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Uncertain information and linear systems
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Studies in Systems, Decision and Control 254
Tofigh Allahviranloo
Uncertain
Information
and Linear
Systems
Studies in Systems, Decision and Control
Volume 254
Series Editor
Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences,
Warsaw, Poland
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Tofigh Allahviranloo
Uncertain Information
and Linear Systems
123
Tofigh Allahviranloo
Faculty of Engineering and Natural Sciences
Bahçeşehir University
Istanbul, Turkey
ISSN 2198-4182 ISSN 2198-4190 (electronic)
Studies in Systems, Decision and Control
ISBN 978-3-030-31323-4 ISBN 978-3-030-31324-1 (eBook)
https://doi.org/10.1007/978-3-030-31324-1
© Springer Nature Switzerland AG 2020
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To My Father
And
My Late Teacher, Prof. G. R. Jahanshahloo
Preface
In this book, I tried to introduce and apply the uncertain information or data in
several types to analyze the linear systems. These versions of information are very
applicable in our applied science. The initial subjects of this book point out the
important uncertainties to use in real-life problem modeling.
Having information about several types of ambiguities, vagueness, and uncertainties is important in modeling the problems that involve linguistic variables,
parameters, and word computing. Nowadays, most of our real-life problems are
related to decision making at the right time, and therefore, we should use intelligent
decision science. Clearly, every intelligent system needs real data in our environment to have an appropriate and flexible mathematical model. Most of the mentioned problems can be modeled by mathematical models, and a system of linear
equations is their final status that must be solved. The newest versions of uncertain
information have been discussed in this book.
This book has been prepared for all undergraduate students in mathematics,
computer science, and engineering involved with fuzzy and uncertainty. Especially
in industrial engineering and applied mathematics in the field of optimization, one
of the most important subjects is the linear systems with uncertainty.
Istanbul, Turkey Tofigh Allahviranloo
September, 2019
vii
Contents
1 Introduction ........................................... 1
1.1 Introduction ....................................... 1
2 Uncertainty ............................................ 9
2.1 Introduction to Uncertainty ............................ 9
2.2 Uncertainty ....................................... 9
2.2.1 Distribution Functions ......................... 9
2.2.2 Measurable Space ............................ 12
2.2.3 Uncertainty Space ............................ 14
2.2.4 Uncertainty Distribution Functions ................ 16
2.2.5 Uncertain Set ................................ 21
2.2.6 Membership Function ......................... 24
2.2.7 Level Wise Membership Function or Interval Form .... 31
2.2.8 Arithmetic on Intervals Form of Membership
Function ................................... 35
2.2.9 Distance Between Uncertain Sets ................. 49
2.2.10 Ranking of Uncertain Sets ...................... 56
3 Uncertain Linear Systems ................................. 61
3.1 Introduction ....................................... 61
3.2 Uncertain Vector and Matrix ........................... 61
3.3 The Solution Set of an Uncertain Linear System ............ 68
3.4 Solution Sets of Uncertain System of Linear
Equations in Interval Parametric Format................... 70
3.5 The System of Linear Equations with Uncertain RHS ......... 92
3.6 Uncertain Complex System ............................ 109
3.7 An Approach to Find the Algebraic Solution
for Systems with Uncertain RHS ........................ 119
3.8 An Estimation of the Solution of an Uncertain
Systems with Uncertain RHS .......................... 141
ix
3.8.1 Interval Gaussian Elimination Method .............. 143
3.9 Allocating Method for the Uncertain Systems
with Uncertain RHS ................................. 154
3.10 Allocating Method for the Fully Uncertain Systems .......... 163
3.10.1 Allocating Method for the Fully Uncertain Systems
(Non-symmetric Solutions) ...................... 173
3.11 LR Solution for Systems with Uncertain RHS
(Best Approximation Method) .......................... 178
3.12 LR Solution for Systems with Uncertain RHS
(Distance Method) .................................. 184
4 Advanced Uncertainty and Linear Equations .................. 211
4.1 Introduction ....................................... 211
4.2 The Uncertain Arithmetic on Pseudo-octagonal
Uncertain Sets ..................................... 211
4.2.1 The Uncertain Arithmetic Operations
on Pseudo-octagonal Uncertain Sets ............... 213
4.2.2 Solving Uncertain Equation as A þ X ¼ B .......... 229
4.2.3 Solving Uncertain Equation as A X ¼ B ........... 230
4.2.4 Solving Uncertain Equation as A X þ B ¼ C ....... 232
4.3 Combined Uncertain Sets ............................. 234
4.3.1 Ranking of Combined Uncertain Sets .............. 245
4.3.2 Distance Between Combined Uncertain Sets ......... 246
4.3.3 Ranking Method Based on Expected Value .......... 247
4.3.4 Advanced Combined Uncertain Sets (ACUSs) ........ 250
Bibliography ................................................ 255
x Contents
Chapter 1
Introduction
1.1 Introduction
Let’s start with a sentence from ‘Albert Einstein’:
As for the laws of mathematics refer to reality, they are not certain, and as far as they are
certain, they do not refer to reality.
