processing numerical series to the fact that this funciion is the
using smuothing method most widely found in the practice of ex
trapolation and interpoiation calcula
tions, and on the other hand, in a majority of instances is comparatively easy to
fit. Functions with a greater number of parameters are far more difficult to fit
and in fact cannat always be done so.
The most common fitting procedures are tal;ing logorithms and the change of vari
ables. Let us examine these procedures from a series of the following specific ex
amples.
l. For finding the parameters oi the exponential function y= axb, a logorithmi.c
transformation is employed of the type: lg y=].g a +b lg x and the change of vari
ables X= lg x, Y= lg y. As a result we have (2.31), where al = b, bl = lg a.
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Thus, having rearranged the experimental points of the proposed exponential depend
ence 3_n the logorithmic ruling, we obtain a linear dependence which can easily be
described and extrapolated and then recalculate the results using the formulas in
verse to the initial transformation of the variables.
2. For the exponential function y= aebX it is also possible to apply logorithmic
fitting: lg y lg a+ b lg e x and the change X x, Y= lg y. We obtain (2.31),
where al = b lg e, bl = lg a. In this instance it is essential to provide for the
rearrangement of the exponential points in a semilogorithmic scale with the subse
quent analysis of the obtained graph.
3. For dependences of the type: a) y= 1/(ax + b) and b) y= x/(ax + b), the follow
ing transformations are used: a) Y= 1/y = ax + b, b) X= 1/x and Y= 1/y. This
gives Y=(X + b)' X= a+ bX. In this instance along the axes of the grid one must
lay off the amounts inverse to the values of the initial variables.
4. I� the proposed empirical dependence has the form y= 1/(a + beX), then the
transfornation of fitting is Y= 1/y, X= eX. Then the coefficients of (2.31) will
be al = b, bl = a.
It is essential to bear in mind that the values of the function parameters deter
mined after the fitting minimize the total squares of deviations for the transformed
values from the linear dependence of (2.31) and do not always correspond to the min
imum deviation of the measured values from the calculated. For this raason such a
_ calculation must be considered only a certain approximation to the trul.y optimum co
eff icient values.
In the case that the empirical formula is assumed to contain three paranteters or it
is known that tne functioa is a threepaxametez one, then by certai,n tzansformations
it is sometimes possible to exclude one of the parameters and the remaining two can
be reduced to one of the fitting formulas.
For example, the initial formula y= axb + c can be fitted after an approximate calcu
culation of the c parameter. For this we select xl, x2 which are moved farther
apart in the empirical serjes and xg which is linked to them by the ratio x3:x1 =
x3:x2. 3or such a choice of independent variables one approximately determ~nes c=
(yly3  y2)/ (yl +Y3  2Y2)� Then by the changing of variables X= lg x; Y lg (y  c),
the initial formula is reduced to a linear one: Y= bX + lg a. In comparing it with
(2.31) we have al = b and bl = lg a.
After defining the parameters it is recommended that for all the points the correct
ness of the approximate calculation of c be tested and in the instance of signifi
cant discrepancies recalculate the parameters.
 It is possible to view fitting not as a meC:hod of presenting initial data but rather
as a method for a direct approximate determindtion of pdrameters for a function
which approximates the initial numerical series. This method is often employed pre
cisely in this manner in certain extrapolation forecasts. We would point out that
the possibility of its direct use for defining the parameters of an approximating
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function is determined chiefly by the type of the initial numerical series and by
the degree oi our knowledge and by our confidence about the type of function de
scribing the studied process.
In the event that the type of function is unknown, f itting must be viewed as a pre
liminary procedure in the process of which, by employin$ various formulas and pro
cedures, the most suitable type of function describing the empirical series is as
certained.
Graphic and numerical analysis of the dependence of two variables could be the sole
method for selecting the fnrm of connection if a set of equally correct analytical
expressions did not conform to the same function graph or ratio. Fig. 2.9 gives
several examples of analytical expressions wiiere it would be impossible to choose
one in using the abovegiven methods of preliminary analysis.
9 ~
I 9=t~
f
R[~
y j ynrclgxy
U 9= 0
x :b
,7/2
a
a
y j ysancya y
A~? $ y= i+$; ~ '
0
~
T '
a
t~b
y
x
x
Fig. 2.9.
Graphic depiction of various analytical expressions
The examination of an empirical series for the purpose of elucidating the optimum
type of function describing it is a broadening of the fitting method or its general
ization. Here it is not essential that the transformations lead to linear forms.
However their results prepare and facilitate the process of choosing the approximat
 ing function in the problems of forecasting extrapolation.
 Source [43] gives a procedure for such research using differential growth functions.
In the simplest case it is proposed that the following three types af diff erential
growth functions be employed.
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1. The f irst derivative or absolute di�ferential growth function
~(t) = y, = a .
(2.32)
On the graph f(t) this is represented by an angular coefficient at each point
of the graph. The 0 (t) = const for the linear law of change of y(t) . For second
order curves (parabolic laws) 0(t) has a linear type of change and for the exponen
tial curves of ~(t) also an exponent. The value of 0(t) depends upon the selected
scale for the measuring of the exponent and time.
2. The relative differential coefficient or the logorithmic derivative
d~ ~ y= d(log y)/dl.
(2.33)
This function can be shown on a graph by constructing it on a semilogorithmic scale.
Then w(t) will be an angular coeff icient at each point. For the exponential de
pendents w(t) = const and for the exponential function w(t) there is a hyperbolic
nature.
3. Elasticity of the function E(r)=._Ldy d(logy) y dt d (log r)
A H aeieod tpamuKO:mfel=d~F~
PJ Ba0vuCnume : l i n l091I I
_ ILJ ebrMflMumb: nn;4'�U....

_ Bo~sod zpa~uKO: cd1= arff (Tal7f
/YO n e y
(2. 34)
Fig. 2.10. Block diagram for calculating
differential growth functions
Key: [translation of only Russian in
appropriate block]
1Start; 2Input; 3Carry out PP
[subprogram]"D"; 4Derivation of
graph; SCalculate; 6Carry out
PP"D"; 7Derivation of graph;
8Calculate; 9Carry out PP"D";
10Derivation of graph; 11End
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In a graph of a dynamic series constructed on a logorithmic scale, elasticity is de
fined as an angular coPfficient at each point. The c(t) = const for an exgonential
function; for the exponential function e(t) has s linear type of change and it is
also liner.r for the combined exponential function.
It must be pointed out that the elasticity of e(t) is a dimensionless amour.t and
_ this makes it possible to employ it in comparing the nat�.,re of changes in differerit
_ processes occurring in the most different possible time scales.
Source [43] gives graphs for differential qrowth functions for all the approximating
functions most used in forecasting extrapolation, including: linear, parabolic,
exponential, logistical, hyperbolic and so forth. An examination of growth func
tions indicates that by their combining it is possible rather uniformly to def ine
th.e type of function producing them and determined by a numerical empirical series.
Thus, for preparing to take a decision on the type of approximating function, it is
possible to propose a machine procedure for calculating the differential growth
functions following the scheme shown in Fig. 2.10. It is best to carry this out on
computers having a graphic data output (graph plotter, teletype, electric typewriter
or display). The set of graphs obtained on the output for ~(t), w(t), e(t) is com
pared with a standard reference table of the type given in [43]. In the block dia
gram of Fig. 2.10, the PP"D" designates a subprogram for differentiating the func
tion of y(t) set by the series {yi, ti}�
2.4.4. The Problem of Choosing the Type of Function and the Ways of Sovling It f or
a Forecast Extrapolation
In the process of smoothing a dynamic series, in fitting it and determining the
functions of differential growth, the type of function describing the initial proc
ess is already approximately determined and sometimes the estimates of this func
tion's parameters are even obtained. For the final choice of the type of function
a retrospective series study carried out in the staga of preliminary processing must
be supplemented by research on the logic of the occurrence of the process as a
whole, including hypotheses on its oc.^.urrence in the future and by research on the
physical essence of the process, possible shifts, jumps and constraints stemming
from this essence.
The basic questions which the researcher should set for himself at this stage are:
1) is the studied indicator as a whole an amount which grows uniformly, diminishes
uniformly, is stable or has an extremiun (or several of them) or is periodic; 2) is
the studied indicator limited above (or below) by any ].imit; 3) does the function
defining the process have a bending point; 4) does the function representing the
process possess the property of symmetricalness; 5) does the process have a clear
limitation of development in time.
_ Depending upon the answers to eacYi of the listed basic questions, secondary ques
tions arise which are of a more quantitative nature or the nature of elucidating
the reasons for the appearance of one or another quality in the process.
For the answer to the f irst question it is essential to bring together information
obtained in the process of the primary processing of the series and namely the
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first derivative ~(t) and the general considerztions on the nature of the process's
development. Obviously for a steadily rising function the graph �(t) should lie
completely in the positive area and its extrapolation would not show a tendency to
cross the abscissa axis in the r'uture.
Now about the essence of the process itself. For example, a majority of parameters
determined by the development of scientif ic and technical progress can be steadily
growing functicns which in a number of instances have asymptotic constraints. Thus
it is possible to speak about the continuous increase in the spe2d of transport, the
rate of data processing, the power of energy units, the distance man penetrates into
space, the increase in the length of human life, the greater labor productivity and
so forth.
Analogous arguments can be given for steadily diminishirag processes. For example,
the shortening of production cycles, the reduction of relative dimensions and
weight of units and so forth.
For stable parameters it is possible to estim3te the amount of their degree of in
stability in the retrospective interval Aymax �r amaX =�ymaX/y� It is essential to
bring out the factors which influence the iristability and analyze their possible
changes in the future.
The most complicated problem in forecasting is the pre3icting of the appearance of
abrupt jumps in the studied process in the future. The basic ways for solving this
in the area of scientifictechnical and economic forecasting are research using
lead methods on patent and scientifictechnical information as well as carefully
planned expert surveys.
The extremums on the retrospective interval are easily detected in examining the
graph of ~(t) at the points it crosses the abscissa axis. The presence of extremums
in the previous development of the process leads to questions about their causes and
 their possible occurrence in the future. Here it is essential to examine the sta
bility of their appearance in the development process and draw a conclusion on the
possible periodic nature of th,:, process.
In the event of a supposition of the monoextremalness of the procesG, the basic
question is to elucidate the point of the extremum and the extremal value of the
forecasted parameter (if the extremum has not yet been passed). This problem can
be solved by the extrapolation of $(t) and by determining the point of its crossing
the time axis. This research must be supplemented by the results of qualitative
and quantitative analysis for the development of factors influencing the achieving
of an extremum by the examined process.
Many economic development processes are characterized by a periodic nature of de
velopment. In truth, these processes basically have a cyclical and not purely a
periodic nature. Thus, the cyclical nature of economic development under capital
ism is generally known. The process of its development contains a number of repeat
ing stages: economic increase, crisis, depression, a new increase and so forth.
Econometric research has shown that over long intervals of time the period of the
cycle can remain relatively stable, although the height of the rise and tiie depth
of the fall can change from cycle to cycle within certain limits.
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If in tne research process it is discovered that a certain indicator depends sub
stantially upon the cyclical development of the overall economic process, then this
fact provides essential information for selecting the type of functinn representing
it in the extrapolation research. In particular, the methods of harmonic analysis
and factorization for trigonometric polynomials (2.16)(2.19) are used very effec
tively for such processes. '
The reply to tne second basic question about the nature of the process is very es
sential for selecting the correct type of f unc[ion which extrapolates the trend.
We have already mentioned at the beginning of (2.4) the mistakes to which extrapola
tion can lead in forecasting with a failure to consider the gossible constraints of
the process. For example, a dynamic series describing an increase in the speed of
transport over the last 100 years is described rather well by the exponential de
pendence [46]. However, consideration of the natural limit of the speed of light
inclines one to choose an Sshaped curve (logistical) for describing the process.
The forecast results using these two curves differ substantially.
In any area of knowledge great attention is given to the problem of studying de
velopment limits. In the area of scientific and technical forecasting it is pos
sible to point out several types of limits examined below.
Absolute limits are unconditional limits where the area of action is not restricted.
Among absolute limits are [50], for example: the speed of light, absolute tempera
ture zero, zera pressure, an efficiency of 1, the temperature for the breaking of
molecular bonds and so forth.
Relative limits occur in a certain area or in terms of a certain object. These in
clude the terrestrial limits such as: maximum speed in the atmosphere, maximum
'epth of the ocean, minimum speed for the orbiting of a sattelite and so forth. It
is also possible to give examples of the relative limits of human capabilities:
maximum gloads, maximum noise level and so forth.
Calculated limits are more particular limits set on the basis of the first two types
of limits and various sorts of transformations and laws linking them to the derived
amounts. For example, the maximum value of efficiency with a set temperature drop
in the Carnot cycle, the limit for loading storage with information ot 1014 bits
per cm3 calculated on the basis of the absolute limit determined by the Heisenberg
formula; the maximum amount of microminiaturization of electronic elements related
to the Uirac constant and defined as lOlb elements in a cm3 and so forth.
An elucidation of the question concerning the limitation of a function to a large
degree is based upon an analysis of the physical and logical essence of the studied
process, its links and the dependences upon the absulute and relative limits in the
investigated area of knowledge. A formal analysis of the diff erential growth func
tion ~(t) can serve as an indication of the limit's existence. In the case of an
asymptotic approximation to the limit, the function ~(t) will obviously move toward
zero.
The secondary questions with an affirmative answer on the existence of a development
limit include: what is the amount of this limit and what is the nature of the ap
proach to the limit value of the studied function.
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The solution of the first question often develops into independent forecast research.
For example, a forecast of the limit computer speed with the set circuitry, the
furecast for maximum power with the given fuel and overall dimensions of an engine,
and so forth. The question is more simply solved in the case that the limit is ab
solute, relative or calculated.
The nature of the approach to the limit is also basically determined by logical anal
ysis of the occurring process. A predominant number of technical and physical in
dicators for the processes move asymptotically toward their limits. At the same
time certain volume indicators in economic forecasting can change over to limit
values at a point. For example, the growth of the production volume for a certain
product with limited capacity of raw material supplies or a sharp decline in the
increase of sales wich the complete saturating of the market. Such processes are
significantly more difficult to extrapolate than are the asymptotic ones since the
nature of their development, as a rule, is determined by external factors and is not
brought out in analyzing the retrospective data about the process.
The soundness of the research and the dep,ree of considering not only the shortterm
but also the longterm trends in the re:earch are of great significance in correct
ly determining the constraints of the indicator and the nature of the move toward
it.
R. Ayres [46] has pointed out that by using disaggregative analysis it is uEally
impossible to predict the appearance of even comparatively simple innovations (if
they alter the configuration of the system), since the limit values for the effi
ciency of any cl_ass of systems (instruments) can be determined by extrapolation only
on the level of individual components. Major inventions usually change the config
uration of the components to such a degree that this in no way can be predicted
ahead of time.
An aggregative approach is also aimed at an analysis and forecasting of a wide class
of systems. In an extrapolation the aggregative approach has been embodied in the
envelope method which will be taken up below. In applying this method the limits
assume a more generalized sensQ and can be set more correctly and thereby more accu
rateiy reflect the future development of the process.
Hence, in setting development limits for any forecasted process it is always essen
_ tial to correctly choose the level of analysis aggregating.
The third question of the listed basic questions about the na.ture of the process
relates to the existence of bending points. This is also very essential in elabor
ating the various forecasts. In a number of studies tiie bending point and its posi
tion on the time scale has been chosen as the criterion for the perspective develop
ment of one or another scientific or technical area. Thus', in [41] for each area
from the possible solutions of a technical (technological) problem a curve is con
structed for patenting dynamics. Then, the presence of a bending point is shown on
the curve corresgonding to each area. In the event that patenting dynamics by the
pr.esent has already passed its bending point, the given area for the development of
technology (production methods) is considered relatively unpromising.
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In formal terms the bending point can be established from the zero value of the
second derivative of an approximating function with the subsenuent change in the
sign of this derivative. On the graph of ~(t), the bending point is shown in the
 form of an extremum so that the determining of it on the retrospective segment
formally does not provide any problem. In the case of significant random fluctu
ations superimposed on the process, it is advisable to carry out its preliminary
smoothing by one of the methods described in 2.4.3.
It is signif icantly harder to predict the appearance and location of a bending
 point in the tuture. Analysis of the essence of a developing process in this in
stance should be carried out in the direction of elucidating the factors which sub
stantially influence the growth rate of the process and then an analysis of the
trends of their change in the future.
An analysis of the bending points is substantially facilitated with the availability
of data on limits in the devQlopment of the grocess and che nature of the move to
ward zero. In this instance the presence of a bending pcaint can be considered set
and its location is determined after chocsing the parameters in the functional de
scripti.on.
As for the fourth question it must be pointed out that among the functions used for
extrapalation in forecasting, only very few possess the property of symmetry.
Basically these are logistical curves with a central symmetry relative to the bend
ing point. Linear develcpment laws possess a symmetry relative to any point but
for them *_his property is not of particular significance.
The for.mal setting of symmetry carL be carried out by an iterative cnmparison of the
final differences of the function to the left and right of certain points whictz
gradually approach a possible point corresponding most cioseiy to L'ne syniifietiy
center. As the symmetry criterion relative to a certain point k(the symmetry
center), it is possible to propose the following expression for the amount of the
standard deviation o` the final differences of an empirical series with a constant
spacing to the right and left of the point:
n /2
SN ? (2.35)
n re~
where nthe total number of points in the segment of the curvs studied for sym
metry.
~ A plus sign is taken for axial symmetry and a mir,.us sign for central symmetry. Then
the presence of symmetry can be determined by the expression
min {Sk} < E, (2.36)
k
where ethe given positive number determining the limit for asymmetry.
T,... o~r;.,,, on rho natiira nf rhP nrocess concerns the elucidation of develop
a.tc too, .:1:~`.....~....
ment cunstraints for the process but not in terms of a:aount but for time. This
78
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involves the elucidation and consideration in extrapolation of moments for the oc
currence of certain events which bring about an end to trie process or its transition
to a different quality. As an example one might give the necessity of endi.ng tle
production process for a certain product by a certain date as set by a higherLevel
plan or forecast.
