Table Of ContentOPTIMAL DECISIONS
Principles of Programming
BY
OSKAR LANGE
Prepared with the Collaboration of
Antoni Banasinski
On the Basis of Lectures Delivered at Warsaw
University
PERGAMON PRESS
Oxford · New York · Toronto
Sydney · Braunschweig
PANSTWOWE WYDAWNICTWO NAUKOWE
WARSAW
Pergamon Press Ltd., Headington Hill Hall, Oxford
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© 1971 Panstwowe Wydawnictwo Naukowe
All rights reserved. No part of this publication may be reproduced*
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the prior permission of Pergamon Press Ltd.
This book is a translation of the original : Optymalne decyzje.
Zasady programowania, 2nd ed., Panstwowe Wydawnictwo Naukowe,
Warsaw, 1967
Translated by IRENA DOBOSZ
(Introduction, Chapters 1, 8 and 9)
and JÓZEF STADLER
(Foreword, Chapters 2-7 and 10-12)
Translation edited by
P. F. KNIGHTSFIELD
First English edition 1971
Library of Congress Catalog Card No. 76-143810
PRINTED IN POLAND
08 016053 0
FOREWORD
In the years 1960-1961 and 1961-1962 I gave lectures in the theory
of programming at a two-year course at the Department of Political
Economy of Warsaw University. In the first year I lectured on the general
principles of programming and in the second—on programming under
uncertainty. As usual, Doctor Antoni Banasinski made notes of my
lectures and helped me in restyling them. Doctor Banasinski prepared
also many numerical examples and in this way my exposition has
assumed the shape of this book.
This is the third book, after Introduction to Econometrics and Theory
of Reproduction and Accumulation, that has been published with the help
of Doctor Banasinski. On the occasion of this "anniversary" I extend
to him my special thanks.
The aim of the exposition of the principles of the theory of program
ming is their synthetic presentation. I attempted to give, first, a general
interpretation of the theory of programming based on the application
of the Lagrange multipliers and then to present marginal and linear
programming as special cases of this general theory. In an abbreviated
way I have already done this in the Appendix: "Mathematical Principles
of Programming" to my book Political Economy, Vol. 1. Now I have
described this approach much more exhaustively with particular consid
eration given to a praxeological interpretation of the method of Lagrange
multipliers. This is consistent with the treatment of the theory of pro
gramming, accepted throughout this book, as part of praxeology—the
science of rational behaviour.
In addition to the exposition of the theory of programming I deal
also with the question of solving programming problems in practice.
In this connection the principles of the simplex method are presented
together with the application of the theory of linear programming to
activity analysis. A separate chapter is devoted to the problem of pro
gramming for multiple objectives which is of great importance for the
political economy of socialism.
IX
X FOREWORD
Almost the whole second half of this book deals with programming
under conditions of uncertainty. In contrast to the general theory of
linear programming this problem has not been dealt with extensively
in literature so far. It is of great practical importance, however, parti
cularly for planning in socialism. I tried to present and to develop the
most important methods used in this field and tie them to the problems
of a socialist economy. It has turned out. along the way, that the theory
of programming under conditions of uncertainty sheds light also on the
methods of procedure of mathematical statistics (the optimal choice
of the confidence coefficients). Moreover, it also has some interesting
consequences in the field of stock control and the pattern of production
over time.
However, probabilistic and statistical methods of programming have
limited practical applications. They not only require the direct or esti
mated knowledge of probability distributions of programmed quanti
ties, but also lead themselves to practical applications only in making
frequently repeated decisions (e.g. in quality control). Many decisions
made in economic planning (e.g. investment decisions related to the
construction of large projects) are in the nature of once-over decisions
in which the law of large numbers does not operate. Under these circum
stances one cannot use programming based on probability calculus.
The resultant gap is filled here by the theory of strategic games which
provided the foundations for making rational decisions under conditions
of a complete and inestimatable uncertainty (ignorance), i.e. such under
which it is impossible to use probability calculus. For this reason the
last part of the book deals with the application of the theory of games
to programming.
