Allahviranloo t uncertain information and linear systems 2019

information, distribution form, random data, fuzzy information and any tions of them.combina-Inherently, uncertainty is complexity with constraints and its theory is useful tomodel the b

Studies in Systems, Decision and Control 254

Tofigh Allahviranloo

Uncertain

Information and Linear

Systems

Studies in Systems, Decision and Control Volume 254

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To figh Allahviranloo

Uncertain Information and Linear Systems

123

Tofigh Allahviranloo

Faculty of Engineering and Natural Sciences

Bahçeşehir University

Istanbul, Turkey

Studies in Systems, Decision and Control

https://doi.org/10.1007/978-3-030-31324-1

© Springer Nature Switzerland AG 2020

This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part

or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed.

The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard

This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To My Father

And

My Late Teacher, Prof G R Jahanshahloo

In this book, I tried to introduce and apply the uncertain information or data inseveral types to analyze the linear systems These versions of information are veryapplicable in our applied science The initial subjects of this book point out theimportant uncertainties to use in real-life problem modeling

Having information about several types of ambiguities, vagueness, and tainties is important in modeling the problems that involve linguistic variables,parameters, and word computing Nowadays, most of our real-life problems arerelated to decision making at the right time, and therefore, we should use intelligentdecision science Clearly, every intelligent system needs real data in our environ-

men-tioned problems can be modeled by mathematical models, and a system of linear

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

of the most important subjects is the linear systems with uncertainty

September, 2019

vii

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 asA þ X ¼ B 229

4.2.3 Solving Uncertain Equation asA  X ¼ B 230

4.2.4 Solving Uncertain Equation asA  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

Chapter 1

Introduction

1.1 Introduction

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 withcertainty, and since it speaks decisive mathematical rules, it does not refer to realityand is far from reality

In fact, uncertainty has a history of human civilization and humanity has longbeen thinking of controlling and exploiting this type of information One of the

gam-bling is said to have been obtained in Egypt in 3500 BC and found similar to thecurrent dice there The gambling and dice have acted an important role in devel-oping the theory of probability

In the 15th century, Gerolamo Cardano was one of the most knowledgeable

solved such problems in numerical form In 1657, Christiaan Huygens wrote thefirst book on probability entitled “On the calculation of chance games.” This bookwas a real birthday of probability

The theory of probability started mathematically by Blaise Pascal and Pierre deFermat in the 17th century, which sought to solve mathematical problems in certaingambling issues

From the seventeenth century, the theory of probability was constantly oped and applied in various disciplines Nowadays, the possibility in most engi-

© 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 bethe most important human knowledge.

In the early twentieth century, on the one hand, Bertrand Arthur William

uncertainty by Werner Karl Heisenberg in quantum physics led to a profoundchallenge in Aristotelian logic On the other hand, the scholars who believed in

three-valued and multi-valued logic as a generalization of Aristotelian logic

progressed, this challenge became deeper in dealing with real-world phenomenamodeling and scientists could not use the ease of use of inaccurate information inreal-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

computations in more than one point In 1951, Dwyer of the University of Michiganbegan to work with intervals, focusing on their role in digital devices, and severalyears later, simultaneously with Warmus in Poland, Sunaga in Japan, and Moore inAmerica, 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 calculatedresult.)

But despite all the efforts made in this regard, the gap between the ambiguities innatural phenomena and its modeling was still clearly visible, and scientists and

modeling becomes closer to reality

In 1937, Max Karl Ernst Ludwig Planck published an article on the analysis of

ambiguity was Planck He did not mention a fuzzy, but in fact explained the fuzzylogic that, of course, was given by the universe science and philosophy wereignored

new fuzzy logic for the sets He considered the fuzzy name for these sets to divert itfrom binary logic

The emergence of this science opened up a new way of solving problems withvague 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 uncertaintysometimes means unknown, known but undetermined, what extent something can

be known, approximate, non-exact, misunderstanding and any other ambiguities inthe words Therefore sometimes it can appear as approximate data, interval

information, distribution form, random data, fuzzy information and any tions of them.

combina-Inherently, uncertainty is complexity with constraints and its theory is useful tomodel the belief degree in mathematical science as a complicated model So, thesystems with uncertain information do have uncertainty in their behavior Withoutinformation on uncertainty, there is a risk of misinterpretation of the results, andmistaken decisions may be made On this basis, unnecessary costs in the real-lifeindustry may be ignored

Considering mentioned above concepts of uncertainty, it will be involved many

but everything can compare with determinacy and it is relative Therefore, tainty or indeterminacy is absolute and important to discuss

opti-mization, intelligent systems, expert systems, decision support systems, forecasting,

uncer-tainty Besides, it is applicable in different concepts like measuring, variable inmodeling, programming, risk analysis, linguistic logic, reliability analysis, mathe-matical set, process and functions, calculus, random variable and so on

In extreme conditions, uncertainty occurs at different levels of knowledge andsciences that are involved the information processing and recognition, such aseconomics, management, engineering, some parts of psychology The concept ofuncertainty is highlighted and has long been of usage particularly in the areas ofdecision making As an additional illustration, a logical decision should be made in

com-bined by undetermined concepts and data, for instance in civil engineering, urbanplanning, and Psychology As it was mentioned before, the number of topics is notlimited to them

probabilities

On the other hand, uncertainty is the situation to doubt or not being able topredict For example: it is the situation where individuals do not make sense of their

uncertainty of the purpose of life can be explained by the perception that theexisting meaning of life has disappeared, as well as the possibilities that mayexplain in the future For instance; waiting time before facing an event with apotential 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 anxiousmood are some of them

Finally, it is claimed that uncertainty acts an important role in the differentsciences