In and of itself the setting of a tin;e restriction on the process and a limitation
point on a time scale can be the result of a forecast or a plan. Nevertheless the
consideration of this factor is essential in extrapolating the development trends of
parameters which describe this process or are linked to it.
Theoretically it is impossible to speak about a process (with the exception of the
scale of the universe) as unlimited in time, and for this reason here it is a ques
tion of a relative limitation on the forecast's lead time. If the time of develop
ment (existence) for a process greatly exceeds the forecast lead time, then it is
possible to speak about a process which does not have constraints in terms of de
velopment time. Otherwise it is essential to determine this constraint and con
sider it in choosing the extrapolating function.
Having analyzed the basic questions arising in the stage of choosing the type of
function for an extrapolatxon of the studied process and the possible approaches to
resolving them, let us move on to an examination of those functions which prefer
ably should be used in a forecast extrapolation and certai.n demands which must be
made on them. In [46] the demands are given which are made on the approximating
curve: morphological simplicity, smoothness, symmetry and mathematical simplicity.
As a whole it is possible to agree with such demands having put symmetry in last
place and mathematical simplicity, having examined this in greater detail, in firsc
place.
What do we understand by mathematical simplicity? This is the minimum possible num
ber of terms in a formula; the minimum possible degree of an indpnendent variable;
a linear ascent of the coefficients; continuity; a minimal number df extremums and
bending points. Such requirements are met by Che standard functions which are most
widespread in extrapolation:
a) Linear y= a+ bt (with b= 0, y= astable state);
b) Parabola y= a+ bt + ct2 (with c> 0a growth function, with c< 0an extremal
function);
c) Step function y= atb;
d) Exponential function y =aebt;
e) Modif ied exaonent y= k ae bt ~
k
f) Logistical (Sshaped) curve y =
1 + be ct
g) Hyperbolic f uncti on y= a+ b.
.._L~
l. I L
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The various sources give different lists of recommended curves. The lists given
 here, in our view, represents that essential minimum which covers apredominant
portion of the needs arising in extrapolating trznds in economic and scientific
technical forecasting.
This list does not include functions making it possible to approximatn periodic
~ prc�cesses. The problem is that harmonic analysis of periodic processes represents
 a rather independent problem. Moreover, in forecasting problems in the area of the
BTS, the necessity of extrapolating periodic processes virtuallq does not arise
(wi..th rare exceptions).
 In a general form, the sequence of steps in selecting the type of extrapolating
 functYon for the set empirical series can be as follows: Smoothing the empirical
series, attempts at linear f itting; in the case of failure the constructing of dif
 f_erential growth functions; in the case of success immediate].y a logical analysis of
the essence of the process using the scheme given in the given section and the
choice of the type of function for the extrapolation.
2.4.5. Calculation of Parameters for the Extrapolating Function
Thus, as a result of solving the previous stages in the forecast research a definite
 type of function has been found for extrapolating an empirical series and its ana
lytical presentation has been given c:ontaining a series (or one) of unknown parame
ters. In the following stage it is essential, in using the empirical series, to
choose (calculate) the values of these parameters which ensure optimum approxima
 tion in a certain sense.
As the optimality criterion ordinarily one uses one or another weasure of the devi
ty R ~,~n~tinn, F.ACh
b
ations of the points in the empiricai series iruru tie SY~+i^X.sym....T
of the possible criteria for approximation optimality has its corresponding method
fer determining the curve parameters. Let us examir_e the basic rosthods having paid
most attention to the least square method as the most widely found.
 The averages method is based upon the minimization of the algebraic total of point
deviations from an approximating curve. In this case the og�'imality criterion is
 written in the following form: n
~
S Z Ji  f (X,, at, (it, am) min, (2 0 and dimiilish without limitation for
= r < 0.
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If nan even number (n = 2p) and the parameters rl, r2, rn are c7Pil' ex,
paired numbers of the type ru = au + Sui and ru+l = au  Sui, where i=Su # 0,
u= 1, 2, . . . , p and au are real and different, then the function F(x) according to
the formula (2.63) is determined, real and continuous on any segment of the inde
pendent variable and can be represented in the following form:
F(.r) fi(x) = Bo F Z e�Cu lz sa (e� cos P,, (x  xe) CF, sin P. (x xo)1.
�=1
where Bp, Bu, vu constant real parameters depending upcn Ao, A1, An; al, a2,
ap; $1, S2, Sp.
As is seen, in this case F(x) will contain terms of a harmonic nature which are am
plitude modulated by the exponential functions.
In the event that all the terms ri, r2, rn move toward zero with a transition
to the limit, the function F(x) can be reduced to the type
F (x)  T� (x) = Ao +At (x  xo) . . A� (X n 1 o)n,
that is, we obtain a step polynomial.
From an examination of the diff erent variations for setting the parameters rl, r2,
rn it becomes apparent that by changing them it is possible to alter the struc
ture of F(x) in rather broad limits, representing it in the furm of a step poly
nomial, a triganometric polynomial, or an aggregate of exponents, or, finally, any
combination of tY:e iisted structures. This is a very enticing property of the FGS
from the viewpoint of using them for trend extrapolation.
It remains to solve the question about the method to seek out the optimum values of
rl, r2, rn. Let us examine a case when the function z(x) is differentiatable
(n + 1) times in a segment containing the initial point x= xo. Then formula (2.64)
 can be reduced to the type
n 81 (x  xe)
r� c.Y) = z tX,l i E zl(Xo) (z. 66)
,1
where zi (xp)the derivative of order j for the initial function at the starting
point;
Dthe determinant described by (2.65) where rl, r2, rn are viewed
as roots of a certain base equation:
~ + an_,r"i ...F . . . _F air . J a� = 0;
~ Qp = 1)^ Ilfo . . . fR;
 a1 _ 1)"' (rlr, . . , rl_l r2r3 . . . i�);
an_1 = 1)"(nl) (rl 4. re + . . . + rh), (2.67)
The functions 8j(xxp) are dete r.nined analogously to (2.64).
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An analytical expression of the remainder in formula (2.63) has the following form:
r ~
R~X~ _ J J n~j~ e~ u z) dz di.
D
rO te
(2.68)
The term of this formula An(tT) can be obtained from (2.65) in substituting the
last n line of the determinant D f cr the line of the exponents of the type
erv(tT), (v = 1, 2, n).
The other term of (2.68) can be def ined from the equation
(2.69)
The idea of an optimal approximation in using the FGS comes down to minimizing the
remainder R(x) and to establishing such valuzs of the parameters ap, al, an1
that the value o� the remainder in each point of the segment does not exceed a cer
tain set amount (the approximation error) and this also determines the type of F(x)
which is the solution to the problem.
In [20] as an example they examine a case of even approximation
R a1 (xx�) (2.70)
Z(x)  Ao ~Al 1) ~ Yo�
where Yo = const. > 0, a< xo < x< xk < b.
If the parameters of A are taken according to (2.66), then we obtain an expression
of the approximation errur in the form
(x  xe)
� r� (.r) (.r) z (xo1  1_ ~ Zt/1 (xo)v . (2.71)
j:t
The unknowns in (2.71) are the parameters rl, r2, rn which can be determined irz
the process of minimizing e(x). With an even approximation the errors values of
e(x) at the extremal points M1, MZ, Mn, at the initial point Mp and the end
point Mn+i = Mk are the same for the modulus and alternating in sign:
e' (Xn) E;' (r,) _ . . . = E' (X.) ~ p; xo 4 x 44 xK;
f, (.t'r) = E (�rn+i n  I , 2, . . . , it; ~ e (xi)
N. j p, 1, 2, nI ~e(X)) .4 N. (2.72)
Depending upon the selected system of base functions and the different values of
the parameters ap, al, an_I in the base equation (2.67) in the process of mini
mization of (2.71) various structures of F(x) can be obr.ained, f.or example, a Taylor
step polynomial, a Chebyshev polynomial or a trigonometric series.
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The other method of even approximation can be termed integral. It is based upon the
use of the formula for the remainder in (2.68):
; t
e(x)=z(x)F(x)=R(x)= J jtip T~dzdt, (2.73)
xe zt
Then the rirst part of the conditions in (2.72) can be expanded on this basis. In
particular, the conditions of the extremums e(x) at points M1, M2, Mn are
written: E'(xl) = E'(x2) = E'(xn) = 0, or considering (2.73)
x, ~n
an(D t )dt= (Xdt O. (2.74)
J
x. ro
The second part of the conditions in (2.72) will assume the form
Ap xC4l i
j J Jn _ i) dT dt. (2.75)
D dz dt ~ L)
t, X, X, x.
If for a certain function of z(x) it is possible to choose a value of n and the
parameters of ap, an_1 in order to f ulfill the condition
n(X) = z(n+2)(X) + an1 zn(x) + . . . + alz( 2) (s) + aoz(l) (X) = 0,
then F(x) will provide an even approximation with a zero error factor. Unfortunate
ly, the nonlineality of incorporating the coefficients ap, anl in the system
of (2.74), (2.75) substantially impedes its practical utilization.
In the instance of a small segment [xp, xk], if n is sufficiently large, it is pos
sible to assume
e 11~1 r~ (1 s) y~ (R '
.
v ~ f r~~  T) ~ ~ . . . 1)1 (2.76)
v=1, 2,~...~ n, xe4 t . 2 I )  1;96 ~f Ti I, . 2 . 93)
If even one of the inequalities is disrupted, then the hypothesis of the random
nature of the deviations in the time series levels from the trend is rejected.
2. The test for the hypothesis of stationarity for the random camponent. ThE basic
condition for the stationarity of a random process is the condition of the depend
ence of the autocorrelation function solely upon the difference of the independent
variables ti = tj = T.
Let iis test the hypothesis that the value of the autocorrelation function does ao,
dFpend upon the choice of the beginning of the observation count but depends just on
 :he amount of the shift of T. ror this for the random component nt(t = 1, 2, n)
we will f ind the value of the autocorrelation function r~, r2, rT (the upper
index is the number of obsPrvations for which the autocorrelation function is cal
cuiated). A formuia for calculafiing an autocorrelation function has the following
form:
~
~114t, s
~
_ ~T~ ,
4 2
r
r= i
Then one of the observation curves is excluded and a t.ew autocorrelation function is
calculated rJ1, rlr1, r T1. In a similar manner one excludes P(P = 0, l, 2,
p) observations and the value of the (p + 1) autocorrelation function is calcu
lat?d. Thus we obtain. r erouos of autocorrelation coeffici.ents and each of rhem
will includP (p + 1) coefficients. For a stationary process, the autocorrelation co
efficients included in the same groiip should be uniform.
The test for uniformity can be carried out in the following manner. For each r(nP)
included in group T, we calculate the amount of the z criterion using the formula
Tn v)
zTp = � 2 1n ~ _,(�v) '
T
Then for this group we calculate
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P
_ Z021p (2.95)
_ z'r _ _ p T _1 .
Mote has proved that the amount
' r
~ ~Zt� _ ` (2.96)
~ _n
Po n
is distributed as X2 with p degrees of freedom. In the event that the amount X2
calculated using formula (2.96) is less than the tabular value of X2 with a set con
fidence level, then the hypothesis of the uniformity for group T of autocorrelation
coefficients can be accepted. If the uniformity hypothesis is canfirmed for all the
groups, then it can be accepted that the random component is a random process th3t
is stationary in the broad sense.
If in the test it is established that a tfine series does not meet even one of Lhese
conditions, it is wrong to apply an equation of the type (2.96) for describing it.
In this case one must move on to higher order differencPS.
r(i 1
0, ~i
n. r
0
t 7 3 4 s 6 7 e gr
Fig. 2.12. Correlogram for determining
the order of an autoregression model
Af ter the necessary tests have been ma3e,
it is essential zo determine the order of
the autoregression model. The first step
in selecting the order of the autoaogression
model is an analysis of the correlogram.
The autoregression proce5s is ciaractexized
by the fact that an autocorrelation furc
tion is diminishing. A diminishing of the
correlogram indicates that the re.lation
with the past is weakening. The autore
gression order is determined depending upon
in what shifts the autocorrelation function
xeaches the greatest amount. Let us ex
p13in this from a.specific example.
From Fig. 2.12 it can be seen that for T> 5(Tthe amount of th2 shift) there is
a damping of the correlogram, that is, the relation with the past is weakening. On
this basis it can be concluded that for thE given example it is not advisable to
~ construct a model above the fourth order. Since an autocorrelation function reaches
the greatest amount in the third and fourth shifts, autoregression models of the
third arLd fourth order are constructed.
After f irst determining the order the parameters of the autoregression model are
found. The autoregression parameters can be determined by a double method. The
first is the direct application of the least square method. Trie condition for the
minimiim dispersion of the deviations in the fixed sample from n observations is
written in the form
 n m z
D(,qr) Z C~,  Z a~8r_~) = min.
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This condition leads to a system of normal equations
ai Z sii I vs Z 8r,bez I a,. Zi bitar~ _
r=M+t r=m+t tM+t
n '
fam+l
n n n
u, Z 8t._A_2,{ az Z 612 4 am Z 8~_~~m =
f=m}1
(2.98)
n rt ~n~
n l Lj 6l. _ t 61._,n . V pt ~ fit4atm , . . .i... pm L.I aim ~
l=m. E 1 1mm+1
` n
~
Tne estimates of the autoregression coefficients can be obtained by another method
from a system of YuleWalker equations .
 ' fl Q, f,llq . . . + /m_lam = pi
' r. . ~ r,Ql + a, + . . . + rm_Iam = p;
(2.99)
rm + rm_lal + rm3af + . . . + e 0,
where ri the coefficient for the iorder autocorrelation.
 For all cases, with the exception of p vaiues, the system of (2.99) is easier to
solve than the system (2.98), since there is a more symmetrical form in the construc
tion. As is known the solution to the system (2.99) comes dow�n to recurrent rela
 tions
ri + a.1. ~ri1 at1, 2r,4 + . a,_1, ~rtrl
I + ai_i, jri F. . a.a, o_irse (2.100)
(s = 1, 2, . . . ~ m);
(Ish  ns1, n ns:a,,, nh 1, 2. . . . ~ S  1). (2.101)
Here as the initial value one uses all =xl. As a result of applying (2.100) and
(2.101), we obtain estimatES for the coefficients of an autoregression model of the
 s order, since asl = al, as2 = a2, asm = am. Having determined the parameters
of the autoregression equation, it is now possible to more precisely set the order
of thE autoregression model. For determining the order of this model the Mann aiid
' Wa13 criteria is employed. It is convenient in the fact that it employs directly
calculable and not estimated amounts.
In order to establish whether a sufficiently high degree of approximation has been
obtained as a result of the approximation by the model (2.92) of the order m, it is
~ essential to determine the deviations which arise in increasing the autoregression
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~
~
% order. ior this autoregression models of the q order are constructed, wi.iere
~ m< qa9
Fig. 2.16. Illustration of the
correlation pleiad method
, As a rule, the picture obtained as a result
~ will contain a number of clusters analogous
i to the clusters in x4 and xi in Fig. 2.16.
In examining such a picture, it is easy to
isolate the vertices of the clusters which
collect a large number of relations. For
level 1 of (2.146) it can be said from
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I'ig. 2.16 that the variables xl, x5, xt, xk and xn_i are related virtually func
 tionally (with r>0.9) with the variable x4; for this reason the analysis and fore
cast can be made only for x4 and the obtained results extended to all the variables
of the cluster through the formulas of functional ratios. An analogous conclusion
can be drawn for the other peak of the cluster xi in Fig. 2.16.
The construction of several such circles of correlation pleiads for the various
 levels of correlation coefficients makes it possible to analyze the internal rela
_ tionships of all the variables in the complex and select for examination a minimum
number of them as determined by the given accuracy of the research and the nature
of the D matrix.
Another approach to solving thz problem of minimizing the system's dimensionality
for an object's variables is given in [5, 39] from the standpoint of information
theory and pattern recognition theory. From these positions the problem of fore
casting BTS development can be represented as the task of recognizing the future
state of an object from the results of furecasting the values of a set of individu
al variables which comprise its description.
The problem of minimizing the demensionality of a description is carried out in the
following manner. On the basis of the retrospective values of the variables, from
them a minimum number is chosen making it possible with the set reliability to dis
tinguish the states of the forecasting object which are of interest to us. The
basic propositions and assumptions in the given examination come down to the follow
ing. For the studied object we know a finite, denumerable set A={A1, A2, Am}
of possible states ca.lled classes. The object is described by the set of X=
{X1, x2, xN} variables, each of which can assume a f ini.te number of values.
It is assumed that all the variables o� xn are statistically independent of one an
other, as otherwise the problem becomes particularly cumbersome for practical use.
!',ccording to Shannon the information content of a certain variable relative to a
set of classes A can be defined as the difference of the initial entropy in the sys
tem and the entropy of a solution for the variable xn
Jn = H(A)  H(A/xn) .
(2.147)
Let the variable xn assume T discrete values acnj (j = l, 2, T). Then the en
tropy of the solution with value j of variable xn can be written:
m (2.148)
Hi (Al�rj Z P(AMlxn1) loK P(Am1Xn/).
According to the Bayes formula .
n(An,) p (Xn j/^m) � P(Am) P (xnll Am)
P (Am/ x.j)  P (xn/1 'N ~ ~ p (AM) p (rnIJAm)
(2.149)
In substituting this expression in (2.14E), we obtain
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x
fl/(A/Xn) n(Xn')
~ P (Am) P (xn//Arn)
?r ~ P~~,~�a F (Xn jllim) lOg M 
I P ( An) P (xnjl Am)
m =1
 ~ I'M M A1 x 1 x I/! (2.150)
p (Xnj ~ l~ ~ mr P ~ n/~/~ni~ ~n~ ~n~n' / ~ :i J m~ 
`M M
 nU1 P(Am) P(X,./lAm) log n ~
l.~ l~ (nn. ) P(�~'n/inm) I'
In these formulas: p(xnj)the probability of the ap?earance of gradation j of
variable n for all classes of Am; p(xnj/Am)the protabili*yo uf the appearance of
value j of variable n for class m.
For obtaining the fu11 entropy of the solution for t:he variable xn it is essential
tu total Hj(A/xn) for al.l gradations of j with weights proportional to the proba
bilities of the appearance of each gradation, that is, p(xnj). As a result, we ob
tain
T
H (AlXn) P (x./) Hl (Al�r.)
