Speech in Plenary Session on
and Analysis of Safety and Risk in Complex Systems
(MA SR -2007)
Saint-Petersburg, Russia, September 4-8, 20071
Professor of Finance & Chairman, Indian Institute
Chief Editor, Finance India
The author would like to thank Professors Aman Agarwal,
Yamini Agarwal and Saurabh Agarwal for their helpful comments on the
earlier draft of this speech. I would also like to thank Mr. P.K. Jain,
Indian Institute of Finance for helping in the preparation of this speech.
1. Invited to be delivered as Opening Speech at the Russian Academy
of Sciences, St. Petersburg, RUSSIA, on 4th September, 2007. (4-8th
© J.D.Agarwal, Indian Institute of Finance
Audio Recording of Speech (Wave Format)
copy of the paper (In PDF Format)
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Chairman I.A. Ryabinin, Chairman of the plenary session. Professor Solojentsev
chairman of the National organizing Committee, Professor G.M. Reinstein
Co-chairman National Organizing Committee; Professor V.V. Karassev Secretary
National Organizing Committee Professor K.V. Frolov Academician (Russia-RAS)
Chairman of International Program Committee and other hon’ble co-chairmen
of International Program Committee, Hon’ble Scientists and Specialists,
speakers, delegates, invitees, ladies and gentlemen, it is my privilege
to have been invited to deliver this opening speech in the International
Scientific School on Modeling and Analysis of Safety and Risk in Complex
Systems (MA SR – 2007) organized by the Russian Academy of Sciences
from September 4-8 2007 in Saint Petersburg Russia.
At the outset, I would like to seize this opportunity to compliment
the organizers of this conference both for choice of the subjects to
be debated and representation of such a large number of scientists and
specialists from as many as 21 countries almost from all over the world,
to share the results of theoretical and practical researches in the
area of quantitative modeling and analysis of risk and develop the connection
between different areas of risk for the construction of overall risk
theory in business and engineering. In my opinion this may be one of
the largest congregations of well known scientists and specialists on
the subject presenting 70 research papers. To the best of my memory
such a wide variety of scientific papers from engineering to business
and finance coving areas risk in bankruptcy, risk in value of exchange,
risk estimates of portfolio investors, wealth effects of financial distress
on industry rivals, bank control and stability credit risk and risk
probabilistic uncertainty of probabilistic measures in portfolio analysis
are rarely handled in one scientific school.
This is a matter of pride for me to be a part of this scientific school,
meeting and knowing about the work of such a galaxy of scientists and
specialist in the historical and one of the most beautiful cities of
the world. I am sure the conference would be a grand success to fulfill
its main goal.
Modeling and analysis of Safety and risk in Complex systems has assumed
new dimensions with increasing complexities in the changing environment
in the world economy. The world has become a global village. Information
Technology, Satellite systems, Telecommunications, means of transportation
have reduced the size of the world in terms of operations. The mouse
of a computer and click of a button can handle transactions worth millions
and billions in no time. Pressing of a button on the computer or in
the laboratories may cause disasters for some, It may cause firms organizations,
banks financial institutions, stock markets and even some economies
to face serious economic and financial problems resulting in to virtual
failure. From the economic point of view the borders of the nations
have lost meaning. Even the cultures are being influenced through the
satellite. As the world is shrinking in size in terms of economic operations,
large systems are becoming more complex and vulnerable than ever before.
Risk which was associated with time in the past has now become an instant
phenomenon. In situations, when extreme steps are taken by some, systematic
risk is overtaken by unsystematic risk.
Modeling, Scientists & Specialists:
Models are symbols of perfection and lead to perfection and make this
world more beautiful, safer meaningful and facilitate living an enormously
good life. Some people may be critical of model builders for them to
be theoretical, living in white marble palaces and engaged in futile
exercises far away from reality. Time has proved and proves that these
very people have not only helped in the advancement of science but also
made the life of the people of the world more comfortable. Models over
the years have helped simple organizations to emerge as mega and complex
systems and facilitated risk mitigation. Today's world is a result of
yesterdays' models and contributions from scientists and specialists
like you, who have gathered here in this scientific school to make this
world great, risk free and people happy, prosperous and feeling safe.