Since the mathematical laws point to reality, this point is not conjectured with
certainty, and since it speaks decisive mathematical rules, it does not refer to reality
and is far from reality.
In fact, uncertainty has a history of human civilization and humanity has long
been thinking of controlling and exploiting this type of information. One of the
most ancient and obscure concepts has been the phrase “luck”. Evidence of gambling is said to have been obtained in Egypt in 3500 BC and found similar to the
current dice there. The gambling and dice have acted an important role in developing the theory of probability.
In the 15th century, Gerolamo Cardano was one of the most knowledgeable
individuals in the field of formal operations of algebra. In his “Game of Chance”, he
presented his first analysis of lucky laws. In this century, Galileo Galilei has also
solved such problems in numerical form. In 1657, Christiaan Huygens wrote the
first book on probability entitled “On the calculation of chance games.” This book
was a real birthday of probability.
The theory of probability started mathematically by Blaise Pascal and Pierre de
Fermat in the 17th century, which sought to solve mathematical problems in certain
gambling issues.
From the seventeenth century, the theory of probability was constantly developed and applied in various disciplines. Nowadays, the possibility in most engineering and management fields is an important tool, and even its use in medicine,
ethics, law, and so on. In this regard, Pascal says: It’s great that science was
© Springer Nature Switzerland AG 2020
T. Allahviranloo, Uncertain Information and Linear Systems,
Studies in Systems, Decision and Control 254,
https://doi.org/10.1007/978-3-030-31324-1_1
1
invented at the beginning to examine luck games, but today it is considered to be
the most important human knowledge.
In the early twentieth century, on the one hand, Bertrand Arthur William
Russell’s theories on the logic of zero and one, and the discovery of the principle of
uncertainty by Werner Karl Heisenberg in quantum physics led to a profound
challenge in Aristotelian logic. On the other hand, the scholars who believed in
Aristotle’s logic, to prove the logic and reject the logic of uncertainty, converge to a
three-valued and multi-valued logic as a generalization of Aristotelian logic.
But the scientists’ efforts to keep Aristotelian logic were fruitless and, as science
progressed, this challenge became deeper in dealing with real-world phenomena
modeling and scientists could not use the ease of use of inaccurate information in
real-world modeling. One of the ways that helped scientists to show that uncertainty
in phenomena was the use of a precise boundary for a range of solutions of a
mathematical model called “interval uncertainty, interval analysis, inclusive
methods” Or concealment methods introduced to the world.
The first attempts to calculate a periodic ambiguity was in 1931 by R. C. Young,
and in his doctoral dissertation, “Quantitative Algebra of Large Quantities,” he cited
computations in more than one point. In 1951, Dwyer of the University of Michigan
began to work with intervals, focusing on their role in digital devices, and several
years later, simultaneously with Warmus in Poland, Sunaga in Japan, and Moore in
America, they founded the intervals seriously. (The interval analysis was introduced
by Ramon Moore in 1959 as a tool for automatic control of errors in a calculated
result.)
But despite all the efforts made in this regard, the gap between the ambiguities in
natural phenomena and its modeling was still clearly visible, and scientists and
researchers also sought to find a new logic that allowed ambiguity, in essence,
modeling becomes closer to reality.
In 1937, Max Karl Ernst Ludwig Planck published an article on the analysis of
logic called ambiguity in Science journal, and in fact, the first person to speak of
ambiguity was Planck. He did not mention a fuzzy, but in fact explained the fuzzy
logic that, of course, was given by the universe science and philosophy were
ignored.
Eventually, in 1965, Professor Lotfi Ali Asker Zadeh, by changing the name of
ambiguity to fuzzy, opened a new way to accept this idea. Lotfi Zadeh published an
article entitled ‘Fuzzy sets’ in the Information and Control journal, which used a
new fuzzy logic for the sets. He considered the fuzzy name for these sets to divert it
from binary logic.
The emergence of this science opened up a new way of solving problems with
vague values and reality-based modeling, and scientists used it to model ambiguity
as part of the system.