In consequence, the book deals with all important problems related
to programming. I hope that it will be favourably received by the readers
and will contribute to strengthening rationality in the methods of plan
ning and management of our national economy.
INTRODUCTION
PRAXEOLOGY AND THE THEORY OF
PROGRAMMING
The theory of programming and its practical applications have aroused
deep interest among economists in recent years.
The theory of programming, to be precise, does not form part of
political economy; it is rather an auxiliary science which may be used
in political economy as well as in other fields of theoretical and practical
research. It can be regarded as part of the general science of rational
activity.
Although it was in political economy that problems of rational ac
tivity appeared first and were recognized as such, their scope has grown
far beyond the boundaries of economics. This gave birth to a separate
science of rational activity—praxeology. The term praxeology was first
used in 1890 by the French sociologist Espinas in his essay on the origins
of technology.1 It has since reappeared more and more frequently in
the literature.
So far, the most systematic exposition of the foundations of this
very young science is given by Tadeusz Kotarbinski in his Traktat o dobrej
robocie (Treatise on Good Work), published in 1955.2 The essential
ideas of the science of rational activity were presented by Kotarbinski
as early as 1913 in his Szkice praktyczne (Essays on Practice).3
Quite independently of Kotarbinski, praxeology was introduced into
economics by the Soviet mathematician Eugene Slutsky.4 The Austrian
1 Cf. O. Lange: Political Economy, Oxford, 1963, Vol. I, p. 189.
2 Tadeusz Kotarbinski: Traktato dobrej robocie, 2nd ed., Wroclaw-Warsaw, 1958.
3 These essays, along with other papers, are in Kotarbinski's Pisma wybrane
(Selected Works), Warsaw, 1958, Vol. I. For the antecedents of praxeology see T. Ko
tarbinski: "Rozwój Prakseologii" (The Development of Praxeology), Wiedza Prak-
tyczna (bulletin), Warsaw, 1962.
4 E. Slutsky : Ein Beitrag für formal-praxeologischen Grundlegung der Oekonomik,
Kiev, 1926, Académie Oukrainienne des Sciences, Annales de la classe des sciences
sociales-économiques, Vol. IV.
1
2 OPTIMAL DECISIONS
economist Ludwig Mises also used the same term, although he erro
neously identified praxeology with political economy and misconceived
its foundations.5
Praxeology is concerned with rational activity. In fact, it may be
defined as "the logic of rational activity". It makes uses of such special
concepts as ends, means, methods, action, plan, effectiveness, efficiency,
productivity, economy, etc., called praxeological categories. These con
cepts may be applied not only in political economy but also in techno
logy, warfare, strategy and tactics, methodology of scientific research
and in other fields. Praxeology defines relationships between praxeologi
cal categories, or praxeological principles of behaviour, which appear
in every field of rational human activity. One of these principles is espe
cially important in view of its role in political economy. It is the economic
principle, or the principle of economic rationality.
The principle of economic rationality can be applied when the end
and means of activity are quantified, i.e. are of the nature of "quanti
ties", or, at least, "magnitudes".6 According to this principle, the maxi
mum degree of realization of an end is achieved when a given outlay
of means yields the maximum effect. In this interpretation, the economic
principle is called the principle of greatest effect or the principle of greatest
efficiency.
There is another interpretation of the economic principle known as
the principle of minimum outlay or the principle of economy of means.
In this case a given {a priori) degree of realization of an end is achieved
with a minimum outlay of means.
It can be proved that the two interpretations or variants of the prin
ciple of economic rationality are equivalent. Indeed, applying the second
variant of the principle of economic rationality, given the amount of
means available, we shall obtain in the final analysis the maximum degree
of realization of the end. For, if less means are used to achieve a parti
cular degree of realization of the end, i.e. if some means are economized,
then the degree of realization of the end can be correspondingly increased
and thus brought to a maximum.