/el
i' M
Z Z P(Am. x�1) log p (A,,, x�)
Iqa1
T /N
~
I
1 #a (Arn, x.l) Iog ~i n (nn, Xnr)�
In substituting the obtained expression and the initial entropy H(A) _
M
_ I p(Am)log p(Am) in (2.147), ;ae obtain the final expression for the information
m=1
~ content of variable n:
n ` ~ /1 (Am) I nP (nm) + GI ..1 P (Am, Xnl) I 0g p (nm, xn,) 
m 1 /1 nti
.41 M
>1 p (nm, X,q) log U p(Am, .1'n1).
~
1=11n 1 m=1 (2.151
where p(Am, xnj)thP joint distributian of probabilities for the values of xnj for
the class Am.
The expression (2.151) is the basic woricing formula for calculating the informati.on
content of variables. Having determined quantitatively the values for the informa
tion r.ontent of the variables, it is possible to rank them according to the decline
in the values of Jn and determine that minimum number of variables which is essen
tial for recognizing the state of the object with the set degree of reliability.
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Let the solution be made by calculating the aposteriori probabilities of the possi
 ble states for the values of the variables xn detzrmined relative to a certain
moment of time in the future. Here it is advisable to consider the variables with
the highest information content out of the entire set. Thus, the route of examin
ing the variables is set by their ranking number from the decline oz Jn. Tiie "Lengt:
_ of the routes that is, the necessary number of variables in tne examination, is de
termined by the given threshold value fcr the aposteriori probability of the class
of states PP.
With the least favorable distribution of the aposteriori probabilities of the hy
_ potheses we have
1 . . P" , I  Pp . . I  pn . (2.152)
1 ' h1  I I ' with the most favorable
1  PI" 0, (l, . . 0. (2.153)
The last coordinate of the minimal route is determined by the Fano inequality:
l.ll(/1) (:1/.c,)  I!(A/.r,)...H(A/,r,)>H(/I)Hp, (2.154)
where Lthe number of steps in the minimum route;
Hpthe entropy of thE solution determined from the distribution of (2.152).
The aboveexamined method of minimizing dimensionality in a description of a sto
chastic forecasting object can be realized in those instances when the problem is
posed from the abovedescribed positions of pattern recognition theory and the
retrospective analysis provides sufficient statistics of the variables and the
states for calculating cor.ditional probabilities.
In the event that in examining the minimal route none of the hypotheses has reached
the given threshold Pp, it is possible to use the following variables of the full
route. Then two variations are possible: either at a certain step the probability
of one of the hypotheses m exceeds P and then one can speak about the forecast of
the object's state close to m, or, if none of the hypotheses reaches the set prob
ability threshold, then one can speak about the appearance of a fundamentally new
state of the object in the future. In lowering the threshold Pp, it is possible to
determine to which of the known this new state is closest.
The designated procedures for determining the information content of variables, for
seeking out the minimal route and for classifying states in examining it can be suc
cessfully carried out on a computer. Experiments in identifying classes of techni=
cal systems have provided positive results.
Furthermore, a method has been provided of multivariate statistical analysis making
it possible to minimize the description of a stochastic object not by discarding
inessential variables but by disclosing certain general characteristics of their
aggregate.
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2.5.3. Certain Information on Factor Analysis
~
Factor analysis in its present form represents a certain area of mathematical sta
tistics [26, 42]. However, its appearance at the beginning of the present century
is usually linked with the names of the psychologists C. Spearman, S. Barthow,
: L. Thurstone and other. Its initial aim was to construct mathematical models for
human capabilities and conduct. Here the results were based on various psychologi
 cal and physical tests and at the output certaiii general indicators or factors were
formed. In this area factor analysis is succ2ssfully employed at present, however
in recent decades it has spread actively into many other areas such as sociology,
economics, geology, meteorology, engineering and so forth. The work of H. Harman
[42] gives over 200 publications on factor analysis and its use in over a score
scientif ic areas. He also points to the great diversity of factor ana].ysis methods
_ and their modifications which are presently known. Let us examine the essence of
 one of them.
Let Xan ndimensional random vector representing a random sample of ineasurements
for an aggregate of interrelated parameters xi; Fa kdi.mensional vector the com
ponents of which are directly unobservable variables (the factors FJ�); Xthe
mathematical expectation of the fector X; Uthe vector of the total of the unob
servable errors and specific factors. According to the basic assumption of multi
factor analysis, each specific measurement of the Xxi vector can be viewed as the
r_otal of the effects of a certain small number of group factors fj (taken with
certain weights aij), the specif ic factor si effecting only the given variable and
the measurement errors ei. Since si and ei are indistinguishable in factor analy
sis, they are ordinarily viewed as the tota'L ui = si + ei.
Further, let Athe matrix of the order n+ k(n> k), the elements of which are �actor
weights aij determining the load of variable i on factor j; mthe number of obser
vations on vector X for which the estimate is made. Let us write the basic idea of
factor analysis in a matrix form:
X=AF+X+U.
(2.155)
For the sake of simplicity let us set all ttie averages at zero: X= 0, that is,
we will further view the unbiased distributions xi. Let us designate the product
AF by Q, then
X = Q+U,
wher.e Qis usually called the general part, and
Uthe specific part of X.
(2.156)
It is assumed that U does not depend on Q and all the ui(i = 1, 2, n) are not
intercorrelated. Here the matrix M(QU') = 0, the matrix M(UU') is diagonal (Mthe
operator of the mathematical expectation; U'the transposed matrix U). Tkien
M(XX') = MI(Q+U)(Q'+U')I = M(QQ')+M(uu')+M(QU')+M(UQ') = M(QR')+M(w (2.157)
If we normalize the X vector for the values of the standard deviations Qi(zip _
xiP/ai) where xipa component of the X vector; pthe ordinal number of a single
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observation of vector X), then M(ZZ') = R is a correlation matrix. In accord with
(2.1.57), this can be shown in the form
R = Rp+U2 = R0+I+112, (2.158)
 where Rthe initial matrix with units on the main diagonsl; Rpthe socalled re
duced matrix; U2a diagonal matrix from the squares of the total specif ic factor
weights and errors; H2a diagonal matrix of the socalled communalities; Ia unit
matrix.
 In factor analysis usually the R matrix determined from (2.158) is subject to factor
 torization. Here the reduced matrix is approximated by the product
Rp = ApAb, (2.159)
wherP Apis taken as the matrix of the factor weights of the order (n X k); Ap
the matrix transposed in relation to Ap.
The factors fl, f2, fk are assumed to be uncorrected. In this instance, the
factor weights can be viewed as coefficients in a linear regression equatLon for
estimating the variables by the factors.
If we disregard U in (2.156), then Ap coincides with A and Rp coincides with R; con
sequently, the factorization will be closer to the original the closer the matrix
of communalities H2 is to one.
In estimating Ap, usually the main component method is employed, the idea of which
 is the following. Since Rp is a real symmetrical matrix, then by the orthogonal
similarity transformation, it can be reduced to a diagonal type
B iRpB = L, (2.160)
hence Rp = BLB', or due to the orthogonality of B
Rp = BLB' 1; (2.161)
here Lthe diagonal matrix comprised of the characteristic roots of Rp considering
their multiplicity; Bthe orthogonal transforming matrix the columns of which are
the eigenvectors Rp which transform the orthonormed system; B1the ma trix inverse
to B; B'the transposed matrix.
From (2.160) we have
Rp = BL1/2L1/2B1 = ApA'o ~ (2.162)
hence Ap = BL1/2, where L1/2the diagonal matrix from the square roots of the eigen
numbers.
The solving of the equation (2.162) of the Ro matrix also comprises the most essen
tial part of the calculation procedures in factor analysis. The geometrically de
scribed transformations are the equivalent of rotating the initial system of coor
dinates in such a manner that the new base axes coincide witt, the symmetry axes (the
main axes) for the distribution of vector X.
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Depending upon the method of determining the communality estimates, a distinction
is drawn between two variations [16]: 1) the communaltiy estimates are considered
equal to one, and this is the socalled closed factor analysis model; 2) the commu
nality estimates are taken below one, calculating them fram empirical data (an open
model). Let us examine a closed model as a simpler method which has proven itself
in a number of practical problems [17].
After determining the factor loads which correspond to the aggregate of unobserv
 able variab les (factors), usually an attempt must be made to interpret them, that is,
a certain useful and generally accessible interpretation of the essence of various
~ aspects of a complex phenomenon as reflected by the isolated factors.
Due to the f act that the procedure of obtaining the loads in fac;:ir analysis does
not lead to a uniform result (with a number of factors greater than one), it is pos
sible to ob tain equivalent sets of loads by their orthogonal transformation. Geo
metrically this will correspond to an additional rotation of the factors in the
measurement space.
As the criteria for locating the optimum (in the sense of interpretation) position
of the factor axes in space, rather many proposals are known. The varimax criterion
has proven ef�ective in a number of actual studies [24, 26] and the sense of this
comes down to reducing the factor loads to the simplest type. The simplicity Vj of
any factor is determined in the given instance as the variance of the squares for
the corresp onding factor weights:
:
V, (n~!)2 Z afl l~ ]In
Z
.
iThe varimax criterion consists of demanding the maximization of the sums
V= ~ V/� max.
 ~
_ For obtaining unbiased estimates,2the values aij are normed by dividing them i:~to
the corresp onding communalities hi. The final varirnax criterion is determined by
the ratio
_ V fn (n;1/It~)2  ( Y arI/h~ 11 ]In ~ J 4 max, 1 (2.163)
while the s olution is written in the form A= AOT, where A0the matrix of factor
weights ob tained by the rnain component method; Tthe orthogonal transforming matrix
= selected in such a manner that the simplicity V of matrix A is maxima,l.
v In the described method the most labor intensive part is the calculatinf; of the
eigenvector s of matrix Rp using the main component method. At present a number of
= machine programs are known realizing the presented method and its modifications.
One such program (compiled by Ye. Yenchenko) is described in [24]. It has been
_ used in ana lyzing statistical complPxes which describe, for example, such an object
" at the network of USSR trunk air routes. In working out the development forecast
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_ for the demand for air passenger traffic in [33], a set of inethods was used includ
_ ing trend extrapolation for short lead times, expert questioning to discover pos
sible shifts in demand, and in addition, factor analysis was used for the network
of Soviet trunk air routes.
A fragment of the network was studied consisting of 12 air routes holding the first
places among the nation`s routes in terms of the passenger traffic volume. As the
characteristics they took the following statistical data on: 1) the amount of an
nual passenger air traffic; 2) the amount of passenger traffic by rail between the
end points of the connections; 3) the population dynamics of the cities which were
~ the end points of the connections; 4) the ratio of the amount of per capita nation
al income to the air fare for the route by years. Thus, the examined stochastic
complex was represenxad by an aggregate of values for 48 variables.
The computer calculation following the described method (carried out by V. L.
Gorelova) showed that the correlation matrix has a very high "rigidity" estimate
(0.797) and the number of essential connections is 63.5 percent with a significance
level of P= 0.01. The calculating of the factor load matrix with the subsequent
rotating of the main factors in accord with the varimax criterion (2.163) made it
pussible to isolate three ma~n factors with a total contribution of 90.3 percent to
the generalized sample variance. The interpretation of the isolated factors led to
their following exglanation in accord with the distribution of the factor loads.
The f.irst f actor (77.2 percent of the generalized sample variants), as in a majority
of factor research, has high loads for virtually all indicators, and somea.*hat
greater for the indicators of the f irst and fourth groups and somewhat less for the
indicators of the second and third groups. It can be interpreted as the generalized
main factor for air passenger traff ic. The second factor (7.8 percent) can be in
terpreted as the rail factor, since the greatest factor loads in it occurred as an
average in the area of the second group of indicators. The third factor (5.3 per
cent) can be termed the population factor, since the greatest factor loads occurred
for it in the area of tha third group of indicators.
The results of the experiment showed how effective factor analysis is as a tool in
studying multivariate stochastic objects. It not only makes it possible to signifi
cantly reduce the dimensionality of the description (from 48 to 3 in the designated
' example) without essential losses of accuracy but also isolates the main factors
which disclose the basic driving forces of the process an.d these can rather easily
be interpreted from the positions of human, logical understanding of its essence.
Having examined the use of factor analysis in the retrospection and diagnosis stages
in forecast research, let us take up the possible method of applying it in the stage
of the immediate elaboration of the forecast (the prospection stage).
The essence of the method of research forecasting based on factor analysis can be
presented in the f ollowing manner.
According to (2.155) and (2.156)
X = AF + U ,
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where U= I H2the matrix of specific factors and errora which is closer to zero
the closer the cocmnunality matrix H2 is to one.
In the closed model which we have used, it is assumed that H2 is a singular matrix
and, consequently, it can be considered that
X = AF.
(2.164)
By retrospective analysis of the ndimensional random vector X the matrix was de
termined for the f actor loads A with a dimension of n x n. In examinir.g this matrix
it was discovered that only a sma.ll number of factors k 5. Then the factor standing in column ,j is given the estima.te yji = 10  yij .
Here the number of points are set in who le numbers. After carrying out the expert
evaluation, the tab le's data are genera lized. For this the total number of points
P
is determined as set by expert k for each line i yi yipx; the total nLmber of points
ra~
P
set by expert k for each column j~y,,K; the resulting total of points Jl~K�
The weight coefficients set by expert k for each factor i are determined from the
following formula
~rK ='Z yr," I ~Z /Z y~,K�
(4.17)
The mean statistica 1 value of the weight coefficient, in being the indicator of the
generalized opinion of the experts, is d etermined from the formula
`m
f'IK//1!.
(4.18)
As we see, the procedure for calculating *_he weight factors using a 10point scale
is greatly simplif ied. At the same time, the use of this merhod requires additional
procedures for estab lishing the reliabil ity of the expert estimates as was poin ted
out in Chapter 2.
The elucidating of the factor priorities from the standpoint of their imgact on the
cost estimates using the just described methods, in addition to the procedural
 difficulties, entai ls also a number of logical and psychological difficulties. The
latter are determined primarily by the absence of causeandeff ect relations between
the factors and the cost estimates.
As was shown in Sec tion 4.1, the opera.ti onal effectiveness factors influence th e
cost estimates of the system functional elements not directly but rather indirectly
through a multiplic ity of intermediate characteristics. Thus, the operational ef
fectiveness factors influence the design characteristics of functional elements and
the physical properties of the employed materials. The latter determine the state
of the production pr ocesses, the means and implements of labor and this, in turn,
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Table 4.6
f
( faeterc
actorsl
"
total
eight
1
~
2
I
p
I
points
coeff .
f ar i
factor
Nao
n
~
oi/
2
y:i
ys,
dto
P
Z uo/
03/
I
I (
I I
I
I
i
Jii
yt,
yi,
yin
P
2j NI/
01/
(
.I
I
I
I
I
I
P
Jri
yrs
yvl
D
~j ya/
~
on/
total
points
P
~ Jt
v
~ Jft
P
~j llti
P
~j yio
r v
I I yi/
influences the labor intensiveness of the work, the skill level of the workers em
ployed in these jobs, the cost of the employed means and implements of labor, mater
ial intensiveness and so forth. Only the last group of characteristics has a direct
influence on the amount of the cost estimates. Under these conditions the attempt
with expert help to set the influen;,e of one or another functional characteristic of
a system's element on its cost estimates is directly tied to the necessity of over
_ coming the ambiguities concerning the multiplicity of intermediate links, their in
teraction and reciprocal influence. The following circumstance must also be pointed
out. Due to the hierarchical nature of the functional structure of technical sys
tems, the design characteristics and the properties of the employed materials in the
functional elements are determined not only by the parameters of the elements them
 selves but also by the parameters of the higher level element in which the given
system is a subsystem.
In addition, a number of design and materials characteristics can be determined by
the influence of functional parameters in the elements of the same hierarchical
level as the cost estimate's ob3ect. This necessitates the creating of an informa
tion basis and the elaborating of special procedures for an expert estimate of the
weights in the conveYSion factor models. Obviously the information basis which
precedes the expert evaluation should be an hierarchical system of interrelated
characteristics which reflects the mechanism of influence of the employed factors
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on the cost estimates through all the intermediate characteristics. Zn having such
a system it is possible to organize a stepbystep expert evaluation, in estimating
in sequence the priorities betw2en the factors which are on the given level in re
lation to the characteristics of neighboring levels.
Here for the estimates on each level or several adjacent levels for which the esti
mates involve the solving of kindred problems, i.ndependent expert groups can be set
up. Each such group will consist of experts from one or two related specialties and
this will make it possible to substantially inc.rease the reliability of the expert
estimates. The estimate reiiability ca*_: 31so be increased by correctly choosing
the methad of disclosing the prioriti.es since the number of estimated factors, Iike
the number of experts in each individual instance, will not be constant.
The hierarchical system reflecting the mechanism of influence of the operational ef
fectiveness factors on the cost estimates and incorporating the main intermediate
links between the characteristics of the various cosubordination levels has beer_
worked out by the authors with the participation of the engineers Yu. A. Teplov and
V. I. U1'yanov. This system called an influence graph for the operational effec
tiveness factors on the cost estimates is shown in Fig. 4.8.
As is seen from the diagram, on the first and second levels of the hierarchy there
are characteristics of the technical system in which the assessed functional element
is a subsystem. Here it is shown that the desi.gn characteristics of a functional
element and the properties of the employed materials are determined not only by the
technical, operational and functional characteristics of the element but also by the
system's analogous characteristics. 'Then the diagram shows the successive influ
et1ce of the design characteristics of the functional elements and the properties of
the employed materials on the complexity characteristics of the production processes
(for example, the method of obtaining stock, the machining method, the method of
connecting the elements and so forth). The designated characteristics of the produc
tion processes, in turn, are related to the basic expenditures which determine the
production cost of a unit of the base parameter for the functional element.
The procedure of expert estimates made using the influence graph should consist of
disclosing the specific influence of the upper level characteristi.cs on the infer
ior level characteristics for all interrelated groups. Here the general influence
of one or several upper level characteristic groups on each group of the related
lower level characteristics should equal one. It is not hard to see that all the
characteristics are interrelated by causeandeffect ties and this sharply reduces
the degree of ambiguity on the intercausality of the factors.