In this book after describing uncertainty, we will cover a different understanding
of the concept of uncertainty in problems as linear systems. The uncertainty
sometimes means unknown, known but undetermined, what extent something can
be known, approximate, non-exact, misunderstanding and any other ambiguities in
the words. Therefore sometimes it can appear as approximate data, interval
2 1 Introduction
information, distribution form, random data, fuzzy information and any combinations of them.
Inherently, uncertainty is complexity with constraints and its theory is useful to
model the belief degree in mathematical science as a complicated model. So, the
systems with uncertain information do have uncertainty in their behavior. Without
information on uncertainty, there is a risk of misinterpretation of the results, and
mistaken decisions may be made. On this basis, unnecessary costs in the real-life
industry may be ignored.
Considering mentioned above concepts of uncertainty, it will be involved many
topics and fields of science. Indeed, there is nothing to compare with indeterminacy,
but everything can compare with determinacy and it is relative. Therefore, uncertainty or indeterminacy is absolute and important to discuss.
Concerning it, every topic like mathematics, finance, statistics, control, optimization, intelligent systems, expert systems, decision support systems, forecasting,
all fields of engineering and other majors are working with the concept of uncertainty. Besides, it is applicable in different concepts like measuring, variable in
modeling, programming, risk analysis, linguistic logic, reliability analysis, mathematical set, process and functions, calculus, random variable and so on.
In extreme conditions, uncertainty occurs at different levels of knowledge and
sciences that are involved the information processing and recognition, such as
economics, management, engineering, some parts of psychology. The concept of
uncertainty is highlighted and has long been of usage particularly in the areas of
decision making. As an additional illustration, a logical decision should be made in
the field of indeterminacy or in the real-life environment that is formed and combined by undetermined concepts and data, for instance in civil engineering, urban
planning, and Psychology. As it was mentioned before, the number of topics is not
limited to them.
There are various definitions of uncertainty in different fields of science and
real-life. Even the same field of science has different assessments on the subject.
Moreover the uncertainty in Human Psychology is evaluated as the difficulties
experienced in the relationship between variables of behavior-situation,
situation-result, situation-situation, decision-making process, behavior-future
probabilities.
On the other hand, uncertainty is the situation to doubt or not being able to
predict. For example: it is the situation where individuals do not make sense of their
own or other person’s behavior where they have social interaction. In addition, the
uncertainty of the purpose of life can be explained by the perception that the
existing meaning of life has disappeared, as well as the possibilities that may
explain in the future. For instance; waiting time before facing an event with a
potential to be harmful.
As a result, uncertainty often has a negative impact on human psychology.
Variables such as intolerance to uncertainty, general anxiety disorders and anxious
mood are some of them.
Finally, it is claimed that uncertainty acts an important role in the different
sciences.
1.1 Introduction 3
Human beings have always observed phenomena and conditions in the world
around them, from phenomena around the universe to phenomena that occur in the
human body. In all of these phenomena, the common aspect observed is ambiguity
in their nature, for example, when speaking of phenomena around humans such as
atmospheric phenomena, with terms such as rainfall, relatively severe precipitation,
heavy rainfall, dispersed precipitation, precipitation Very severe and …. Or if we
want to point out the phenomena in the human body, the amount of hormones
secretion in the body depends on such factors as time, sex, activity, lifestyle, diet
and many factors. As you know, in the corner of all the phenomena of the universe,
ambiguity is an integral part of every phenomenon.
When we investigate in nature, it is observed that all decisions that are made in
nature are also carried out on an uncertain system, for example, when a tree wants
to get its water and solubility, depending on the amount of humidity in the air, the
amount of warmness, the amount of wood and many other factors of water and
solos.
For a long time ago, the human brain was able to understand the ambiguity and,
based on its inference system, could easily understand and decide on the ambiguity
and, with time and experience, needed for each subject of the brain deduction
system improved and better decisions have been taken.
Since humans evolved and technology began to evolve, scientists demanded the
design and planning of devices that could automatically perform a series of actions,
but it was observed that this is a problem and cannot trains computers and controllers as the human brain to decompose things and then makes decisions.
Some great scientists believed that there are so many things that humans can do
easily, while computers and logical systems cannot do them, and the reason for this
is, the logical system is not an intelligent one. To design such a system its logic
should be familiar with uncertain concepts. So the uncertain logic is the basic logic
for a new technology that could explain what a particular phenomenon is making.
The important point is how to process and formulate human knowledge and
information in such a way that the meaning of a phenomenon is well identified and
appropriate decision is made. With reviews and studies on human decision making,
it has been observed that many professional decisions are based on the individual’s
experience and training.