The models may either evolve over a period or are a result of scientific
thinking. Its not just adding another brick on the wall already constructed
but born out of creativity, innovation, need to improve and of course
a scientist's secret urge to make his/her humble contribution to his
field of specialty to be a living human, across cultures, religions,
geographical boundaries, many many years ever after his/her death. His
desire to be referred, quoted and becoming a base for further discoveries
in the world.
I salute these scientists and specialists (past and present) who have
given this world so much at the cost of their own comfortable and luxurious
living. A man has only one life. He has a right to enjoy it like everyone
else, but, these scientists and specialists instead of spending and
enjoying in opera spend their life in laboratories, in libraries, on
their computers and dump themselves in their chairs for major part of
their day, month after month, year after year, in their lifetime. The
time, they are expected to be in intimate embrace with their wife and
justifiably to have a feeling of oneness is spent embracing scientific
issues building models, and merged in scientific equations. Even in
their sleep, their sub conscious mind is working on some problem, discovery,
or complexities of the model to seek solutions. Many times in the midnight
their wives discover them in their study or lab lost in their own world
seeking solution to some complex problem to make this world what it
would be tomorrow while others are living their life comfortably and
enjoying. In my young age my wife often found me missing on the bed
and in a huff discovered me on my table. She was generally very kind
to go to the kitchen and make a cup of tea and quietly keep it on my
table in support of my work. Of course there were no computers (Laptops)
at that time. Now she is tuned to the extent that when my children or
relative ask for me she would with a sweet smile say "where else,
see him in his study".
Most of the scientists are not expecting praise, position, publicity,
power or money for their discoveries and contributions. They are satisfied
if their models and discoveries are useful to the society and in their
field of specialty. Result of the sacrifices of scientists and specialists
and their models in the complex system are clearly visible.
Modeling & Analysis of Safety:
Safety is a matter of major concern for the state, corporates and people.
The world is faced with several odds from time to time. The inequalities
of incomes and wealth and consumption pattern amongst people and between
nations have widened. A part of the world, unfortunately, suffered from
extreme hunger, poverty, deprivation, underdevelopment, illiteracy,
diseases such as HIV/aid and lack of the most basic necessities of life
despite the best efforts of international agencies, the national governments
and NGOs. Appropriate models for safety and security against these evils
prevailing in the society, are required at global level.
Another major area of concern is safety from natural disasters such
as earthquakes, floods, droughts, Tsunami etc. Japan where large numbers
of earthquakes occur has devised mechanism to construct buildings which
are earthquake prone. Similarly models for early forecasting of some
of these natural disasters and facilitating early timely action have
been evolved by scientists and specialists.
Another major area of concern, is models for safety of plants, factories
and buildings. The fall of buildings, or collapse of a machine, aircraft
crash, train accidents or sudden shut down of a factory etc may cause
major catastrophe. Why should a train accident occur or a buildings
fall. At time short circuiting of electricity or leakage of gas or use
of outdated and very old machines or buildings, or human error is the
cause. Probabilistic models using Monte Carlo simulation can help avert
some of these catastrophes. Union Carbide gas leakage in 1984 in Bhopal,
India, was one of the worst catastrophes resulting in the loss of thousands
of lives and making thousands incapacitated. The commendable works of
E.D. Solojentsev and I.A. Ryabinin deserve special mention in this regard.
They have developed excellent logical Probabilistc models (LP models)
for managing safety and risk management in both business and engineering.
These models can be extended to other areas equally and efficiently.