In addition, the link between any two groups of factors can be established by an
isolated independent expert group. The latter provides an opportunity to set up the
_ necessary number of uniform expert groups, each of which will consist af the repre
sentatives of related specialties. Thus, the relationship between the functional
characteristics of the system and the element can be set by specialists in systems
theory; the relations between the functional and design characteristics and the
physical properties of the employed materials will be set by designers; the rela
 tion between the physical and production properties of the materials by production
engineers and so forth.
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15 fPlldoBoie u e+amepuoanncre 3ampumo,
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Fig. 4.8. Influence graph of operational effectiveness factors
 on cost estimates
ICey: 1Operational effectiveness factor; 2System's technical
 and operating characteristics; 3System's functional
characteristics; 4Technical and operational character
istics of element; SFunctional characteristics of element;
6Scrength characteristics of element; 7Design charac
teristics of element; 8Physical and chemical properties
of materials; 9Characteristics of production processes;
10Technological properties of materials; 11Labor in
tensiveness; 12Worker skill level; 13Cost of materials;
14Consumption of materials; 15Labor and material ex
penditures; 16Production cost of a functj,onal element
for system
The organizational diff iculties of conducting such an expert evaluation are appar
ent, however there obviously is no other reasonable alternative for constructing the
conversion factor models. At the same time the conversion factor models are a very
valuable and at times irreplacable economic forecasting tool since, as a rule, the
number of the characterisCics of the technic^?. systems influencing their cost esti
mates is great while the number of prototypes comprising the initial statistical
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aggregate in the majority of instances is insignificant. This circumstance i.mposes
constraints on the possibility ot constructing the statistical dependences which
will be taken up in the following section.
 4.2.3. Interpolation and Extrapolation Methods for Statistical Depeiidences
 The methods of interpolatiun and extrapolation of statistical dependences are based
on the assumption that there are quantitative relations between the value estimates
of the BTS functional elements and their characteristics as well as the basic fac
tors describing the state of the individual stages of the BTS life cycle and the
environment. The task usually is to disclose these ties, to select the main factors
out of the multiplicity of them, to localize the ties which cannot be estimated
quantitatively and to coordinate the entire aggregate of determining factors by a
mathematical model which would reflect the basic patterns of the studied phenomenon.
The models which satisfy these requirements have been named the mathematical eco
nomics cost models and at present are the basic forecasting tool for the cost esti
mates.
The essence of forecasting using the mathematical economics cost models is that by
the statistical dependences which reflect the influence of the indicators of inter
est to us on the cost estimate, the probable changes of the cost estimates arP de
termined when the in.dicators assume new values different from those which were ob
served in the initial statistical aggregate. Here, if the new value of the indi
cator does not go beyond the limits of the observed range of changesy the forecast
is of an interpolation nature. Otherwise the set statistical dependence between
the cost estimate and the changeable indicator is extrapolated for its new values.
_ In the BTS economic forecasting problems, of greatest interest is the extrapolation
of statistical dependences since the characteristics of developmental processes
_ evolve. However in the problems af optimizing the BTS parameters, the need arises
also for interpolation calculations, if the optimizatian process is carried out
under a certain compromise scheme. In this instance, the optimal variation of a
system can include functional elements the individual parameters of which will be
somewhat below the parameters of the nearest prototypes. In order to assess the
� expenditures on such functional elements, an interpolation of the statistical de
pendences is employed.
The mathematical economics cost modzls can be of two types. The model can be repre
sented by one general multiple regression equation reflecting the influence of the
entire aggregate of factors simultaneously. This is the simplest method of model
ing the cost estimates and it provides very rough forecasts. More dependable fore
casts are achieved by employing composite, mixed models based upon factor modeling
of the cost estimates. The factior modeling method consists in modeling the influ
ence of the individual groups of related factors on the cost estimates separately
in accord with the place, time and nature of th.is influence. The thereby obtained
local modela or submodels are brought together according to definite rules into the
overall mathematical economics model.
The use of factor modeling methods becomes possible with a sufficiently full and
reprasentative amount of statistical information encompassing the various aspects
in the developmental process of the cost estimate's object over large intervals of
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time. Since these conditions are not always feasible, at times one must resort to
multiple zegression equations.
At the same time it is essential to point out that reg3rdless of what the final fdrm
is for showing the mathematical economics model of the cost estimates or the form of
the multiple regression equation or composite model, the modeling of th: cost esti
mates must start by dividing the general process of forming the cost es+_imates into
individual fragments, the elucidation of the composition of influencing factors and
the establishing of quantitative indicators reflecting the influence of the factors
on the cost estimates.
For this reason the constructing of one or another mathematical ecanomics model for
cost estimates must be carried out according to a general scheme (Fig. 4.9). The:
entire sequence of events represented in the basic block diagram for modeling the
cost estimates can be divided into a series of independent stages: logical modeling
(blocks 12), the forming of the initial information file (block 3), logical
statistical analysis or selection of influencing factors (blocks 45), mattiematical
modeling (block 6), forecasting (block 7) and forecast verification (block 8).
The logical modeling stage envisages the carrying out of preliminary research aimed
at showing the possible states of the cost est;.Lmate's object, the stages of it:s
life cycle and of the environment. In this stage a logical scheme is constructed
making it possible to compile a general notion about the formative mechanism of the
cost estimates under the influence of the developmental processes.
 The logical modeling of the states of the cost estimate's object (block 1.1) is car
_ ried out in the following sequence: the class of systems is established which are
tn include the cost estimate's object; the system prototypes are selected which are
identical in terms of their basic purpose and performed functions; the system is de�
 composed with a model being constructed for the internal structure and the place and
role of the object in the process of the system's functioiling are determined; the
parametric series of the object's prototypes is set; the basic functional character
istics of the object, their internal and external relations are set; the object is
_ decamposed with its schematic diagram, composition and purpose of the basic struc
tural elements and other design characteristics being determined; the relations are
established between the basic functional and design characteristics; the object's
model is constructed showing its structure, internal and external relations.
In analyzing the functional characteristics, particular attention must be paid to
their ties with the elements of the same and adjacent hierarchical levels of the
sysr_em. This is essential for fully encompassing the factors which influence the
development of the cost estimate's object, as the latter is often determined pre
cisely by the external reZations. It is very impartant ta trace how these relations
influence the design characteristics as the design features of the object ultimately
have a direct impact on the cost estimates (see Fig. 4.8). The logical modeling of
the studied life cycle stage of the cost estimate's object (block 1.2) includes a
solution to the following basic problems: constructing a model for the internal
structure of the process and reflecting the basic developmental stages of the object
i in the given life cycle stage; thE establishing of the basic characteristics in the
state of the processes inherent to the given stage and their quantitative estimate;
cozstructing a logical model reflecting the reciprocal influence of the state char
acteristics of the life cycle stage.
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modeling and forecasting cost estimates for the BTS functional elements
[See the key on foilowing page.]
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_ [Key to Fig. 4.9 on preceding page]
 1Logical modeling of state of development processes; l.iModeling state
_ of cost estimate object; 1.2Ncodeling state of stage of object's life
cycle; 1.3Modeling state of environment; 2Logical modeling of forma
tive process of cost esr_imates; 2.1Modeling influence of object's
characteristics; 2.2Modeling influence of characteristics of stage;
2.3Modeling influence of environmental characteristics; 2.4Modeling
combined inf luence for characteristics of developmental processes;
3Forming the file of statistical information on the developmental
prehistory of the cost estimate's object; 3.1Determining information
sources; 3.2Elaborating forms of information carriers; 3.3Collection
and systemati.zation of information; 3.4Assessing reliabi.lity and uni
formity of information; 3.5Correcting and adjusting information;
4Logicalstatistical analysis of characteristics in developmental
process; 4.1Analysis of state and development of object's character
istics; 4.2Analysis of state and development ef characteristics for
life cycle stage; 4.3Analysis of state and development of environmental
characteristics; SLogicalstatistical analysis of formative process of
cost estimates (selection of influencing factors); 5.1Research on in
fluence of state characteristics of life cycle stage; 5.2Research on
iZfluence of environmental characteristics; 5.3Research on influence
of object's characteristics; 5.4Research on combined influence of
characteristics of developmental processes; 6Elaboration of mathemati
 cal economics model of cost estimate; 6.1Elaboration of model reflecting
 relation of cost estimates with object's characteristics (first submodel);
6.2Elaboration of model reflecting relation of cost estimatas to char
acteristics of object and li.fe cycle stage (second submodel); 6.3Elabor
ation of model reflecting relation of cost estimates to characteristics
of object, life cycle stage and environment (integral model); 6.4Analy
sis of models and setting of constraints on change in variables; 7Fore
casting of cost estimates; 7.1Extrapolation and interpolation of depend
 ences; 7.2Setting of confidence intervals for forecast estimates;
 8Verification of forecasts
The internal structure model of the process characterizing the designated 1{.fe cycle
stage should have sufficient detailing in order to isolate the permanent elements
in the process which do not depend upon the properties and particular features of
the cost estimate's object or upon the elements the composition of which varies
from object to object. This will make it possible subsequently to isolate from the
totaZ expenditures th3t portion which is most related to the change in the charac
teristics of the cost estimate's object.
The logical modeling of the state of the environment (block 1.3) envisages the fol
lowing: establishing the sources and nature of the external effects; estab.lishing
the characteristics of the state of the sources and a quantitative estimate of
these characteristics; constructing a model which shows the links of the environ
ment with the characteristics of the developmental processes. These problems are
solved separately for the cost estimate's object and the stages of its life cycle.
Such a dividing makes it possible to form a clearer notion of the mechanism of the
environment's influence on the cost estimates. This ultimately facilitates the
choice of the indicators which ref lect this influence in constructing the cost
model.
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The concluding phase of the logical modeling stage is the constructing of a hypo
thetical model for the forming of the cost estimates in each life cycle stage of
: the system's functional element (block 2). For this purpose one first constructs
_ isolated models (blocks 2.1, 2.2, 2.3) each of which reflects the influence dis
closed in the previous stages on the cost estimates for the characteristics of the
object, the life cycle stage and the environment.
As a result of uniting these models, an integral model is constructed for the form
ing of the cost estimates (block 2.4). The integral model should not be the mere
total of the isolated models. In constructing it it is essential to consider the
interrelations between the characteristics of the different groups of factors in
forming the cost estimates.
Regardless that the integral model is to a certain degree approximate it serves as
 a good basis for further research. Using it it is possible to establish a prelim
inary list of problems wl:ich the researcher wi11 encounter in the pracess of con
 structing the cost model. Mos t importantly, the logical modeling stage makes it
possible to draw up a list of characteristics from which it is possible to start
forming tY:e statistical informa tion file on the developmental prehistory of the cost
estimate's object.
The formation of the file of in itial statistical information (block 3) is an intera
tion process with a direct link and feedback with all subsequent modeling stages.
This stage consists of a number af repeating operations: determining the informa
tion sources, working out the f orms of information carriers, the collection and
systematization of information, assessing the reliability and uniformity of the in
formation, its correcting and the clarification of the block (3.13.5).
Tlie iterative nature of the pro cess of forming the data file is exp.lained prlmarily
by the fact that the logical modeling makes it possible to obtain only a rough
sketch of the scheme for forming the cost estimates. As a consequence of this, the
data content corresponding to the initial list of characteristics is subsequently
adjusted as the research is deepened. Moreover, such operations as assessing the
reliability and uniformity of the information can be carried out only in the process
_ of statistical analysis or in constructing the cost models. Due to the designated
factors the formation of the initial information file can be considered complete
after the final variation of the model is obtained.
In the general instance the f i le of initial statistical information is a time
systematized series of prototyp es for the system's functional element each of which
rias its corresponding actual cost estimates, functional and design parameters, as
well as characteristics of the studied life cycle stage and the environment. The
designated characteristics ref lect the state of the developmental processes for the
cost estimate's object over the examined period of its prehistory. In possessing a
sufficiently representative series of state characteristics it is possible to move
 on to studying the process of f.orming the cost estimates.
However, it is advisable beforehand to carry out a logical statistical analysis
of the very characteristics of the state of the object's developmental processes
(block 4), bearing in mind the realization of the following basic goals: estab
lishing the integral quantitative indicators which accumulate the influetice of the
 corresponding group of characte ristics on the cost estimates; dividing the overall
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aggregate of characteristics into dynamic which have certain time trends and the
stationary; the static which are not related to the developmental processes; the
continuous and the discrete. Tue first is essential in order to narrow the range
of examined indicators and to level out the existing contradictions between the in
dividual characteristics; the latter is required in order to clarify the basic
tasks and to choose the research methods.
The integral indicators can be obtained directly by the correlation and regression
analysis methods (see Section 2.5) and on the basis of expert estimates. In the
latter instance the integration of the characteristics can be attained by employing
estimates for the relative weights ot the individual characteristics. Here two
methods can be employed for constructing the integral indicators: additive and mul
tiplicatiye. In the first instance the integral indicator's model has the form of
the weighted total of the individual charactaristics:
m
PE _ i1 prPce
in the second, a weighted product
pi: = ri p"~.
i~
(4.19)
(4.20)
where pithe normed value of characteristics i;2
ftthe relative weight of ciaracteristics i in the process of forming the
cos t es tima tes f r~ (it = I~�
For obtaining the integral indicators it is essential that the synthesized charac
teristics be expressed using a finite number of quantitative measurements. How
ever, in a number of instances the state characteristics of the developmental proc
ess reflect purely qualitative features which cannot always be reflected by a finite
number of quantitative estimates. Moreover, it is essential to bear in mind that in
forecasting the cost estimates on the basis of integral indicators, the need arises
of compiling forecasts for each synthesized characteristic. This r_an cause insur
mountable difficulties if special models and methods are not worked out for fore
casting the individual state characteristics of the developmental processes. For
this reason under certain circumstances, for simplifying the procedures of cost es
timate forecasting, unformalized methods can be used for choosing the integral in
dicators. These methods are based upon the propoaing and subseqt;ent checking of the
_ logical hypotheses about the relationships of the individual characteristics and
their impact on the cost estimates.
~ In using the unformalized methods for synthesizing the characteristics, the use of
the following basic rules can be helpful: the movement from cause to effect, since
_ LOne of the passible methods for norming the characteristics is given on page 109.
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the latter can be determined quantitatively; the movement from effect to cause if
, the same conditions are satisfied; the choice of indicators which determine certain
states of the examined processps but are not related to the latter by causeand
effect ties; the localization of the influence of factors which cannot be estimated
_ quantitatively by the creation of isolated models; the establishing of indicators
for adjusting the nrodels. Let us illustrate the use of these rules in the given
sequence using specif ic examples.
The characteristics of the state of the environment (block 1.3, Fig. 4.9) tn which
the macroeconomic factors belong, include: rhe level of the material and technical
base of the national ecor_omic sectors; the industrial and national economic manage
ment system; the level of specialization and cooperation, the structure and prin
ciples for locating the industrial sectors.
In the process of developing the BTS, all factors progress, that is, the material
and technical base of the sectors is improved, new highly productive equipment and
new production processes are introduced, the level of automation and mechanization
rises, the forms of the organization of production and labor, the management levels
and so forth are improved. In accord with the changes in domestic and foreign
policy, the state confronts the national economy with new tasks the implementing of
which necessitates a reorganiza',:ion of the management system.. New industrial sec
tors arise, while the forms of specialization and cooperation, the structure of the
sectors, the principles of their location and so forth change.
A quantitative esCimate for the state of even one aspect of this process would re
quire the use of more than a score indicators. Even ifl this instance there would
be no absolute certainty that all the particular features of the influence of these
factors ar: the cost estimates had been considered. The elaboration process and the
assessing of the adequacy of the integral indicator's model require a great deal of
time and effort and the model is obsolete before it has been obtained.
Moreaver, its use for forecasting purposes will be of dubious value as it is essen
 tial to elucidate the state of the entire aggregate of indicators over the fore
casted period and this requires the presence of forecast models for each indicator.
At the same time, a change in the macrovariables can have a noticeable impact on the
forecasting results of the cost estimates even over a short period of time. The way
out of this situation can be found if one expresses the influence of the macroeco
; nomic factors by the consequences of r.::ose processes the action of which they re
 flect.
The processes af scientific and technical development in the national economic sec
tors lead to a rise in social labor productivity and this ultimately is expressed in
a decline in the cost of industrial products. Consequently, the introduction of an
_ indicator for the reduction of costs into the general mathematical economics model
will make it possible indirectly to consider the effect of the macroeconomic factors
on the cost estimate level in the forecasting. Thus, the action of the factors
which act as a cause of a certain process can be considered b}� using an indicator
which reflects the consequences of this process.
The use of the second rul.e is characteristic for modeling a process involving a
change in the cost estimates under the influence of the design features of BTS
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functional elements. Among the design features, for example, oile could put: the
geometric shapes of the design elements, the mechanical properties of the employed
materials, the classes of precision and roughness in piece working, the design
scheme of the article and so forth. Each of the designated characteristics reflects
a certain aspect in the mechanism of forming the cost estimates. However, a whole
series of characteristics reflects the quality features of the object and cannot be
expressed by a finite number of quantitative measurements. Where this is possible,
their number is extremely great. From this it follows that the attempt to synthe
size suc.h characteristics encoutrters as many difficulties as was the case with the
environmental characteristics. At the same time, if one turns to the factors which
determine the need for various design solutions, one will see that they are com
pletely caused by the characteristics of system operational effectiveness.
Thus, an increase in the payload of a passenger aircraft involves an increase in its
overall dimensions and design weight. If constraints are imposed on these character
istics, then the need arises for using lighter and stronger elements making it pos
sible to increase the effective volume of the cargo and passenger cabins. This, in
turn, involves a rise in the mechanical properties of the structural materials, it
complicates the configuration of the design elements and so forth. In precisely the
same way an increase ln aircraft speed requires the use of heatresistant high
 alloyed steels and alloys, a change in the geometric shape, the complicating of
aircraf.t systems and so forth.
TY:us, if an aircraft's basic functional charaeteristics are incorporated in its cost
model, one can thereby establish the influence on the cost estimates of the design
f eatures which ar.e the consequence of a change in the functional characteristics.
The third of the formulated rules for selecting the indicator.s i:; widely employed in
analyzing the microeconomic factors. With the overall development level of the
material and technical base, specialization, conc2ntration and c.3operation of the
serialprodsction and developing enterprises, the s[ate of the production proc:esses
f or the functional elements is largely determined by the scale of their serial aut
put.