In the training section, writing a proper model and formulating issues is much
more tangible, but in the experience section of the complicated design of the
appropriate model and formulation it is very difficult to work, and the question is,
how can an experience be introduced into the education system as a pattern?
In this regard, many studies and activities have been carried out and it has been
observed that important information sources come from two sources in practical
systems, one of the sources is the expert knowledge that defines their knowledge to
the systems with natural language and another one is mathematical measurements
and models derived from physical rules.
The main task of an uncertain system is to transform human knowledge into
mathematical models. In fact, an uncertain system has the ability to transform
human knowledge into mathematical models, and indeed the brain of an uncertain
4 1 Introduction
system is the knowledge-based system and knowledge of experts with the rules of
‘if-then’, are introduced as verbal expressions, and the second step is to combine
these rules into a single system in order to achieve a specific goal. We believe that
we should seek to build models that model ambiguity is as part of the system.
Some of the applications and relationships of other sciences with uncertain logic
and systems are application of uncertain logic and systems in agriculture, mobile
robots, archeology, medicine, medical engineering, and civil engineering.
One of the important reforms in the agricultural industry is the sustainability of
agricultural systems and the usage of uncertain information and logic is important
for assessing the stability of a system in three areas:
• Proper definitions of system stability indicators,
• Proper measurements of system stability indicators,
• The easy decision-making process for system stability.
Since the concept of sustainability is inherently obscure, so with three areas of
mentioned above activities we can reach sustainability in the agricultural system. In
Aristotle’s logic, when the sustainability of an agricultural system was investigated,
various dimensions of the system were eliminated due to the consideration of
deterministic sets and could not declare the degree of correctness of the sustainability of the desired indicator, but using logic with uncertain information, all the
indices, even with the smallest degree of membership is taken into account and the
system is stable.
In this regard, an assessment of the groundwater level for land under cultivation,
soil evaluation in the field of cultivation, re-cultivation of crops, and many other
issues with uncertain logic can be improved.
As another advantage of uncertainty is to improve agricultural tools using
uncertain controllers in three following domains,
• Planting and planting machinery, etc.,
• Have greenhouse irrigation systems and …,
• Automatic harvesters, harvester blades, and combines, etc.
Moreover, application of uncertain logic in the field of selling agricultural products
is in product grading and Product Marketing.
Application of uncertainty in mobile robots:
In the construction of mobile robots, due to the many advantages that uncertain
systems have for the control, such as ease of implementation of the controller,
reducing computing complicities, acceleration in response time, controller flexibility, robust nature of uncertain controller, successful industrial and laboratory
applications. Therefore, these systems are very used in the construction of mobile
robots. Some uses are as follows:
• Control the position of the mobile robot,
• A strategy to avoid dealing with barriers,
• Exact Navigation,
1.1 Introduction 5
• Motion tracking,
• Genetic uncertain control in the mobile robot.
Now some applications of uncertain systems in archeology:
Many archaeological data are vaguely and imprecise, so uncertain logic is very
useful in archeology.
In fact, this logic is a way of analyzing data that results from a lack of information and uncertainty in archeology, and no longer need to remove such data and
substitute data in ancient analyzes.
In fact, when archaeologists deal with data on objects or buildings that are
inaccurate or uncertain, or the evidence and relationships that exist between variables, are inaccurate, or when there is no consensus between archaeologists, the
uncertain logic can create these faults and gaps according to the following methods.
• Uncertain confidence
• Uncertain Inference System Design,
• Use uncertain statistics.
Therefore, uncertainty is also well used in the following cases.
• Restoration of semi-demolished monuments,
• Determine the history period of discovered objects,
• Restoration of discovered objects and ….
Application of uncertainty in medicine:
The uncertain logic is an inefficient tool for describing the behavior of complicated systems. These characteristics lead to it being used in the modeling of
biological devices as well as the basis for the diagnosis and treatment of diseases. In
medical sciences, an exact diagnosis of a doctor is one of the most important
treatment processes. The modeling of decision making processes that physicians
eventually identify is divided into two main parts.
• The first part includes data collection, including laboratory data, radiology,
patient examinations, and also general patient information, including the history
of illness, patient history, and so on.
• The second part of the information collected from the first section that is
examined by the physician and ultimately, through the inference process and
decision-making process, reaches the final diagnosis.
On the other hand, the existence of some properties in medical science makes it
necessary to use uncertain sets and computations.
The systems with uncertain information and logic are well used in the following
applications,
• The nature of a disease,
• Patient information collection process,
• The disease information collection process,
• Boundaries between signs and symptoms of diseases,
• Diagnosis of the disease,
6 1 Introduction