Third, we need to develop models of safety against disasters which are
man made such as terrorist attack on the World Trade Centre and the
most recent devastating forest fire in Greece. A strict monitoring of
air surveillance through proper modeling might have helped averting
such events and providing safety of the twin towers and also life of
Fourth, Modeling and analysis of Safety against volatility of international/national
markets, money laundering, corruption, foreign exchange fluctuations,
bribing is of paramount importance. East Asia crisis of 1997-99, Harshad
Mehta Security scam of 1992 in India, Enron and Arthur and Anderson
episode in U.S. and massive money laundering taking place in the world
economy are some of the examples. There are models available but not
effectively used to monitor such situations. Some of the models even
require refinements using extensively probabilistic models, simulations
and stochasticity in the models. Scientists and specialists are involved
in bringing refinements but at times the models fail because there is
not enough support from the business or political leadership in implementing
Complex systems is therefore often used as a broad term encompassing
a research approach to problems in many diverse disciplines. Design
and development costs for extremely large systems could be significantly
reduced if only there were efficient techniques for evaluating design
alternatives and predicting their impact on overall system performance
metrics. Complex Systems is a new approach to science that studies how
relationships between parts give rise to the collective behaviors of
a system and how the system interacts and forms relationships with its
environment. Complexity theory takes its roots into Chaos theory, which
has its origins more than a century ago in the work of the French mathematician
Henri Poincaré. Chaos is sometimes viewed as extremely complicated
information, rather than as an absence of order. The point is that chaos
remains deterministic. With perfect knowledge of the initial conditions
and of the context of an action, the course of this action can be predicted
in chaos theory. As argued by Prigogine, Complexity is non-deterministic,
and gives no way whatsoever to predict the future. The emergence of
complexity theory shows a domain between deterministic order and randomness
which is complex. When one analyses complex systems, sensitivity to
initial conditions, for example, is not an issue as important as within
the chaos theory in which it prevails. As stated by Colander “The study
of complexity is the opposite of the study of chaos. Complexity is about
how a huge number of extremely complicated and dynamic set of relationships
can generate some simple behavioural patterns, whereas chaotic behaviour,
in the sense of deterministic chaos, is the result of a relatively small
number of non-linear interactions”. Due to the systems' analytical intractability,
simulation is the most common performance evaluation technique for such
systems. However, the long execution times needed for sequential simulation
models often hamper evaluation. The slow speeds of sequential model
execution have led to growing interest in the use of parallel execution
for simulating large-scale systems. Widespread use of parallel simulation
is easier now than ever before for integrating parallel model execution
into the overall framework of system simulation. One drawback to widespread
use of simulations is the cost of model design and maintenance. The
simulation environment, developed at UCLA attempts to address some of
these issues. It consists of three primary components: a parallel simulation
language called Parsec (parallel simulation environment for complex
systems), its GUI, called Pave, and the portable runtime system that
implements the simulation algorithms.
Risk & Uncertainty in Complex Systems:
Risk management is managing (preparing for) future uncertainties. A
leading example, originating from finance, is the problem of choice
among mutually exclusive investment opportunities or portfolios having
uncertain returns. Uncertainties are risks. They are the unknowns associated
with future events. The decisions we make today create the risks that
we must manage tomorrow. Risk management and decision-making is effectively
the same thing. Both involve the dismantling of choices so as to understand
uncertainty of outcomes associated with a particular option. High quality
decision-making serves the purpose of risk management. If decisions
include a thorough consideration of uncertainties, then future risks
are simplified or minimized.
The problem of risk and uncertainty in economics is not new. The treatment
of uncertainty in decision-making is traced as far back as 1738 with
Petersburg Poradox1. But in the work of Knight2
(1921) risk and uncertainty have been recognized as the pertinent areas
in economics. Knight stated that the problem of risk and uncertainty
has been recognized and discussed primarily in three connections (i)
insurance, (ii) speculation, and (iii) entrepreneurship. Knight pointed
out that uncertainty must be taken in a sense radically distinct from
the familiar notices of risk from which it has never been separated.
According to him, ‘risk’ refers to those cases where a quantity is susceptible
of measurement, while ‘uncertainty’ refers to the case of non-quantitative
type. The work of Knight has primarily been recognized as referring
risk to a situation where the probability of occurrence of each outcome
of a decision is known while uncertainty has been recognized as referring
to situation where the probability of occurrence of each outcome of
a decision is not known. Miller3(1977) is of the opinion
that no such attention is paid to this distinction today because in
either case the future is unknown.
Every decision is a balance between what we believe to be true and what
we are forced to predict. Every decision involves the analysis of available
information and ultimately the selection of a choice among alternatives
with varying degrees of uncertainty. It follows that to better our information
and the more balanced and thorough analysis, the higher can be the quality
of our decisions. Better decisions are informed, reasoned, and balanced.
Making better decisions means living with less risk. Poor decisions
are risky. They are made without a full understanding of what might
go wrong. It is not that if you make a poor decision you are necessarily
going to be proven wrong, but that the decision was made without a full
understanding of the uncertainties or risks involved. With poor decisions,
risks are understated and returns exaggerated. High quality decisions
imply a complete understanding of the uncertainties involved. Making
high quality decisions involves recognizing what risks you are taking.
It is about making informed decisions. Risks are the residual uncertainties
left behind when decisions are made without perfect information. But
of course we never have perfect information. We therefore make decisions
based on what we believe to be true and take a chance (accept the risks)
of being wrong. High quality decisions are decisions in which the magnitude
of the risk of being wrong is understood
There are three sources of uncertainty inherent in decision-making.