For example, wi.th a high general level of inechanization for the production processes
at a specif ir enterprise, the mechanization level for the manufacturing processes of
the cost estimate's objects can be below the average due to the small scale of out
put. Th2 larger the scale of output for the functional elements the more preferred
it is to deepen specLalization, increase the level of automation, the equipping of
the production processes and so forth.
Thus, the scale of production, without being linkEd to the designated factors by
causeandeffect ties, determines the possible states of the production processes for:
the individual subsystems. Consequently, the incorporation of indicators for the
output scale into the model will make it possible to reflect the impact of the cor
responding group of factors on the expenditures level when the modeling of the ef
fect of their direct indicators causes certain difficulties.
If the influence of the individual factors is only qualitative, it is possible to re
sort to isolated modeis. This is how they proceed when one functional element com
bines the functions of several special elements in the system. For example, a
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boostercruise engine combines the functions of the takeoff and main engine. With
the same characteristics such a combined engine raill differ from an ordinary cruise
engine in a number of design and technological features and many of "these features
cannot be given a uniform quantitative esti.a ste. In this in,tance, a mathematical
economics model is worked out separately for each engine subclass: liftoff, cruise
and boostercruise.
The last of the named rules is employed in those instances when the influence of the
 factors is a discrete one, that is, certain conditions influencing the structure and
level of the cost estimates change in abrupt shifts. Thus, in converting to the new
planning and economic incentive system adopted by the September Plenum of the CPSU
Central Committee in 1965, there were changes in prices for many types of raw prod
ucts and uraterials, the sources and forms of paying bonuses to industrial workers
 and so forth. In ehis instance the mathematical economics models obtained on the
basis of retrospective i*formation and encompassing the period preceding the change
over of the enterprises to the new system would not be able to reflect the particu
lar features of that period for which the cost forecasts were being made. From this
viewpoint, forecasting accuracy could be increased by incorporating in the model cor
rection factors which would consider the corresponding changes in the cost estimate
structure.
From what has been stated it can be seen that the stage of selecting the quantita
tive indicators which reflect the inf luence of factors that determine the basic form
ative patterns of the cost estimates is linked to carrying out piofound comparative
and semantic analysis of inforu!ation about the actual expenditures, to studying the
particular features of the extant prototypes of the object and to studying the
organizationaleconomic conditions of thei;r creation, production and operation as
well as the environment characteristies. The results of this complex of research
are represented by a set of hypotheses on the nature of the impact of the designated
factors on the cost estimates. The advanced hypotheses require an experimental
verif icat ion.
The hypotheses are verified by the methods of mathematical statistics and probabil
ity theory. This stage is called selecting the influencing factors (block 5). Be
low we give the procedure for selecting the influencing factors for constructing
cost estimate models in the form of multiple regression equations.
The choice of the influencing factors consists in establishing a certain range of
quantitative indicators which during the prehistory period of the object's develop
ment had a determining influence on the process of cost estimate formation. Here
it is assumed that the prototypes of the functional element represent a particular
selection from a certain general aggregate of elements the state characteristics of
which are distributed normally relative to their averages X1, X2, ~..,2XP, and 2hat
the spread of these properties can be described by the variances Q1, v2, oP,
where pthe number of indicators describing the entire aggregate of studied state
characteristics.
If the expenditures in the general aggrPgate are distributed normally with the aver
age X1 and the variance Qi, then the essential influence is considered to be the one
_ of that indicator out of the total set X2, X3, Xp_1, the variance of which ex
 plains a certain portion of the general variance Q1. However due to the absence of
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data on the general aggregate, under real condition~ onj operate2 with the estimates
of the averages xl, x2, xP and the variances S1i S29 SP calculated on the
basis of sampling data.
In this regard an estimate is introduced for the essentiainess of the observed sta
tistical relations, that is, an estimate of to ;ahat degree the relationships estab
lished on the basis of sampling research refiect the state of the general aggregate.
The estimates of essentialness or reliability are made with a certain predetermined
~
degree of conf.idenr_e in the correctness of the advanced hypotheses. This confidence
is expressed numerically by the probability that our estimates will be within the
limits of a certain confidence interval the width of which depends upon the amount
of the mean square deviation S, the size of the sampling n and the set probability
P, and does not depend upon the shape of the distribution curve of the sampling data
[30]. In particular, the true mean of the general aggregate is estimated by the in
terval
X  tg {!n < X < X ly n , (4.21)
where the criterion for estimating the significance of the xandom value X with
the se~level of fiduciary probability 01) and the number cf degrees of freedom v=
n1.
Formula (4.21) is read as follows: with a probability equal to,T, it is possible to
assert that the mean general aggregate X is within the limits k� ty(S/n). The use
of the significance estimate criteria plays a large role in studying the process of
expenditure formation as it makes it possible to isolate the regular from the random.
Before moving on to an estimate of the quantitative effect of the indicators chosen
on the basis of preliminary analysis, it is essential to make certain that the exam
ined aggregate of objects is uniform from the viewpoint of the quality features
which can be described by quantitative indicators. For example, it is essent;al to
establish how much the cost estimates are influenced by the des3.gn and production
features caused by the combining of the functions of liftoff and cruise engines into
one propulsion unit. In fornal terms it is essential to answer the question: are
the liftoff, cruise and boostercruise engines a part of a general over.all aggre
gate from the viewpoint of the influence of their design and production differences
on the level of the cost estimates?
Since Pach of the designated engine varieties is characterized by certain expendi
tures, the task is to disclose how essential is the difference between the expendi
tures on each engine group. For these purposes we can employ the method of estimat
ing the differences between average values [30].
The essence of this method is in testing the hypothesis that the two independent
particular aggregates with a volume nl and n2 have been taken from the same normally
distributed general aggregate having a mean value x and a variance Q2. If this is
the case, then the difference between the particular values xll and x12 should not
differ substantially from zero. This socalled zero hypothesis is tested with the
Student criterion t:
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xis ni"s
nl~n3 ~ t;~'' . (4.22)
where xll and x12average expenditures for each of the compared engine groups;
ssquare root of the full estimate for the variance of the differ
ence xll and x12.
The criterion tI? is found from tables with a number of degrees uf freedom v=
nl +n2  2. If the condition of (4.22) is not satisfied, it follows from this that
the influence of the design and production features of the compared engine varieties
on the cost estimates is so great that their joint study can lead to distorted ideas
about the influence of the remaining factors on. the cost estimats formation process.
In addition to the described method, similar problems can be solved using rank cor
relation methods (see, for example, [47]).
 The correspondingly grouped prototype. aggregatQS for the cost estimate object are
subsequently studied for the purpose of establishing the quantita t ive ties between.
expenditures on each stage of their life cycle and the quantitative indicators re
flecting the inf luence of the individual groups of factors.
The relation between expenditures and any indicator reflecting the influence of one
or a certain aggregate of factors is established by paired correia tion coefficients
, for linear relations and by correlation indices if the relation is nonlinear (2.84).
In the latter instance it is also possible tn use a correlation c oefficient but with
the stipulation that the natural variables are replaced by their nonlinear functions
(see Section 2.5.2).
The significance of the correlation coeff icients and indexes is es timated from (2.86)
or (2.87). If the conditions of (2.86) and (2.87) are satisfied, the influence of
the given indicator on expenditures is considered significant. However, under real
conditions one must deal not with one but with several indicators. Here it is es
sential to bring out which of the designated indicators has had a significant im
pact on the expenditures during the prehistory period of the system's development.
The basic difficulty in solving this proL3em is that between the indicators them
selves there are definite ties called a covariation of variables [47]. Thus, there
is a relation between the speed and range of an aircraft, the weight and power of a
radar, the specific thrust and specific weight of an engine and so forth. In all
instances it is important to determine to what degree one or another indicator in
fluences the expenditures if the remaining ones are fixed. For this purpose a par
tial (pure) correlation coeff icient (or index) is used:
~ r,o:a~...tnurYV.3t...cP1i
 ri2.31 ...n 1 (4.23)
Y~~  ~ip.~...(v~l~  r4P.3/...IPUJ
, where r12.34.., pPartial correlation coefficient between indica tor xl and x2 with
x33, x49 xp as constant.
The partial correlation index is employed in the case of nonlinear relations and is
f igured using the same f ormjla under the condition that the variab les are expressed
by the corresponding functions.
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The significance estimate for a partial cozrelation coefiicient ox index is carried
out using the Fisher z ceiterion
z=I 111 ~F' ria.u...v 2  f12.34 p > zo,oe (4.24)
with degrees of freedom v= n 2(p  1) [ 47
 This criterion makes it possible to select out of a multiplicity of indicators those
the influence of which on expen3itures are determining and to begin to construct
multiple regression equations (block 6).
For modeling cost estimates it is possible to use�linear and nonlinear multiple re
gression equations of the sorts (2.119)(2.122). Thus, cost estimate modeling neces
sitates the choice of a form of relation between the variables.
The nonlinearity of a majority of dependences between the cost estimates and the
 factors of their formation is a general logical prerequisite for choosing the type
 of mathematical economics cost model. This is due, in particular, to the existence
of sensitivity thresholds in the cost estimates to a change in individual indicators.
For example, an increase in the production scale of functional elements leads to a
decline in their production cost. However the rate of this decline is not con
stant, since however great the scale of output costs always maintain a certain value
which diffe.rs significantly from zero. Moreover, the sensitivity of the cost esti
mates through an increase in the functional characteristics of the system elements
rises as the characteristics approach their limit amounts.
 The list of arguments in favor of nonlinear models could be extended, however this
is better put off until the following sections where we will examine the particular
features of forming the cost estimates by the life cycle stages. But here we would
add that in employing general multiple regression equations as a model of the cost
estimates the nonlinear models are preferable, since they are capable of simultane
ously describing linear and nonlinear relationships. For example, if the actual
relation between variables is a linear one and a logarithmically linear model (2.120)
has been chosen, the parameters of the variables linked to the cost estimates in a
linear manner will be close to one. Thus, if in (2.120) the parameter b2 = 1 and
the model has the form
xl = blx2x33x44,
then this shows that the relationship of the variables xl and x2 with the fixed xg,
x4 is a linear one. Of course it must be remembered that here there is a certain
simplification of the relationsnips. Moreover, what has been said cannot be extended
 to models (2.121) and (2.122). For this reason the final 3udgment about the type of
dependence must be made on the basis of the statistical criteria for estimating the
model after calculating the parameters of the regression equations (see Section
2.5.2).
The examination of the methods for constructing and estimating the multiple regres
sion equations shown in Chapter 2 makes it possible to point up certain properties
of them which must be considered in modeling the cost estimates.
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1. The relationship between the dependent variable and each indep2ndent variable
is described by the same mathematical function.
2. The number of model variables is limited to the quantity of objects for which
statistical information exists (to the volume of the initial aggregate). Thus,
from (2.138) it can be seen that a ratio p.6, n 1 should be maintained between the
number of variables p and the volume of the initial aggregate n. Otherwise no con
clusions can be drawn on the model's reliability.
3. An increase in the number of independent variables leads to a drop in the model's
sensitivity to a change in each individual variable, since the overall variation of
the dependent variable is broken down into eversmaller components. From (2.130) it
can be seen that the multiple regression coefficients are linearly dependent upon
the correlation coefficients. The latter, even with the complete absence of a re
lation between the variables, are always different from zero due to the existence
of unobservable estimate mistakes. As a result, with the addition of a new variable,
the amount of each of the regression coefficients will diminish, since the correla
tion coeff icients are interdependent. Moreover, if the dependence between the vari
ables cannot be strictly linearized, the estimates of the mean variances and, con
sequently, the correlation coefficients are greatly biased and this leads to the
distorting of the model's parameters.
The listed properties of regression equations pose the fol3owing basic problems:
the choice of the configuration of the mathematical economics cost model in which
the specif ic features of forming the cost estimates would be reflected by unique
analytical functions; the reduction in the aumber of simultaneously modeled factors.
These problems can be solved using factor modeling, that is, by constructing com
posite mathematical economics models of the cost estimates.
The composite structural model should consist of several submodels, each of which
can function independently under the conditions of the known constancy of the other
inf luencing factors. In this instance each submodel is constructed with certain
 fixed values of the indicators which are independent variables of other submodels
and is expressed by a mathematical function which best corresponds to the character
istics of the described fragment of the cost estimate formation process.
The choice of the influencing factors and the modeling of cost estimates for the pur
pose of obtaining a composite model must be carried out in the following sequence:
a study of.the influence of the state indicators of the life cycle (l.c.) stage; the
eliminating of the influence of local (static) relations; studying the influence of
the state indicators of the environment; eliminating the influence of discrete and
continuously uperating characteristics; studying the influence of the object's char
acteristics; synthesis of the submodels and estimating the accuracy of the composite
model.
In considering that the influence of the individual groups of factors is modeled
separately, the composite model can consist simultaneously of simple and multiple
regression equations which reflect the statistical ties, of time trend models as well
as elements inherent to the comparative models. In other words, factor modeling
makes it possible to employ a wide range of �orecasting methods for the cost esti
mates. For this reason the composite model is the most flexible forecasting tool
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for the cost estimates as it possesses high sensitivity to the characteristics of
the BTS developmental processes.
The concluding stage in modeling the cost estimates for the BTS functional elements
is the setting of constraints on the changes irr the independent variables. The
establishing of constraints for the change of the independent variables is required
to determine the acceptable limits for applying the mathematical economics models
expressed by multiple regression equations in forecasting the cost estimates.
The constraints on the changes in the independent variables can be divided into in
ternal and external. The internal constraints are imposed on changes in the inde
pendent variables within a range of observed values in the initial statistical ag
 gregate, and the external ones are imposed beyond this range. The problem of set
ting the constraints is directly linked to the procedure of forecasting the cost
Astimates and to the calculating of errors and setting the confidence intervals.
For this reason all the listed questions will be examined simultaneously.
As was pointed out, multiple regression equations are determined by breaking down
the general variance of the dependent variable into components, each of which is ex
plained by a variance of a certain independent variable. The mean measure of varia
bility for the dependent variable, with a change in any independent variable, is
characterized by the corresponding parameter of the equation. Each parameter of a
multiple regression equation expresses a quantitative relation between the dependent
and independent variables under the condition that all the remaining variables re
main unchanged.
The designated property
caused by the fact ttiat
ence of other variables
ables is achieved by co
the formula proposed by
ficient
for the parameters of multiple regression equations is
in calculating the amount of a certain parameter, the influ
is eliminated. The eliminating of the influence of vari
nsidering their joint paired distributions as is seen from the
E. Yule [47] for determining the multiple regression coef
bi:.a4 ...p = r,,.31 ...v (F1.31 p
Q ~
O.,d ...n
(4.25)
where r12.34...ppartial correlation coefficient determined from (4.23);
Q2�34���P' 02.34...p"mean square deviations of dependent and independent
variables.
The partial correlation index employed in the case of nonlinear relations is calcu
lated from the same formula under the condition that the variables are expressed by
their nonlinear functions.
From (4.23) and (4.25) it follows that the multiple regression coefficients maintain
their force only within the areas of the joint distribution of the independent vari
ables in the limits of the observed range of their change in the initial statisti
cal aggregate. In other words, the domain of existence of the function expressed by a
multiple regression equation is restricted to the intersection areas of the subsets
belonging to the sets of the possible combinations of independent variables.
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xJ t mKp (XimOr, XPmin)
/ o
/ o ~i o
0
~ ~ ~
0
~ o
o �
xlmin o 0
Iqn0(x;min~Xpinux~
~ua~in XOwar XO
X~
Fig. 4.10. Domain of existence for Fig. 4.11. Domain of existence for
function expressed by regression equa function expressed by regression equa  tion with two independent variables tion with two independent variables
with a oneway link with feedback
Such domains are shown in Figs. 4.10 and 4.11 for equations with two independent
variables with a oneway (Fig. 4.10) and feedback (Fig. 4.11) correlation link with
_ the following conditions: 
The two variables xi and xp are independent variables of the multiple regression
equation xl = bi +bj xi +bpxP; '
The initial statistical aggregate from the data of which the equation parameters bl,
bj and bP were calculated contains n pairs of values of xj and xpi, where i = 1, 2,
n;
The values of the variables lie within the limits set by the system of inequalities
x/ min < .rI < X/ mexi
XP mla < xp G Xp max�
(4.26)
The rectangles shown in Figs. 4.10 and 4.11 have been formed by the intersection of
the verticles with the abscissas xp max and xp min and the horizontals with the or
dinates xj max and xj min� Obviously the area of the rectangles contains a multi
plicity of all the possible combinations of the variables xi and x within the limits
set by the system of inequalities of (4.26), while the areas limitEd by the dotted
lines contain only those combinations of independent variables which are encountered
in the initial statistical aggregate. The multiple regression coefficients have
been determined precisely in these areas.
~
From this it follows that the shaded areas of the rectanbles represent zones with
undetermined (unpredictable) interpalation errors. And the greatest errors in fore
casting must be expected when the point with the coordinates of m(xj; xp) coincide
with one of the critical points mkp and mK p._which are most distant from the center of
of the area of the joint distribution of the variables. Thus, interpolation errors
 can reach very impressive amounts if the appropriate constraints are not imposed on
the changes in the independent variables.
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From what has been said it follows that the constraints defined by the system of in
equalities of (4.26) are necessary but insufficient disciplining conditions for the
interpolation f orecasts.
The disciplining conditions can be considered sufficient if to the constraints of
(4.26) one adds a constraint which would be an area of acceptable values for one
variable when the other assumed a certain new previously unobservable value.
 Such an area can be described if it is assumed that the joint distribution of two
variables which are the independent variables in the regression equation xp and xj
is a normal one.
~
i s l~r~l xp  xp s
f (�CVI xi) 2n 1  i2aXPcrX1 (Ixp )
I
~ Xi  Xi )2 xp _ xp X1 X/
r
ONl (Ixp crxl I
where rcorrelation coefficient for values xp and xj.
In crossing the distribution surface with planes parallel to the plane xpOx~ and
 projecting the sections on plane xpOxj, we obtain a family of similar and uniformly
distributed ellipse with a common center (xp, xJ), the equations of which have the
f orm
s i a�n  ~n ~ xlxl 2 xv  xp xl  XI
7,
u r' [ ( aXP ) + ( axJ ) arv Qx/ (4.27)
 where athe fiduciary probability.