The principle of economic rationality is sometimes defined in yet
another way, namely as a procedure leading to the maximum realization
5 L. Mises: Nationaloekonomie. Theorie des Handelns und Wirtschaf tens, Geneva,
1940. For critique of Mises see O. Lange: Political Economy, éd. cit., pp. 239-240.
6 The difference between the concepts of "quantity" and "magnitude" will be
explained later.
INTRODUCTION 3
of an end with the minimum input of means. A definition of this type
is fallacious, because it leads to a contradiction. This will be proved
later on, when the principle of economic rationality is expressed in
mathematical terms. However, the fallacy of the third definition of the
economic principle is obvious even intuitively.
We have said before that the end which we wish to achieve, using
the principle of economic rationality, must be a "quantity" or a "magni
tude". Let us clarify these concepts. A phenomenon is of the nature of
a quantity when it is measurable, i.e. when it can be uniquely expressed
by a number. But there are phenomena which, though not measurable,
can be ordered, i.e. uniquely arranged in a certain way according to
their order of magnitude. We say that such phenomena are o fthe nature
of magnitudes. For example, stars are classified by brightness, or miner
als by hardness. Classification by order of importance is often used
in physiological research, where some phenomena are arranged by,
say, the degree of pain they produce, etc.
In order to apply the principle of economic rationality it is sufficient
that the end which we desire to achieve should be of the nature of a mag
nitude. The principle of economic rationality required that we should
be able to tell whether the end has been achieved to a greater or lesser
extent as compared to an initial or any other given level. This remark
is of a general nature, since for any calculation which involves maxi
mization or minimization of a certain variable it is sufficient that the
variable should be a magnitude, i.e. that its various values should be
uniquely arranged. From the foregoing it is obvious that every "quan
tity" is also a "magnitude", but not vice versa.
As stated before, the principle of economic rationality has played
an important part in the development of political economy and was
first formulated by economists. This is not accidental, because the prin
ciple of economic rationality is closely connected with economic de
velopment. The development of a commodity-money type of economy,
and especially of the capitalist mode of production, led to quantifying
economic phenomena; monetary calculation took the place of calcula
tion in physical categories. In a natural economy the object of economic
activity is to satisfy a multiplicity of needs (food, clothes, leisure, defence,
etc.) by various means (bread, bricks, timber, metal, etc.). In a capitalist
economy, where all economic calculation is carried out in terms of
money, the object of economic activity is uniquely quantified; it consists
in achieving money incomes. Since the means used by capitalist enter-
4 OPTIMAL DECISIONS
prises (wages, depreciation of equipment, savings, etc.) are also quanti
fied in monetary units, the outlays on these means become commen
surable.
Thus, in capitalist economy, the ends and means of economic activity
became quantified in commensurable monetary units. There emerged
a definite, uniform object of economic activity—monetary profit. This
contributed to the rationalization of economic activity. The old methods,
based on custom and tradition, were replaced by well defined methods
of rational economy based on calculation and its tool, book-keeping.
The principle of economic rationality—a historical product of the
development of the capitalist economy—is generally confined in capital
ism to individual enterprises/In socialism it affects the whole national
economy.
The principle of economic rationality is related not only to economic
activity; it finds its applications in other fields of human activity as well.
For example, in technology, it helps in deciding how to achieve the
maximum speed of a vehicle or the maximum resistance of a bridge,
etc., given available means. Of course, the technological problems which
are to be solved on the basis of the principle of economic rationality
may also be stated otherwise; the aim of activity is, say, to achieve a given
efficiency of a machine with a minimum outlay of means.
Similarly, in warfare strategy and tactics, the problem may be one
of achieving a given end with the smallest outlay of material resources
and a minimum of casualties, or, conversely, to achieve the maximum
strategic or tactical effect for a given outlay of resources and casu
alties.
There are a great many examples of this type. But in some fields of
human activity the principle of economic rationality is not or cannot
be applied. This occurs primarily when, as we mentioned before, the
ends and means of action are neither quantities nor even magnitudes.