Known-unknowns are areas of uncertainty that are recognized and integrated
into the decision-making process. This is the stuff about which we know
and we should be concerned. In deciding which type of new car to purchase,
the known-unknowns include future repair costs (after the warranty period
is over) for one brand versus another or residual value (sale price)
after a number of years of use. These are issues that we know to worry
about but do not know just how worried to be.
Unknown-unknowns are risks (uncertainties) that are relevant to decision
but not included in the analysis. This is the stuff that is”off our
radar”. Your level of knowledge about a particular situation determines
unknown-unknowns. The more informed you are, the fewer (or more obscure)
The third element of risk, analytical-bias, relates to imperfections
in our understanding and analysis of choices. This is the stuff of habit,
prejudice, and mental laziness. Analytical bias may be referred to as
the unknown-unknowns of the known-unknowns. It is a consequence of being
human. Every analysis is a reflection (to some degree) of the person
doing analysis. Analytical bias is the difference between what we believe
to be true and what is actually true. It is the difference between fact
A large number of problems in production planning and scheduling, location,
transportation, finance, and engineering design particularly in complex
systems, require that decisions be made in the presence of uncertainty.
Uncertainty, for instance, governs the prices of fuels, the exchange
rate fluctuations, the availability of electricity, and the demand for
chemicals. A key difficulty in optimization under uncertainty is in
dealing with an uncertainty space that is huge and frequently leads
to very large-scale optimization models. Decision-making under uncertainty
is often further complicated by the presence of integer decision variables
to model logical and other discrete decisions in a multi-period or multi-stage
setting. Decision quality is measured in terms of residual risk left
behind as a result of imperfections in knowledge and analysis. The highest
quality decisions have little or no uncertainly, the poorest are based
on ill informed guesswork; they are “a stab in the dark.” So, to understand
the quality of a decision one must understand the weaknesses in the
inputs that have gone into the decision-making process and in the approaches
taken to reaching decision. The degree to which various sources of risk
in a decision are considered determines the quality of the decision.
Decision quality is risk management.
The effect of risk and uncertainty on asset prices, on rational decision
has increasingly engaged the attention of economists and other researchers.
Modeling & Analysis of Safety & Risk in Complex Systems:
Another area of concern or managing risk and uncertainty in complex
systems is capital investment and portfolio investment. Most of the
literature in capital investment and portfolio decisions under risk
and uncertainty has mainly followed three trends: (1) Simplistic Approach,
(2) Portfolio Theory Approach and (3) Mathematical Programming Approach.
The first type of approach is to use a simple criterion by suggesting
a simple modification in the deterministic criterion. Some of the capital
budgeting decision techniques that have been suggested in the past and
belonging to this type are payback, risk adjusted discount rate, and
certainty equivalent approach. This approach although in practice is
highly tractable, cheap, quick and easily understood, but as Wilkes
very rightly stated is less intellectually satisfying.
The Portfolio Theory Approach includes techniques such as probability
distribution simulation approach, decision tree analysis, utility theory
and sensitivity analysis to measure the return and variance on capital
employed, as a surrogate measure of risk. The portfolio analysis approach
so far gains only limited acceptance
Therefore a more recent approach has been to treat some of model parameters
as random variables. The major contributions of this approach, referred
as Mathematical Programming Approach, have come from Naslund (1968),
Beck (1967) and Benhard (1969). Most of these approaches were based
on an erroneous assumption that firms pursue single goal while firms
pursue multiple goals and conditions of certainty never exist. A stochastic
goal programming model and fuzzy goal programming models developed by
me could provide a possibly good solution.
Capital Investment in the production area involved selection of fixed
assets under conditions of uncertainty regarding future product demand.
The fixed assets are generally chosen with consideration given to their
income generating capacity; however that consideration is frequently
secondary to other decision criteria, such as the desire to satisfy
demand or perhaps minimize excess production, achieve quality control
standards, space utilization, meeting environmental requirements etc.
Over the second half of the 20th century, optimization found widespread
applications in the study of physical and chemical systems, production
planning and scheduling systems, location and transportation problems,
resource allocation in financial systems, and engineering design. From
the very beginning of the application of optimization to these problems,
it was recognized that analysts of natural and technological systems
are almost always confronted with uncertainty. As such, the traditional
approach to capital budgeting, i.e. maximization of net present value,
break down when the complexities of production environment are considered.