, As a general statistic which is calculated from the values of many variables, it is
possible to use the statistic Ta, which is related to a Fisher d,istribution in the
following manner:
Ta = 2 1)
a~
n2 P (4.28)
where Fa the statistic having 2 and (n  2) degrees of freedom.
` The radiuses of the ellipses will change depending upon the value of the fiduciary
probability a. From equation (4.27) it can be seen that the e]lipse is determined by
five parameters xp, xj, QXj, QXP, r.
The symmetry axes of the el]ipse form with the Oxp axis the angles determined by the
equation
2iaxpax'
tg 2yp  s s � (4.29)
_ aXp  Qsl
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The equation of 2he ellipse assumes a canonical form if the coordinate axes coincide
with the symmetry axes of the ellipse. Let us designate the variables in (4.27):
xp  Xp , tX _ x/X/. � .
xn qxp ~ / _ Qx/
Let us move the beginning of the coordinates to point (xp , x) and turn the coordi
 nate axes to angle as determined by the equation of (4.291. In this case, the
equation of the ellipse will be expressed by the formula
T ' ( ixp I Ixl  Z~xptx,r) _ ~a� . (4.30)
\
In a standardized scale, the center of the ellipse is at the start of the coordi
nates (tx,, = 0, tX. = 0) and the axes of the sllipse are directed along the bisec
tor of th~ coordinate angles: the first and third angles for the first axis and the
second and fourth for the second axis.
The coordinates for the ends of the first axis are:
A, (T. YI 2 Ta 2
_ A,CT~Y~+` T Y~+r
Z ~ T. 2 1�
The coordinates for the ends of the second axis are:
I r 1 r
Bl(T~~ 2, TaV 2
 B2(TaY~ 2 r, TaYF 2 rY
With r> 0, the first axis is the .ajor axis of the ellipse and the second is the
niinor one. The greater irl, the nire the ellipse is extended along the major axis.
If r= 0, that is, the random amounts xP and xj are not correlated, then the ellipse
is turned into the circumference of the radius Tp, and the equation of this circum
2 2
P + tXj = Tp~,.
ference is tX
The transition to the variables xP and xi is carried out according to the following
formulas:
ap =iXDUxp + Xp; Xi  l,clQxi + Xi.
~ Thus, the constraints for a model with two independent variables are determined by
the system
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~ xv mio < xp G zP m.x;
aG/ min < X, < Xi max; (4.31)
11 T a (1
~xp XP F ( xl Xf ) 1 2 xO x0 Xi
O;
P ~s I QxP I
_ In the general case, for n variables, the equation for the function limiting the
confidence arez can be written in the foilowing manner:
. 7.a _ XQ,lXr;
X=IX,XJ. XsXI,...,X�X�).
The matrix XT is the transposed matrix for the X m.atrix
XX,
X r  X,  .X,
X~  Xn
The inverse matrix Q 1 is calculated in the fallowing manner:
s
~xiQ,~l,c7 � � � Qxtxn
QxIxA � � QXn� �
(4.32)
Consequently, if there are n independent variables, then the constraints imposed on
 their change will assume the form:
( Xt min < Xl .4 xl msxi
.YS m1n 4 Xi 4 Xt max;
Xn mIn 'G xn 4C Xn mAxi
14, XQ lXT.
~
(4.33)
In the event of a curvilinear regression, instead of a correlation coefficient a
correlation ratio is used (2.84). The center of the ellipse is located at the point
with the coordinates xPC, x' jc which represent the coordinates for the center of
gravity of the curve and are calculated by the formulas:
6 b
x r xP ~ ~_.1 fXn. ~ r xII + l~xP' x/12dx .
a ,
~ ~ ~ ~
' lc = a b (4.34)
n, b x
J Y ~~(fxy. x~)sd.Y f V~+ (/xp, xl)sdX
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_ The values of the
the given curve.
 is shown in Fig.
xd
xjc
With the formulated constraints on the change in
the independent variables of regression models,
the probability of the occurrence of unpredictable
interpolation errors is minimized.
In observing the disciplining conditions expressed
by the corresponding system of constraints on the
changes in the independenz variables, it is pos
sible to establish the degree of accuracy of the
forecast estimates using the multiple regression
equation (see Section 2.5.2).
The accuracy of forecasting cost estimates is in
Fig. 4.12. (:onfidence area for versely proportional to the error of the indi
curvilinear regression of two vidual pzediction (2.140) and is determined by the
 variables confidence interval of (2.143).
As was already pointed out, the closer the new values of the independent variables
come to the limits of the observed (in the initial statistical aggregate) range of
 changes the greaL�er the forecast errors, since the errors of the regression coeffi
 cients are equal to zero with the equality of the independent variables to their
averages. However, in the space of the observed comb3nations of independent vari
ables, the nature and strength of influence of an individual variable on the cost
estimates are not uniform. As this is so, the greater the danger is that tha nature
and strength of influence of the variable on the cost estimates will change if the
var.iable goes beyond the limits of this space.
The question of to what degree the relations change between the cost estimate and
the lndependent variables cannot be solved by formal methods and for this reason
the imposing of external constraints represents largely a conceptual problem. Its
solution depends to what degree the values of the independent variables differ from
their limit states near which the probability of disrupting the established ties is
increased. In this regard the imposing of external constraints becomes possible if
for each variable values are set for the limit states as their existence is beyond
dispute. Thus, for the BTS elements it is essential to know the theoretically
achievable limits of their functiona.l characteristics. Then, proceeding from an
_ analysis of the time trends of the functional characteristics, it is possible to
formulate the external constraints.
An indispensable condition for formalizing the external constraints should be a suf
ficient distance of the extreme limits of the accepta`ole changes in the functional
characteristics from their limit states. Here an important di.sciplining condition
should be the constraints related to the areas of the reciprocal correspondence of
functional characteristics, that is, constraints of the type (4.27) should be satis
 factory. In particular, in extrapolating a statistical dependence of two independ
ent variables xj and xP (see Fig. 4.10), in assigning new values xP > xp max or
xp < xp min, it is essential that the extrapolation value of xj remain within the
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integration limits correspond to the beginning and end points of
ThQ confidence area for a curvilinear regression of two variables
4.12.
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limits of the conf idence area of (4.27). In observing all the necessary precautions,
the extrapolation mistakes can be commensurable with the interpolation errors on the,
nearby boundaries of the existence domains of functions expressed by the multiple
regression equations. For calculating the extrapolation errors it is possible to usQ
formula (2.140), and for calculating the confidence intervals, (2.143), as an ae
sumption on the possibility of extrapolation in principle presupposes apriori that
the errors of the forecast estimates are unsystematic and are subordinare to a norm
al distributian law.
In forecasting cost estimates using composite mathematical economics models which
are a linear and nonlinear combination of statistical dependences, the total fore
cast errors is calculated from (2.144) and (2.145).
Ttie forecast for the value estimates of BTS functional systems, like any other fore
cast, requires verification. The verific2tion of cost forecasts is also particu
larly essential in extrapolation when there are fears that the established depend
ences can be disrupted.
The verification of forecasts for mathematical economics models can be carried out
using a duplicate forecast made by a different method. For the purposes of verifi
cation of cost forecasts it is most effective to use the conversion factor method if
the integral model of the mean conversion factors includes indicators which could be
selected in the process of logical and statistical analysis of the formative process
of the cost estimates (block 4 in Fig. 4.9). Then the mean conversion factor model
and the mathematical economics model of the cost estimates will be comparable, since
they will contain the same indicators. The difference in the regression coefficients
from the weight coefficients set by experts will indicate the basic sources of dis
crepancies in the results of the forecast e$timates and this can help in determining
the area of search for a better model if the decision is taken to carry out a re
peated cycle of analyzing and modeling the cost estimates. Thus, the use of the
conversion factor method for the purposes of forecast verification, in addition to
carrying out verification per se, can help clarify the mathematical economics models
of the cost estimates.
In concluding an examination of the forecasting methods for cost estimates, we would
like to draw attention to the following. The mathematical economics models, un
doubtedly, are the most objective and flexible tools in forecasting the BTS cost es
timates. In reflecting the general patterns in the change of the cost estimates
under the influence of the characteristics of the system developmental processes,
the mathematical economics models make it possible to assess the consequences and
effectiveness of the decisions taken to control theae prQCesses. Using the mathe
matical economics models, along with selecting the optimum BTS parameters, it is
possible to choose the variations for the processes of creating, serial production
and operation of their functional elements. However, all these merits are realized
only in the instance that the model has been correctly constructed in carrying out
sll the logical and formal procedures examined by us.
The modeling process, as one can see, entails great expenditLres of time and re
quires the involvement of highly skilled specialists. For this reason, as was al
ready pointed out in 4.2.1, the use of mathematical economics models should be
justified by the forecasting goals. Among these goals we would put the choice of
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 optimum parameters for systems, production processes and so forth, that is, the
solving of such problems when it is important to know the influence of one or aa
other parameter on the cost estimates. But in those instances when the influence of
the individual parametsrs on the cost estimates is not ma3or, the time trend extrap
_ olation methods can be saccessfully employed for forecasting purposes.
As for the accuracy of the estimates, as we have seen in forecasting using the '
mathematical economics models, this depends largely upon how correctly the con
straints have been set for changes in the variables of the mathematical economics
_ models. Since the procedure of setting the constraints is not always formalizeable,
there is always the probability of the occurrence of unpredictable errors. Thus
in individual instances the extrapolation errors using mathematical economics models
can be comparable with the errors of time trend extrapolation.
4.3. Basic Patterns in the Formation and the Forecasting Methods for the Costs of
NIR and OKR of Large Technical Systems
Scientific research work and prototype design work (NIOKR) which are frequently
linked by the counon term "creation," are the two most important stages in the life
cycle of a BTS. Precisely here, in these stages, start the processes of scientific
and technical development of systems and these determine the evolution of the ap
pearance of the systems and their functional properties and the means and methods of
materializing these properties in the broad sense of this word. NIOKR includes a
series of events in the system's life cycle from the genesis of the initial idea
(concept) for creating the system up to the construction and development of the
prototype.
Scientific research work (NIR) includes research on the processes of the external
and internal functioning of the systems as well as the physicochemical processes
occurring in the subsystems and functional elements. Along with research on the
systems and their elements, the NIR is carried out for the purpose of seeking out
 new design materials, fuel and other energy resources, production processes and the
methods of organizing, planning and controlling the creation and production of the
systems.
On the basis of the tactical and technical requirements formulated considering the
results of ttie predesign scientific research on the systems, i,heir prototype design
work (OKR) is carried out. The OKR encompasses a range of design work and the
building (manufacturing) and testing of the prototypes of the systems and their
elements.
The increased complexity of scientific and technical problems related to the crea
tion of modern technology is a reason for the constant increase in the absolute ex
, penditures on NIOKR and increasing their proportional amount in general industrial
expenditures. Thus, in U.S. industry in 19401965, the volume of expenditures on
 NIOKR rose almost 200fold while their proportional amount in the iiation's budget
increased from 0.82 to 15.4 percent (Fig. 4.13). Here in the total volume of allo
cations on NIOKR, around 90 percent is taken up by expenditures related to the cre
ation of large technical systems such as: aviation, missile and missilespace com
plexes and thermonuclear weapons.
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zno ~r,T or
In the general NIOKR expenditures, the highest
0o30mNa1ne, .ro MNoKO ,
proportional amount is taken up by OKt. This
~_~y~^�~ 4
2
is explained by the high material and labor
~
pam na NMOXP
3 rs ~
intensiveness of manufacturing system proto
x
~
types, by the complexity of their testing pro
E
E
grams and by other factors. However, a char
~ ' 4
?
acteristic feature of recent years has been
4100.  4_
the more rapid growth rate of outlays on NIR
in comparison with expenditures on OKR. The
~
more rapid growth rate of NIR expenditures.has
i su ~,o ~
been an objective pattern of the presentday
scientific and technical revolution, since the
scientific potential which ensures the in
o:
creased rates and the continuity of scientific
1940 1945 1930 9ss Iseo 1965
and technical development is created mainly as
a result of scientific research.
Fig. 4.13. Dynamics of expendi
The increased outlays on the NIOKR for large
tures on NIOKR in the United States
technical systems, in outstripping the growth
Key� 1Expenditures on NIOKR;
�
rates for NIOKR expenditures in general in
2Proportional amount of
dustrial outlays, gives important significance
expenditures on N70KR; 3Ex
to the questians of forecasting production
penditure growth index;
costs for the NIOKP. of the BTS. The methods
4Proportional amount of ex
of forecasting the production costs of scien
penditures on NIOKR
tific research differ substantially from the
forecasting methods of the cost estimates em
ployed in the remaining stages of the BTS life
cycle. This is determined both by the adopted practice of calculating actual expen
 ditures on the NIR as we11 as by cer
tain characteristic features of sectorial NIR.
NIR is carried out by sectorial scientific research institutes (NII). The end
product of the NII is the solution to a certain scientific problem related to a rise
_ in the operational effectiveness of
the systems, to an improvement i.n the functional
characteristics and design of the system elements and to an improvement in the
processes of creating and producing
the systems and the methods of planning, organ
izing and managing these processes.
Along with theoretical research, the NII also
carry out experimental work which is
done for tha purpose of testing the results of
theoretical research on mockups and
models.
In terms of its character the scientific research work conducted by the NII is
divided into three types:
1) Fundamental research consisting in the solving of broad general theoretical
problems related to the creation of the system as a whole or a range of uniform
articles comprising various aircraft systems;
2) Exploratory (preliminary) research conducted in the aim of disclosing the pos
sibility and advisability of solving various problems at the given moment and choos
ing the most rational areas of research;
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3) Applied Yesearch aimed at solving particular problems involving an improvement
in the quality of designs, production processes, the organization of production for
a certain type of product and so forth.
The exploratory, fundamental and applied research create the scientific potential
for carryingout OKR and serial production of the systems and their functional ele
ments. Here a majority of the NIR results is used in worlcing aut and producing a
series of system generations. Thus at each moment of time the NII are solving
problems related to the development prospects of the sector. The designated diver
sity and perspective orientation of the NIR are the reason that the attempts at
modeling NIR expenditures, depending upon the characteristics of the systems or the
individual stages of their life cycle, have not been crowned by success.
At present the methods of an indirect estimate for the costs o'r NIR have become
widespread. An indirect estimate for the cost of the NIR related to the creation
of BTS presupposes a forecasting of these expenditures proportional to a certain
cost estimate which is sensitive to a change in the system characteristics. Con
sidering that scientific research by its nature comes closest to the processes of
OKR and, in addition, is financed from the same source, the state budget, the NIR
expend:itures are set proportionately to the OKR costs. Here the share ot expendi
tures on sectorial NIR which should be put against the costs ef the OKR of a spe
cific system is determi.ned frcrm an analysis of the existing proportions in the sec
torial budget allocations f or the creation of new tec.hnology.
In the general instance, expenditures on the creation of a system functional e.le
ment are determined by the formula
Cniokr  Cokr (1 + Knir) ,
(4.35)
where Cokrthe costs of the OKR for a system element;
Knir.d Proportionality factor charact2rizing the ratios existing in a given
sector between NIR and OKR costs.
F:s was already pointed out, the growth rates for the NIR and OKR expenditures are
not r_onstant over time. This is reflected in the fact ttat the proportionality co
efficient ICnir has a certain time trend and for this reason the forecasting of NIR
expenditures with the known Cokr Pzesupposes the modeling and extrapolation of a
time trend for the proportionality factor.
If a f unction reflecting the time trend of a proportionality factor is differenti
able, f or forecasting the expenditures it is possib].e to use the following formula
[34):
0 dKnir (T )
Knir  Knir dT (4.36)
where K.nirthe for the NIR and OKR cost ratio in the base period TO;
dKnir(T)
aT the gradient for the index of the NIR and OKR cost ratio;
ATthe time gap; AT = T To .
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The correct determining of the possible ratios for the NIR and OKR expenditures in
the future has a marked impact on the results of the forecast estimate for the cost
of creating the BTS. For this reason it is essential to show great attention to an
analysis of the time trends of the proportionality factor ICnir in order that the
function describing these trends correctly reflects the gener.al patterns in forming
the NIOKR expenditure structure.
However, there is no doubt that the accuracy of forecasting the OKR costs for the
BTS has the basic impact on the results of the forecast estimate, as the OKR holds
the largest proportional amount in the general expenditures on creating the systems.
The model for the OKR costs of a BTS functional element can generally be expressed
by a multiple regression equation in which the independent variables represent the
characteristics of the functional element and the process of its OKR. For a loga
rithmically linear form of dependence, a model for OKR costs is written
Cokr = bi II �rb~� (4.37)
J "'1
The model (4.37) is a very approximate reflection of the procass involved in forming
the cost of an experimental subject. A multiple regression equation does not con
sider, and indeed cannot consider, all the particular features of OKR and these par
ticular features, as will be shown below, have a substantial impact on the process
of OKR cost formation.
The OKR of systems and their functional elements is carried out at prototype design
bureaus (OKB) and is characterized by three basic stages: designing, the manufac
turing of prototypes, testing and adjustment.
The first stage includes the work of designing the prototype, carrying aut experi
ments and working out the working drawings and technical documents required for
manufacturing and testing the prototypes. In the second stage work is done to manu
facture the prototypes, as well as to design and manufacture special fittings and
tools. The third, concluding stage of the OKR provides for the carrying out of ex
perimental adjustments and testing for both the system as a whole as well as its
individual elements.
Each of the designated stages is carried out to a certain degree by an independent
functional complex of the OKB. Designing is carried out by the designing complex
(PKK), the manufacturing of ptototypes by the production complex (PK) and testing
 by the testing complex (IK). For this reason the OKR process for a system element
can be represented as the process of the sequential transformation by each function
al OKB complex of its specific inputs into specific outputs (Table 4.7). The spe
cific input of the PKK is the information input, for the PK it is the material, and
for the IK the ob,ject (the latter in Table 4.7 is shown as part of the material in
put).
 The specific information input is the concept of the system (element) represented in
the form of a technical designing requirement or tacticaltechnical requirements
(TTT) for the system (element). This is the organizing specific input of the OKB.