In a natural economy, for example, there exists a multiplicity of ends
and means which are not commensurable. A similar situation, though
much more limited in scope, may arise both in capitalist and in socialist
economies.
The use of means in accordance with the principle of economic ration
ality is called the optimum use. Optimization of the use of means may
consist then in maximizing the end or minimizing the means, i.e.:
(i) in achieving maximum realization of an end with a given outlay
of means, or
INTRODUCTION 5
(ii) in achieving a given degree of realization of the end with a mini
mal outlay of means.
The use of means otherwise than optimum is called waste. The concept
of waste is a praxeological category. Waste occurs in economic or other
activities when (i) the means are so used that the maximum degree of
realization of an end is not achieved, or (ii) for a given effect (the degree
of realization of the end) the outlay of means is larger than necessary.
The principle of economic rationality is thus the general praxeo
logical principle of behaviour. It is the principle of rational behaviour
in a situation when the end and the means are of the nature of magni
tudes.
Programming, which is the subject of our exposition, is concerned
with a special type of rational activity; the science of programming,
or the theory of programming, forms part of praxeology. It constitutes
the mathematical theory of application of the principle of economic
rationality. The theory of programming, having emerged as a separate
science by the end of World War II, is still a very young science.
It was virtually created simultaneously with another science, called
operations research. Developed in response to the requirements of mili
tary operations, the two sciences are closely interrelated.
Operations research came into being in Great Britain in the early
years of World War II, when research groups, made up of scholars
representing various disciplines, were set up to work out scientific
methods of logistics and military operations. One of the problems they
dealt with was to determine the optimum number of ships in convoys.
The practice of warfare had proved that large convoys were easier to
defend against attacks of enemy aircraft and submarines. On the other
hand, the larger the convoy the more slowly it would move, its speed
being determined by the speed of the slowest vessel; also, the more ships
in the convoy, the greater the number of ship defects which slowed
down the whole convoy, thus reducing its safety. There was an urgent
need of finding a "compromise solution" to the problem which consisted
in defining the optimum number of ships in a convoy. Other problems
were to determine "the optimum route" for ships carrying war supplies
so as to minimize losses in ships and cargo, etc.
All these problems involved maximizing the realization of an end or
minimizing the outlay of means. Most of the methods originally con
ceived to solve military problems proved to be easily applicable to peace
time problems, including rational business management.
6 OPTIMAL DECISIONS
The science of programming was developed in the United States
during the last war to tackle problems of military supplies, logistics, etc.
One of the problems was to find the best location for warehouses intend
ed to store food and war supplies. A location close to the theatre of
hostilities facilitated rapid delivery of supplies to the place of destination
but exposed the stores to a greater danger of destruction by enemy
action. A similar problem involved the choice between a dense network
of small warehouses and a scattered network of large ones.
Those were all problems in optimization; the techniques used in sol
ving them have since been successfully applied in peace-time economic
activity.
Prior to World War II, already in 1939, the Soviet mathematician
L. V. Kantorovich published his Matyematicheskiye metody organizatsii
proizvodstva (Mathematical Methods of Production Planning and
Organization),7 in which he gave an exposition of the basic ideas of
the science of programming in relation to problems of organization of
production and transport. His further works in the same field were
published in 1942 and 1949. Kantorovich's early works aroused some
interest among mathematicians rather than among economists. Recently,
however, the results obtained by Kantorovich and the possibilities of
their application in practice have become the object of increasing atten
tion. In 1959, L. V. Kantorovich published Ekonomicheski raschot opti-
malnogo ispolzovaniya resursov.8
Operational research and the theory of programming can be regarded
as part of praxeology, because the scope of their applications is not
restricted to economics alone. However, since the two sciences are at
present chiefly used in the field of economic activity, they are of special
importance to political economy and other economic sciences.
7 Kantorovich's paper was published in English in: The use of Mathematics in
Economics (V. Nemchinov ed.), Edinburgh-London, 1964.
8 The English translation, published by Pergamon Press Ltd. in 1965, appeared
under the title: The Best Use of Economic Resources.