In order to deal with this problem, a hierarchical optimization model
with ability to reflect multiple conflicting goals is necessary. Moreover,
in order to adequately describe the decision environment problem of
uncertainty and multiple conflicting goals are introduced using a chance
constrained integer goal programming model.
Beginning with the seminal works of Beale (1955),Bellman (1957), Bellman
and Zadeh (1970), Charnes and Cooper (1959), Dantzig (1955), and Tintner
(1955), optimization under uncertainty has experienced rapid development
in both theory and algorithms. Today, Dantzig still considers planning
under uncertainty as one of the most important open problems in optimization
(Horner, 1999). Approaches to optimization under uncertainty have followed
a variety of modeling philosophies, including expectation minimization,
minimization of deviations from goals, minimization of maximum costs,
and optimization over soft constraints. Early attempts to formulate
the capital budgeting problem employed mathematical programming. In
particular linear programming was used for maximizing net present value
of project selected subject to some budget constraint by H.M. Weingartner.
In these models multiple conflicting goals were not assumed to exist
or it was assumed that the benefits from these goals could be easily
translated into monetary benefits and incorporated into the existence
of objective function.
Theories and models, both mathematical and non-mathematical, developed
in the West erroneously postulate the attainment of single objective
of profit maximization and assume conditions of certainty. Firms in
general pursue multiple objectives and conditions of certainty never
exist. It is also generally seen that a decision maker faces a number
of potentially serious problems choice among a large number of alternative
projects, financial and non-financial constraints on capital resources,
critical manpower problems, unreliable or income relationships. Each
of these problems together with many others requires special consideration
and may pose different constraints on decision making. All the techniques,
theories and models based on single goal hypothesis, are outdated, irrational,
misleading and give erroneous results, as firms and people pursue multiple
goals with priority structuring incorporating risk and uncertainty.
Maximum return with minimum risk is the dictum which everybody follows.
Unfortunately, the existence of multi-conflicting goals measured in
incommensurable units may preclude this translation. For this reason
Agarwal (1976) developed capital budgeting model dealing with multiple
conflicting goals, within a hierarchical optimization framework and
ordinal priority structuring. While this model provides for inclusion
of multiple goals, they do not reflect the problem of uncertainty of
future product demand. However, by employing chance-constrained capabilities
as a supplement to regular Integer Goal-Programming Model, this deficiency
can be compensated for. This solution approach to the capital budgeting
problem in production area provides practical advantage over linear
programming. It allows inclusion of multiple conflicting goals measured
in incommensurable unit.
Agarwal (1978) model on Stochastic Goal Model for Capital Budgeting
Decisions under Uncertainty takes care of uncertainty situations as
well as multiple goals. The model which is primarily an extension of
his Goal Programming model for Capital budgeting decisions (1976) is
primarily a mathematical programming model but involves various quantitative
techniques particularly probability theory, times series iterations,
mean variance approaches, simulation and sensitive analysis to incorporate
and handle parameters affected by uncertainty. The information about
some of the models developed by me, is available in my books and articles
and also on our website: http://www.iif.edu
At the end, I must take this privilege to thank the organizers particularly
Professor E.D.Solojentsev and Professor Aman Agarwal for giving this
honor to me to share my views on the subject with such august gathering.
I wish I were have been one of you - scientists and specialists, presenting
my own work. With this, I have attempted to perform my most earnest
duty assigned to me by the organizing committee.
I once again salute the scientists and specialists who significantly
contributed through their modeling and discoveries to make this world
a place for better living for all.
It has been my earnest desire to visit Russia ever since 1966 when as
a young boy, I got an opportunity to meet and spend about two days and
was highly impressed by a Russian Scientist visiting India. I am glad
40 years later it is fulfilled today.
I must also thank the members of this august house, the scientists and
specialists, ladies and gentlemen, for their very kind patient hearing.
I am sure each one of us would be wiser and wealthy with the research
work sharing with other scientists during the international scientific
Once again I would like to congratulate the organizers and the Russian
Academy of Sciences for organizing International Scientific School and
giving all of us an opportunity to share our seminal work. I wish the
conference a grand success.