In the designing process, in addition, scientifictechnical information is employed
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218
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which forms the basis of knowledge in this area. The embodiment of the concept in
the design of a system (element) occurs by mental processes which generate new in
formation in the form of design decisions. The variation of the plan which satis
fies the TTT forms the inforniation specific output of the OKB.
The specific input for the production complex is the material input in the form of
the subjects of labor which change their properties under the impact of the means
and implements of labor. The prototypes of the system (elQment) are the result of
this effect and they form the specific object output of the PK.
The specific input of the testing complex is the system (element) with its initial
level of ambiguity relative to the conformity of the functions and the ability for
them to meet the TTT. This ambiguity is minimized or completely surmounted under
the effect of the range of monitoring and metering equipment employed in the testing
process. A system which carries out the specified functions with the required level
of effectiveness forms the specific object output of the IK and the OKB as a system.
The distinction of the specific inputs and outguts determines the uniqueness of the
basic OKR stages and this must be considered in forecasting the OKR expenditures for
BTS. The predominance of information processing the generating processes gives the
design stage an exploratory nature and this explains the high degree of ambiguity in
the process and its end result. The ambiguity of a design process is particularly
great in the early stages of elaborating the system's design. These include the
elaboration of the prelimina.ry project, the technical requirement and the technical
proposal.
A technical development requirement gives the technical, operational and production
requirements made on the system and its subsystems. A technical requirement estab
lishes the basic purpose, the f lighttechnical characteristics of the article to be
developed, the conditions for its employment as well as the composition and basic
characteristics of the subsystems and elements.
A technical proposal is an aggregate of design documents which contain the techni
cal and technicaleconomic feasibility studies for the elaboration of the system on
a basis of analyzing the technical requirement and the different variations for the
 possible solutions for the articles as well as a comparative estimate of the solu
tions considering the design and operational features of the tobedeveloped and
existing articles as well as patent materials.
In the abovelisted stages, the design studies are particularly closely tied to the
NIR, as in the process of the preliminary studies new problems are frequently
brought to light the solving of which necessitates special NIR. The results of
this research cycle are considered in the f urther working out of the design.
 The degree of ambiguity in the result is noticeably reduced only in the draft de
sign stage. The amount of information in this stage increases significantly and
_ the data from the preliminary design stage are clarified and established by experi
mental data. Experimenting is conducted by creating mockups of the individual sub
systems and elemenrs in the system.
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A draft study is the first stage in which the design parameters of the article to be
designed are defined and the design appearance of the system to be designed is form
ed. On the basis of the draf t design, in the process of technical designing, the
_ working components of assembly units, schematic diagrams for fuel supply, electri
cal equipment and so forth are worked out.
In the stage of technical designing, quesrions arise which are analogous to the
questions of draft designing, but the number of variations for solutions is substan
tially reduced, since some of them were rejected as a result of mockup construction
in the draft design stage.
The stage of working designing is characterized by an extensive work front to cre
ate the drawings for the article, its individual units, assembly units and parts.
On the basis of the working drawings, directive methods are elaborated for manufac
turing the prototype of the system (element).
But sti].1, regardless of thE rather thorough elaboration of the design, the ambigu
ity about the conformity of rhe system (element) to the TTT remains high until the
carrying out of fullscale testing, and for this reason in a numbex of instances the
failures discovered in the testing lead to the halting of experimental subjects even
before they are complete. The reason for the premature halting of OKR at times can
be found in the miscalculations made in elaborating the preliminary project and the
TTT for the system. Th.e low scientific and technical gotential contained in the TTT
leads to the obsolescence of the sytem which is in the OKR stage. Obviously under
the conditions of a high degree of development ambiguity, the mistakes leading to
the halting of work are one of the common patterns in the OKR. For this reason, in
forecasting the expenditures on the OKR of the BTS it is essential to consider the
estimate for the average probability of successfully completing the OKR..
The average or mean probability of success u is determined by the ratio of the cost
of the successfully completed work over the past period of time Tk  Tp to the total
cost of all the work performed over the same period [34]:
Tic
f Coxr (s) dz
Tp
it ' (4.38)
r CuNh (T) LIT
TO
In the event of the continuing of the OKR f.ollowing the testing results, the design
documents and prototype undergo the corresponding changes and the testing is re
peated. The number of such cycles is rather difficult to predict, and for this
reason the testing process, like the designing of the system, is characterized by a
high degree of ambiguity. This significantly complica.tes the process of modeling
CKR costs for systems and their elements.
The process of manufacturing the prototypes is more determinQd in comparison with
the first and third stages, in a number of instances it possesses common traits
with the stage of serial production but also has a number of specific features.
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In contrast to the first and third OKR stages, the process of prototype manufacture
on the input and output has mainly material flows. Prototype production requires
raw products, matPrials, semifinished goods and preassembled articles but its end
result is the prototypes of the articles being designed.
Experimental enterprises are classified in singleunit type of production, although
_ they differ from classic singleunit production in a certain focus of the production
_ process and specialization of production on a uniform group of articles. In addi
tion, in singleunit production there is no change in the technical specifications
for the order and no supplementary work and this is characteristic for experimental
, or prototype production. The limited range and scale of output also tell on enter
prise size. Experimental plants are significantly smaller than serialproduction
 plants in terms of the number of employees and the productive capital.
The manufacturing of ne*a articles in units (or experimental batches) causes a num
ber of particular features in experimental production such as the lower equipping
of production with special tools and fittings, and, consequer.tly, the small capacity
of the tool shops; the use in the production process basically of universal equip
ment, the high skills of production workers and the consolidated elaboration of
production methods.
a) Y c
~
E~
�Q
:E
v�
z=
Z3,
~ ~zr
e
~
b
~
'O L'0
~ y O
= L
(b
Fig. 4.14. Dynamics of expenditure
indexes for developing articles de
pending upon size of experimental
batch
Key: aIndexes for change in ex
penditures; bFor one proto
The designated features of experimental pro
duction increase the production cost of proto
types in comparison with costs in serial pro
duction. However, the dynamics of expendi.
tures on manufacturing articles in the process
of experimental and serial production shows
conmmon trends: the costs of each subsequent
speciinen are less than the previous one. In
other words, in experimental production costs
are influenced by the degree of developing the
design and the manufacturing methods of the
4rticle. This is one of the most important
features of serial production (this question
will be examined in detail in the following
section). As a consequence of the influence
of the degree of production development on the
costs of prototypes, the cost_, of the experi
mental batch increase more slowly than the
sizes of an experimental batch. The latter
tells also on the behavior of OKR expendi
tures with a change in the number of proto
types.
r_ype; cFor an experimental The cost dynamics. of prototypes J1 of the ex
batch; dExperimental batch perimental batch J� and the total OKR expen
ditures for articles J, depending upon the
number of examples manufactured from the start of prototype production, are shown
in Fig. 4.14. The designated patterns are apparent even with thP comparatively
small experimental batches. In a number af instances the experimental batches reach
significant sizes and then the similarity of experimental production with serial
production is further increased.
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The entire amount of OKR [34] is carried out un the first prototype complex which
includes the flying prototype, the models for static and dynamic testing and re
peated load testing (for each subsequent flying model, starting with the second,
there is only the stage of manufacturing and testing with the necessary amounC of
rework and adjustment). In this manner consideration is given to the invariance of
_ design expenditures to the size of the experimental batch. A model for this type of
OKR costs has the following conf iguration:
"+1~ x~~ ~4.39)
CoKr _ � ~Yo 0
= where yo the proportional amount of conditionally fixed expenditures in the costs
of the f irst experimental complex;
Yo the proportional amount of variable expenditures in the costs of the first
prototype set;
nothe number of examples in the experimental batch;
an+lelasticity coefficient for variable expenditures in relation to size of
exterimental batch determined empirically;
 &okr1(x)the costs of the first experimental set of aircraft expressed in the form
of an equation of dependence upon the vector of the functional and design
characteri.s tics .
The specific weights of the conditiona??y fixed and variable expenditures are de
termined on the basis of analyzing the time trends in the OKR cost structure over
the previous period of time.
The model (4.39) is most effective for forecasting the OKR costs of the system ele
ments when the number of examples in the experimental batch is comparatively small
and their purpose is controlled by the adopted testing system as occurs in the de
velopment of aircraft. In manufacturing prototypes in large batches and with sig
nificant fluctuations in the batch size (the latter often depends upon the degree
of originality and newness of the articles), it is essential to consider the impact
of the scale of experimental production on prototype costs. For this purpose it is
essential to make the process of forming expenditures in the stage of manufacturing
the prototypes into an independent object of modeling.
In considering that the number of pratotypes directly or indirectly influences the
testing costs, the OKR cost model is best shown in the form of the total of the
particular expenditure models for each stage.
The cost model for the experimental batch generally is expressed by the following
 formula:
L'�" C" (x, /tu) /1o,
(4.40)
where Cnothe average cost of the prototype expressed in the form of a dependence
upon the characteristics of the functional element and the number of prototypes.
The dependence of prototype costs upon the size of the experimental batch no can be
approximatsd by the step function
C�o a,n~^,
(4.41)
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where al and an the equation parameters determined empirically and An < 0.
This dependence can be expressed in a dimensionless form if as the base one selects
the amount of the prototype costs found from (4.41) with a certain fixed size of
the experimental batch. In the given instance the most suitable base is the proto
type cost corresponding to the arithmetic average from the amount of the experiment
al prototype batches of the functianal element:
n
 no nor/n~ (4.42)
where nthe number of prototypes in the initial statistical aggregate.
Then the relative influence of the experimental batch on prototype costs is express
ed by the index
! = C /C
C~~ , (4.43)
 "o "o
0
where Cnothe cost of a prototype from a batch equal to no.
= In substituting the values Cno and Cno calculated from (4.41) for no and no, respec
tively, in (4.43) we obtain the dependence of prototype costs upon the size of the
experimental batch in a dimensionless form
J r )in
.
Cn�ne
~
(4.44)
Now, having expressed the influence of no by Jcno, (4.40) can be written in the
following form
x
Can = CA� (X ) lo 610)n� (4.45)
0
Thus, the cost model for the OKR of functional elements which are characterized by
large ranges in the change of the scale of experimental production, can be repre
sented by the following equation:
( l~` 1 (4.46)
Cacr ='i CA(X) Cno (X) tio l rto ! nCt~X~
where Cddesigning costs;
Cttesting costs.
The establishing of the dependence of design costs Cd upon the characteristics of
the functional elements, as was alread,y pointed out, is a difficult task due to the
high degree of process ambiguity at this OKR stage.
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For the system functional elements where among the characteristics it is possible to
isolate a base parameter that characterizes the geometric dimensions of the element
(for example, the weight of the aircraft frame, absolute engine thrust and so
forth), designing costs are approxi.mately expressed by the equation of dependence
upon this parameter. However the choice of the base parameter is possible not for
all the functional elements of tha systems while the modeling of designing costs
using other parameters requires a further breaking down of the designing process
into smaller items, since the sensitivity of designing costs to the various parame
ters is not the sarne for all the elements in the internal structure of this process.
Thus, the obtaina.ng of a good model for desi.gning costs for a rather complicated
functional element can take up a good deal of time and effort even for an experi
enced research collective.
At the same time, an analysis of the OKR expenditure structure shows that, regard
less of the tendency for an increase in designing costs for complicated systems,
their proportional amount in overall OKR costs is relatively slight and for certain
functional elements is 25 percent. Naturally, under these conditions, even major
errors in design expenditure forecasting will not have a substantial impact on the
accuracy of the overall OKR cost estimate. For this reason, in a number of in
stances, when the obtaining of a reliable mathematical economics model of forecast
ing costs requires the carrying out of a complex range of research, it is possible
to permit a certain simplification. For example, as an adequate model for the.form
ing of designing expenditures one can adopt the average designing costs calculated
from actual expenditures for the designing of the prototypes of functional elements:
S
cd.
(:d t;;r ~ Cd~ Cd  t;~ _ . (4.47)
where SEdthe estimate of the mean square deviation of actual expenditures from the
arithmetic average.
The procedure for forecasting OKR costs can be further simplified if the pr.oportion
al amount of designing expenditures Ycd in the total OKR expenditures of a function
al element is sufficiently stable or a certain time trend for this indicator is
known. Then the OKR costs are determirLed from the formula
~uic~ ' n + Ct) I 1  yC  (4.48)
A
&
The forecasting of testing costs Ct ts somewhat facilitated by the fact that in the
testing process a large amount of energy resources is required. For this reason,
 with other conditions being equal, the testing expenditures will be sensitive to a
change in the capacity of the energy sources and to the consumption of energy re
sources per unj.t of capacity. However, due to the ambiguity o� the testing process,
testing cost models usually introduce a large error factor into the total error of
 an OKR cost forecast in comparison with the expenditure models for experimental pro
duction.
The OKR processes for the BTS are constantly being improved and this is one of the
manifestations of the factor of an increase in cocial labor productivity. 1Koreover,
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this factor influences the costs of the products of past labor consumed in the de
velopment of the systems. In this regard, in forecasting the OKR expenditures, it
is essential to consider the trend in the decline in ORR costs related to the growth
of social labor productivity. The influence of this factor can be considered by
using the cost time trend coeffici.ent in the sectors producing the BTS and their
functional elements:
KT = (1 + O.O1W)Te T� (4.49)
where Tnithe average annual reduction in the costs of industrial product consumed
in the OKR process,
Te the year of carrying out the OKR of the system's elements which are the
objects of the cost estimate;
To the year of drawing up the cost forecast.
The total cost of working out the system S is the total expenditures on working out
the individual system elements considering the possible use of results from the de
velopment of elements in other systems and the overall development cost of the sys
tem:
~
Conr S'_ ~ CoKN /YuHI' 1+
i
(4.50)
where Cokr jthe OKR costs for element j of the system;
Yokr jthe proportional amount of expenditures on developing element j re
lated to the development costs of the system being designed;
Cop gthe costs for the general development of the system.
4.4. Basic Formative Patterns and Methods for Forecasting the Costs of Serial
Production of the BTS and Their Functional Elements
4.4.1. Particular Features of the Serial Production Process and Cost Formation
The process of serial production for the functional elements of the BTS can be con
ditionally divided into three basic stages: 1) tYie development af the first serial
models or the stage of production preparation; 2) the development of serial produc
tion or the development stage; 3) fu11scale serial output or the stage of Qstab
lished serial production.
The first stage encompasses the period from the start of the preparations for pro
ducing the article up to the output of the first serial model. At this stage or
ganizational and technical measures are carried out related to preparing the enter
prises to turn out the new article. Production preparations for the new article in
clude the fullowing measures: the rearranging and reorganization of existing shops
and sections; the elaboratian of serialproduction drawings for the product and pro
duction f ittings; the elaboration af serial production methods; the manufacturing
of the required amount of production fittings (the first stage of fittings); the
testing of the initial materials; the manufacture and testing of individual struc
tural elements and units of the article; the manufacture, assembly and testing of
the first serially produced model.
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It is essential to point out that a portion of the work related to production prep
arations for the new articles is carried out in parallel with their serial output.
The Iatter is explained by the need to shorten the overall production preparation
 cycle in the aim of accelerating the output of the first articles. The output of
articles of the head series and their subsequent putting into operation are needed
to eliminate design shortcomings the discovery of which is possibly only under ordin
ary operating conditions. As a consequence of this the final adjustment of the
article's design is made at a serialproduction enterprise in the process of dis
closing and eliminating the existing shortcomings. The designated circumstances
necessitate the manufacturing of the prototypes under conditions where the produc
tion process for manufacturing the article is equipped with a:ninimum range of spe
cial production fittings and without which the output of the article is essentially
impossible.
Thus, the conditions for turning out th.e first serialpraduced models of the article
have the following particular features: production instability and a relatively low
technological level; low equipping of the produr_tion processes; imperfect forms for
organizing production and the work areas; the absence of work skills for the workers
and insignificant experience of technical personnel; the ,absence of technical stand
ards for labor intensiveness; the incompleteness of the article's design and so
forth.
T.he listed characteristic traits inherent ta the moment of completing the first
serial production stage are the reason for the relatively high costs of the first
articles. Subsequently, as serial production is developed, article costs decline
substantially and in a number of instances by the end of the second stage are 1520
percent of the initial.
 The decline in costs at the production development stage is achieved by introducing
measures aimed at raising the organizational and technical level of production. At
this stage the serial production drawings and the serial production methods are
 finally elaborated; the manufacturing of the production set of fittings is f ully
completed (the level of equipping in a number of instances reaches 9095 percent);
the production areas and lines are determined; technical standards for labor inten
siveness are introduced (the proportional amount of technical standards reaches 70
75 percent); adjustments and improvements are incorporated in the artiCle's design;
the workers gain work skills and the technical personnel gains experience in manu
f acturing the new article.
The production development stage is characterized by an increase in product output
per unit of time. By the end of this period, the enterprises reach a steady pro
ductien program and for this reason, in speaking about developing the production of
BTS functional elements, it is essan*_ial to bear in mind not only the process of
developing the design and the production methods for the articles but also reaching
designed output scale.
Thus, the production development of new articles is a dual process. On the one hand,
the production development of new articles is accompanied by a rise in the organiza
tional and technical level of se.rial production (the qualitative aspect), and on the
other, in keeping with the development of serial production there is an increase in
 the quantity of articles produced per unit of time (the quantitative aspect). The
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first factor influences primarily the labor and material intensiveness of the arti
cle. The action of the second factor is manifested in the fact that with an in
crease in product output per unit of time there is a decline in the share of the
shop and generalplant expenditures and in the expenditures on carrying out special
testing and settingup outlays. Expenditures decline on production fittings per
unit of article.
The two mentioned aspects of the process of cost formation are interrelated and the
basic is the quantitative aspect. Thus, the production scale has a direct impact
on the optimum level of outfitting zhe production processes in manufacturing the
articles. Under the conditions of producing large batches of articles, the outlays
on production fittings become more advisable, as here the increase rate in the ex
penditures on serial production provided by the increase in production outfitting
outstrip the growth rate of the expenditures on outfitting. The higher the scale
of product production the more the optimum level of outfitting is felt and thereby
better prerequisites are created for producing the expenditures on the serial pro
duction of the articles.
The output scale ultimately predetermines the possibilities of organizing mechanized
and automated production, the use of specialized production equipment and the intro
duction of advanced production processes.
The product output scale also has a substantial impact on the level of specializa
tion and cooperation. The deepening of specialization and the widening of the in
terdepartmental and intrasectorial ties become effective only under the conditions
of largeseries production making it impossible to employ the most progressive means
and methods of specialized production.
~
~
0
u
The scale of output, in characterizing, in addi
tion, enterprise pioduction capacity, is an in
direct indicator of the production concentration
level in the sectors specialized in producing
the BTS functional elements.
~
a
The dynamics of cost reduction and the reaching
u
i ~ ~
0
of the designed product output scale are shown
o
~
in Fig. 4.15. The degree of the serial produc
tion development of the design and the reaching
a'
prnduction tu11
of the designed product output scale influence
development output
not only the level and dynamics of the cost de
production period
cline but also the ratio of the individual ex
penditures included in its structure. Fig. 4.16
Fig. 4.15. Change in product
shows a typical change in the cost structure
costs and output in the process
occurring under the influence of the designated
of serial production
factors. As is seen from the graphs, in the
process of the production development of the
article there is a rise in the proportional
amount of expenditures on wages and
materials and a decline in the share of the
shop and general plant uutlays and
the direct aggregate expenditures (expenditures
on special fittings, testing and so
forth) in the full production costs.
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~
a~
a
x
a,
~
co
r.
0
.,4
W
0
a
0
$4
a
io _
t
on 
~
30 y~ 
?
`
?o s.,..~
\~_~43
f0 Q=
x_x_xsx_x_x_X_,, 6>.Lx_xx_xx_x
production period
Fig. 4.16. Change in proportional
amount of basic expenditures in
full product costs in the process
of developing their serial production:
The third stage or the stage of established
serial production is characterized by the
following basic features: a further im
provement in the design and production
methods of the articles; the broadening and
improving of the production set of equip
ment and f ittings; the improving of the
organization of production and labor; a
rise in the automation and mechanization
level of the production processes; the ex
tensive introduction of rational methods
for producing the initial stock and so
forth. The given range of organizational
and technical measures is carried out over
the entire serial production period. Dur
ing this period one most strongly feels
the impact of the f actors related to the
rise in social labor productivity.
3Special fittings; 4General plant In the stage of fullscale serial produc
exgenditures; SWages; 6Testing tion, when the reserves for cost reduction
brought about by the newness of the article
and by other particular features of the production development stage have been
basically exhausted, a further decline in production costs occurs as a consequence
of improving the technical means and organizational forms of serial production and
the other factors which determine the growth of social labor productivity. However,
in contrast to the second stage of serial production, during thi.s stage the inten
sity of the cost decline does not exceed 42 percent per quarter.
1Materials; 2Shop expenditures;
Research on the particular features of serial production and the nature of the
change in expenditures during its various stages has made it possible to draw the
following conclusion. The pracess of forming the costs of the BTS functional ele
ments is shaped under the influence of three basic factors which characterize the
organizational and economic conditions of serial production of the systems: the
organizational an.d technical level of production; the degree of the production de
velopment of the design and the serial output of the articles; the production scale
of the articles.
The organizational and technical level of production accumulates the influence of
both the macro and microeconomic factors and the impact of the latter on the cost
of new articles to a signif icant degree is determined by their output scale. In
_ this regard, the task of establishing the indicators which characterize the scale
of product serial output moves to the forefront.
The scale of serial output can be estimated using the following basic indicators:
the number of articles manufactured from the start of serial production N; average
daily output calculated for a certain production period q; product output per unit
of time (quarter or day) reached by a certain moment of time in the designated
period q.
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The number of_articles N manufactured since the start of production and the average
daily output q characterize the scale of serial production only when combined with
the third indicator, the length of the period t during which N articles have been
produced or an average daily output q has been reached. The daily output q, in con
trast to the first two indicators, is an independent characteristic of the output
scale at the given concrete moment of time during any segment of tha serial produc
tion period of the articles.
4
k
v
'b
a
~
4J
N
O
U
4.
9
, ~
a
~
~
Z, o
~
r1
a~ c0
b
~
b
.4
4J
m
O
U
j9
8 W
e ,1
O
4 m
td
b
0 Y 4 6 B !0 17
quarters
Fig. 4.17. Inf luence of absolute
product outgut scales on c3st
_ dynamics and level
Fig. 4.18. The inf luence of the ratio
of the initial and established product
output scales (1) and (2) on their cast
dynamics
The influence of daily output on the cost level and dynamics is illustrated by the
examples of Figs. 4.17 and 4.18. Fig. 4.17 characterizes the dependence of cost
dynamics and level upon the absolute serial output scale under the conditions of
established production. In Fig. 4.18, cost dynamics are compared with the product
growth rate. From the diagrams it is obvious that product costs decline more in
tensely the higher the scale of established production and its increase rate in
the process of reaching serial output. The more intense rates of cost decline with
an increase in the difference between the initial and established output are ex
plained by the fact that the output of the first serially produced examples, inde
pendently of their quantity, occurs under approximately equal organizational and
technical production conditions. Conversely, under the conditions of established
serial productlon the organizational and technical level will be higher the greater
the output scale under the same conditions.
Thus, the process of cost formation can be conditionally divided into two parts:
the formation of cost dynamics and the formation of the cost level.
Cost dynamics are chiefly determined by the particular features of developing serial
output while the level is determined by the output scale of developed articles and
by their design and production features which are determined by the oper�atinnal ef
fectiveness factors. By the cose level one understands the cost of articles corres
ponding to the start of established serial production. It is felt that under these
conditions the influence of the factors characterizing the processes of reaching
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serial production has been virtually exhausted and that subsequently the process of
serial production will occur with an established level of production equipping and
other organizaLional and technical characteristics of serial production. The start
of established serial production is usually linked with the moment of producing the
socalled serially developed product. Here it is assumed that the cost of a serial
ly developed product, in contrast to the cost of the previously produced articles,
shows a relative resistance to a change in the output scale and the other character
istics of the product production development processes.
Establishing the cost of a serially produced product is essential for achieving com
parability in terms of the degree of serial production development for prototype
articles of a system's functional element when these comprise the initial statisti
cal aggregate. The costs of articles which are compared from the viewpoint of the
degree of developing a design and the reaching of the designed output scale are used
as the initial base in studying the influence of the functional and other character
istics of a functional element. In addition, the costs of a serially produced arti
cle make it possible to establish the influence of its level and output scale under
the conditions of established production. DeteLmining the costs of a serially pro
duced article comes down to setting the moment of moving from the production de
velopment stage to the stage of fullscale serial production. The time interval
(ordinarily the ordinal number of a quarter, starting from the beginning of serial
output) at which this transition is made has been given the name of the "serial out
put point."
Above it was stated that the process of reaching serial production of products has a
quantitative and qualitative aspect. Here the quantitative aspect which is the
reaching of the designed output scale largely determines the qualitative aspect of
the production development process, that is, a rise in the organizational and tech
nical level of production. The conclusion arises that it is possible to speak about
the moment of transition from the production development stage to full serial produc
tion only proceeding from the dynamics of daily output. Obviously the time interval
by which daily output reaches a relative stable level will correspond to the serial
output point.
~q _start of develoed prod.
k   
0 4o I ti'v'l ~ 4~
,J ~ I I
~ I ~o I 1~ I ~o
0
0 7 ~i 6 d 10 12 '4,
t
quarters
, Fig. 4.19. Influence of development
rate of output production program on
duration cf serial development period
The relative stabilization of daily output
is achieved at various moments of time
depending upon the pace of developing the
production program. The development pace
of the production program for turning out
various articles can change in a rather
broad range. Fig. 4.19 shows the most
characteristic curves for the change in the
daily output of functional elements of a
system. From Fig. 4.19 it can be seen
that the time intervals corresponding to
the moment of re.lative stabilization in
daily output vary between 8 and 15 quarters
counting from the start of production.
This shows thar the determining of the
serial output point is a necessary condi
tion for eliminating the influence of the
degree of product development on costs.
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The serial output point, depending upon the pace o� developing the production pro
gram, can be determined if one imposes certain constraints d on the relative rate
of increase in daily output per unit of time.
The time trend for daily output is best described by the dependence
q(1) _ d, arctg (d,t I d,) d4, (4.51)
where tthe time counted from the start of serial output (in quarters);
di, d2, dg, d4equation constants determined empirically.
The relative increase rate in daily output can be determined as
q' (t) : qmaX, (4.52)
d
where q'(t)the derivative function of (4.51); q'(t) = dl 2 2;
1+(d2t+d3)
qmaX the function maximum of (4.51).
The function maximum is found from the condition
liin y(1) = dl arLtg oo dd, (4.53)
hence
n
4~~�X = d, 2 d,.. (4.54)
In reducing (4.52) to the set value of d and solving the obtained equation
(q'(t):Qmax  d) for t, we determine the ordinal number of the qua�rter corresponding
_ to the beginning of developed production, according to the following formula:
r L y I d, didz
~ ~ 1 2 a (nd, I d,)
(4.55)
The concrete amount of the constraint d imposed on the relative increase rate of
� daily output from the viewpoint of the comparability of various articles from proto
 types which are in the initial statistical aggregate is not of fundamental signifi
cance. It is important that it is constant for all the compared articles. However,
 this amount should be rather small in order that the serial output point is to the
right of the area of intense output growth. In the given example (see Fig. 4.19),
the serial output point was set at d= 0.01. It is not difficult to see that under
 these conditions the growth of daily output is virtually halted.
The setting of a serial output point makes it possible to select with sufficient
soundness the values for the costs Co and the daily output qo for serially produced
articles and this _Qlays a major role in forecasting the costs of BTS functional
elemer.ts. Cost forecasting usually starts by determining tne probable amount of
costs for a serially produced article. After this the possible cost changes are
determined over the entire extent of the serial production of the articles, that is,
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a forecast is made for the cost dynamics. However, research on the cost formation
processes must start by analyzing the cost dynamics of articles in the process of
their development and for this reason it is most convenient to start an exposition
of the cost forecasting methods from the cost dynamics `orecasting methods.
4.4.2. Forecasting Production Cost Dynamics of BTS Functional Elements
The forecasting of cost dynamics consists in determining the possible changes in the
' costs of a specif ic article depending upon the assumed conditions of developing its
~ serial production. Above it was pointed out that a decline in costs is a character
istic feature in the stage of developing the production of products. However, this
reduction occurs at different rates. The establishing of the regular changes in the
cost reduction rate also creates the necessary prerequisites for farecasting cost
dynamics.
From the preceding material (see 4.4.1), it is obvious that developing a new func
tional element represents a process of the adaptability of an enterprise to the
_ serial output of the product. The qualitative aspect of this process is the develop
 ing of the design per se and its quantitative aspect is reaching the designed out
put scale. These are interrelated aspects. And the primary one is the quantitative
aspect since the possibilities (and often the advisability) of increasing the organ
 izational and technical level of production are restricted to the designed output
scale. Proceeding from this, the methods of forecasting cost dynamics are based on
the a;ssumption that there is a quantitative relationship between the cost reduction
rate and the product output scale. For this reason the methods given below differ
only in the principles for estimating this relationship.
Fig. 4.20 shows the graphs for the increase in the total number of articles manu
factured since the outset of serial production and the corresponding graphs for the
change in costs. As is seen in the figure, product costs decline more intensely
than total product output grows. In the general instance this dependence can be
~ expressed by the multiple regr.ession equatian:
C= 1~Nnalba (4.56)
J
l=4
where Nthe total number of articles produced since the start of series production;
tthe time over which N articles were preduced;
xjcharacteristics of the system's functional element;
bl, b2, bPequation constants determined empirically.
The model of (4.56) expresses the dependence of product costs upon factors determin
ing both cost dynamics and level. A characteristic shortcoming of the model (4.56)
is that it does not meet the demand of product comparability in terms of the degree
of serial production development. As was shown (see 4.2.3), the parameters of *_he
multiple regression equation are calculated under the conditions of eliminating the
inf luence of other variables and for this reason express the dependence between the
dependent ar.d independent variables when all the remaining variables are held on
the level of their averages. This means that the constants b4, b5, bp express
the cost dependence of a system's functional element upon the element's characte:r
istics with N and t corresponding to their average arithmetic values N and t.
~
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k
a~
~
~
u
~
0
U
rI
b
~
u
cd
FOR OFFICIAL USE ONLY
However, as is seen in Fig. 4.19, the con
ditions of comparability for each individu
ally taken article change very substantial
ly (in the designated example to changes by
almost double) and depend not upon the ab
solute amount of N but rather upon the rate
v of developing output. Thus, the conformity
~ of N and t to the comparability conditions
fcr even one article of the initial statis
tical aggregate can be the result of only
0. a random coincidence and in any event can
o not be extended to all remaining articles.
~ The given circumstance can lead ta major
(11SLVL LlUlls Vl 6tiC pivuuct i.vot a',aa"..Y.�..aid.�...^.^.�
~
'd upon product characteristics, as these de
0
pendences will be influenced by the degree
a
of serial production of the article.
Naturally it is very difficuZt to judge the
reliability of forecasts made using naodels
of the type (4.56).
A more attractive method from the desig
o ~ 4 6 a 10 12 t nated viewpoint is one based upon the use
quarters of a multiple regression equation to model
the cost index JCo which represents the
Fig. 4.20. Dynamics of product costs ratio of the article's cost observed in
each time interval of the studied period
depending upon output size of serial production to the cost of the
serially produced article. In this in
stance the requirement is observed of product compatibility in terms of the degree
of serial production and a model describing the cost dynamics has the form
,Tco = b1Nb2tba.
(4.57)
The model (4.57), in comparison with (4.56), provides more dependable forecasts,
however the effectivene ss of its use is reduced as a consequence of the existence of
internal constraints on the change in the independent variables which are inherent
to multiple regression equations (see Section 4.2.3).
In terms of the problem of forecasting cost dynamics using the model (4.57), to the
ordinary constraints examined in Section 4.2.3, one must add the specific con
straints related to the transformations of the initial statistical aggregate. Tt:ase
occur in calculating the parameters of the multiple regression equation of (4.57).
The initial statistical aggregate consisting of n prototype articles of a function
al element contains n pa irs of values (C1, NI); (C2, NZ), Ck; Nk) for each time
interval of serial prod uction (t = 1, 2, k). At the same time for calculating
the parameters of a mul tiple regression equation it is essential that for each ele
ment Ct of set SC there be the corresponding fully determined value of Nt for set
SN in each time interva 1 t. For achieving this correspondence it is essential to
average the values of C1, C2, Ck and their corresponding values of N1, N21
Nk in terms of the number of articles n in the initial aggregate for each t(t =
 l, 2, k).
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_ Graphs for the time change in the averaged values L and I~ in Fig. 4.20 are depicted
with a broken line. It is not hard to see that the aggregate consisting of nk val
ues of C and N is transformed into a series where for every value of N there is a
uniform corresponding averaged value of C. If one considers that the pairs of
series (Nt, CL)i (i = 1, 2, n) differ in terms of the lengtn of the period for
starting up serial production tp, the averaging of N and C must be carried out
within a certain predetermined value of this period, for example, the average dura
tion of the period for starting up serial production for the studied aggregate of
articles:
I 'I
~
lu = n ~ Ioi�
Under these condiCions, in considering that the function C= F(N, t) is determir.ed
in an area bounded by an ellipse which is described by the equation
 N r t
1~
R' a 2 a Q 11
II ~~tl fN Nla the required aircraft fleet can be determined using the
f ormu la :
NsE _ I , (5.38)
 KzGcom GfVcr.eTg
where KZaircraft load factor.
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For the calculated number of aircraft one determines the required number of engines
using the formula
TsE
NdE = NSEnd tdE .
(5.39)
The presence of technical parameters for the system elements and quantitative data
on the demand for them make it possible for the methods given in Chapter 4 to de
termine the cost parameters of the system, as follows: expenditures on NIOKR, the
cost of manufacturing and operating the system which, in turn, as component elem2nts
becomes part of the ATS economic effectiveness criterion.
The system's economic effectiveness criterion with the approximate calculation
method is the minimum of reduced expenditures for the annual volume of its work:
n 
_ T.
T` ~ Z C Cw r  Z ltjCioi E)'
C. N' (i__
1i
Cn X N (1 E)r`Z�"o _ ni xIN ~ ~ E~r~~`'o _F
LJ
n
1
+ P, +,i; ~ P,; ; hL) A% = min,
(5.40)
where Dcannual praductivity of system`s central element, tonkm per year;
Tcservice life of system central element;
 csystem central element;
jelement of system central element;
mi number of considered elements of syatem central element;
Ccotexpenditures on operating system central element during year t;
Gjotexpenditures on operating system element in year t;
Zcokrexpenditures on working out system central element, rubles;
Ncthe size of manufscturing batch of central element, pieces;
TcZokro lead in making expenditures on.OKR for central el.ement to the system's
operation, years;
Zjokrexpenditures on working out system element;
Mjsize of manufacturing batch of system element; '
TjZokro lead in making expenditures on OKR in relation to system s operation,
years;
Pcprice of system central element, rubles;
Pjprice of system element;
Kiother capital investments into operation of system (not counting cost of
system elements).
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The expression (5.40) can also be given in the following form:
Z = ZydAg,
_ where Zydreduced expenditures per unit of system work;
Zyd = Cqd + EnICy,d ,
where Cydproportional cost of a unit of work;
 Kydcapital investments per unit of work.
(5.41)
(5.42)
The calculations using the formula of (5.40) provide a differentiation of expendi
tures for the individual aircraft elements: engines, equipment and so forth. In
existing practices calculations are isolated for just one element, the engines, and
for this reason we will divide the aircraft conditianaliy into two elementa:
a) engines; b) all remaining aircraft elements.
Let us designate the aircra�t by the index tu, the first element, the engines, by
the index d and the remaining elements by pt. Considering this (5.40) will assume
the f orm
 x T~
p~ r~a (Cplo t~ ldCder) ~ I1 E)~ +
r
(I + E)r
_ I En l7 Qtok~ (I E)T ~Za~._a ~ 11~~~Ok~" ( I +
E) Tca
PPl + 1144. 0 1 ko1. 1.. /