Agarwal, J.D.,”Capital Budgeting Decision under Risk and Uncertainty”,
Doctoral Dissertation (pp.168) submitted to University of Delhi, 1976
and published in 1988 by Indian Institute of Finance.
Agarwal, J. D. “A Goal Programmed Model for Capital Budgeting Decisions”,
presented in operations Research Society in India Conference held at
Indian Institute of Management, Bangalore, in December 1978 and published
in Arth Vijnan - The Journal of Institute of Economics and Politics
, Pune, Vol.22, No.3 & 4 ,September - December, 1981, pp. 299-313.
Agarwal, J.D.,”A Stochastic Goal Programming Model for Capital Budgeting
Decisions under Uncertainty,” Finance India,Vol.1, December 1987,pp.1-17.
Agarwal, J. D. and Chandra Prakesh Gupta, “A Fuzzy Goal Programming
Model With Additively Structure to Capital Budgeting Decisions” , Finance
India , Vol. V, No. 2,June 1991,pp. 161-172.
Agarwal, J. D. and Chandra Prakesh Gupta, “Capital Budgeting Decisions:
A Fuzzy Programming Model”, Finance India, Vol. III, No. II,
Arrow, K.J. , “Alternative Approach to The Theory of Choice in Risk
Taking Situations”, Econometrica, Vol.19,1951,pp 404-437
Bernoulli, D., “Exposition of a new theory of the measurement of risk”,
Econometric, 1954, 99. 23-36, Translation of a paper, originally
published in Latin in St. Petersburg in 1738.
Charnes, A., W. W. Cooper and G. L. Thompson, “Constrained Generalized
Medians and Hyper medians As Deterministic Equivalents for Two-Stage
Linear Programs under Uncertainty”, Management Science, Vol.
12, No. 1, September., 1965, pp. 83-112
Chranes, A. and W. W. Cooper, “Chance Constrained Programming” , Management
Science, Vol. 6, No. 1, October 1959, pp. 73-79
Cilliers, P. (1998). Complexity and Postmodernism : Understanding
Complex Systems, Routledge, London
Dantzig,T, and Henri Poincaré , “Critic of Crisis : Reflections
on His Universe of Discourse”, New York, 1954
Egeerton, R.A ., “Investment Decision under Uncertainty”, Liverpool
University Press, 1960
Kiel, L. Douglas and Euel W. Elliott, “Chaos Theory in the Social
Knight, Frank H., “Risk, Uncertainty and Profit”, University
of Chicago Press, Chicago, 1921, reprinted 1971.
L.A.N. Amarala and J.M. Ottino, Complex networks — augmenting the
framework for the study of complex system, 2004.
Markowitz , H , “Mean Variance Analysis in Portfolio Choice and Capital
Markets” ,Basil Blackwell Inc. , New York,1987.
Markowitz , H., “Portfolio Selection” , Journal of Finance, Vol.
Markowitz, H.M., “Portfolio selection, Efficient Diversification
of Investment”, John Wiley and Sons, 1959
Merton , R. “A Simple Model of Capital Market Equilibrium with incomplete
Information” ‘ Journal of Finance, Vol. 52,1987,pp.483-510.
Miller, M.H., “Risk, Uncertainty and Diversification of Opinion”, Journal
of Finance, September 1977, pp. 1155
Ryabinin, I.A., Reliability and safety of structure-complex systems,
Saint Petersburg: Politechnika, 2000
Sharpe , W. “Capital Asset Prices : A Theory of Market Equilibrium Under
Condition of Risk” ,Journal of Finance, Vol . 19, 1964,pp. 425-442.
Sharpe, William F., “Capital Assets Prices With and Without Negative
Holdings”, Finance India, Vol. V No. IV, December 1991, pp. 469-486
Solojentsev, E.D., The system of automated debugging of complex objects
– volumetric engergetic machines, Control systems and machines,
Solojentsev,E.D., Scenario Logic and Probabilistic Management of
Risk in Business and Engineering, Springer.
Solojentsev, E.D., Karassev V.V., Risk Logic and Probabilistic Models
in Business and Identification of Risk Models, Informatics, 2001;
Tintner, G, “Stochastic linear programming with applications to agricultural
economics”, Proceedings of the Second Symposium in Linear Programming,
White,D. J.,“A Bibliography on the Applications of Mathematical Programming
Multiple-Objective Methods” , The Journal of the Operational Research
Society, Vol. 41, No. 8 (Aug., 1990), pp. 669-691.
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