Human beings have always observed phenomena and conditions in the worldaround them, from phenomena around the universe to phenomena that occur in thehuman body In all of these phenomena, the common aspect observed is ambiguity

in their nature, for example, when speaking of phenomena around humans such asatmospheric phenomena, with terms such as rainfall, relatively severe precipitation,

want to point out the phenomena in the human body, the amount of hormonessecretion in the body depends on such factors as time, sex, activity, lifestyle, dietand 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 innature 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, theamount of warmness, the amount of wood and many other factors of water andsolos

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 ambiguityand, with time and experience, needed for each subject of the brain deductionsystem improved and better decisions have been taken

Since humans evolved and technology began to evolve, scientists demanded thedesign 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 con-trollers as the human brain to decompose things and then makes decisions.Some great scientists believed that there are so many things that humans can doeasily, 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 logicshould be familiar with uncertain concepts So the uncertain logic is the basic logicfor a new technology that could explain what a particular phenomenon is making.The important point is how to process and formulate human knowledge and

appropriate decision is made With reviews and studies on human decision making,

experience and training

In the training section, writing a proper model and formulating issues is muchmore tangible, but in the experience section of the complicated design of the

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 beenobserved that important information sources come from two sources in practical

the systems with natural language and another one is mathematical measurementsand models derived from physical rules

The main task of an uncertain system is to transform human knowledge intomathematical models In fact, an uncertain system has the ability to transformhuman knowledge into mathematical models, and indeed the brain of an uncertain

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

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 logicand systems are application of uncertain logic and systems in agriculture, mobilerobots, archeology, medicine, medical engineering, and civil engineering

One of the important reforms in the agricultural industry is the sustainability ofagricultural systems and the usage of uncertain information and logic is importantfor 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 ofmentioned above activities we can reach sustainability in the agricultural system In

various dimensions of the system were eliminated due to the consideration ofdeterministic sets and could not declare the degree of correctness of the sustain-ability of the desired indicator, but using logic with uncertain information, all theindices, even with the smallest degree of membership is taken into account and thesystem is stable

In this regard, an assessment of the groundwater level for land under cultivation,

issues with uncertain logic can be improved

As another advantage of uncertainty is to improve agricultural tools usinguncertain controllers in three following domains,

• Planting and planting machinery, etc.,

• Have greenhouse irrigation systems and …,

• Automatic harvesters, harvester blades, and combines, etc

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 uncertainsystems have for the control, such as ease of implementation of the controller,

flexi-bility, robust nature of uncertain controller, successful industrial and laboratoryapplications Therefore, these systems are very used in the construction of mobilerobots Some uses are as follows:

• Control the position of the mobile robot,

• A strategy to avoid dealing with barriers,

• Exact Navigation,

• 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 veryuseful in archeology

In fact, this logic is a way of analyzing data that results from a lack of mation and uncertainty in archeology, and no longer need to remove such data andsubstitute data in ancient analyzes

infor-In fact, when archaeologists deal with data on objects or buildings that areinaccurate or uncertain, or the evidence and relationships that exist between vari-ables, are inaccurate, or when there is no consensus between archaeologists, theuncertain 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:

com-plicated systems These characteristics lead to it being used in the modeling ofbiological devices as well as the basis for the diagnosis and treatment of diseases Inmedical sciences, an exact diagnosis of a doctor is one of the most importanttreatment processes The modeling of decision making processes that physicianseventually 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 isexamined by the physician and ultimately, through the inference process and

On the other hand, the existence of some properties in medical science makes itnecessary to use uncertain sets and computations

The systems with uncertain information and logic are well used in the followingapplications,

• 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,

• Treatment of the disease,

• Treatment methods,

• Determine dosage and …

Application of fuzzy logic in medical engineering:

Since many medical devices could not have realistic simulations in a real andnatural condition to provide patients with real simulations, uncertain logic has beenvery effective in improving the performance of these systems and medical devicesand has been able to give a much better result Therefore, they have shown theirperformance well for the following purposes,

• Removing additional noise when recording electrical events of the heart, brain,ear, stomach, etc.,

• Moving devices into the body in order to maintain the patient’s naturalconditions,

• Producing smart and sensitive drugs for some of the body excitements,

• Making micro-robots that can enter a specific dosage of the drug at certaintimes,

• Determine the intensity of sports devices and …

In all civil engineering applications, components of the process at different stages ofthe project including design, planning, construction, operation and so on, involvevarious sources of uncertainties Most parameters of the system such as geometry of

They are intrinsically uncertain or random parameters Thus the safety of the systemscannot be achieved by deterministic models using the average values of the parameters

as usual Consequently, engineering decisions regarding all these processes should bebased on more realistic and probabilistic models considering the uncertain nature ofthe parameters in the processes

Other applications in civil engineering are,

• Determine the level of risk of structural changes and risk analysis,

• Predict the durability of building materials such as concrete and …,

• Anticipating the strength of building materials against atmospheric and ronmental factors,

envi-• Detection and prediction of the demolition of materials,

• Evaluation of corrosion of materials according to various factors,

• Forecasting the type of destruction and its level,

• Predict long-term material behavior,

• Identify patterns of destruction and corrosion,

• Determine the best type of materials in different areas and …

Chapter 2

Uncertainty

2.1 Introduction to Uncertainty

uncertainties The concept of uncertainty can appear in several forms Sometimes it

is as an uncertain set, an expected value or interval data, random data and bination of all of them A different point of view of it, can be described as amembership function entitled fuzzy data and combination of membership, proba-bility and distribution functions In this section, we are going to have discussionsabout all of the mentioned above data

com-2.2 Uncertainty

uncertainty distribution functions But before it, we should talk about uncertainvariable in uncertainty space To this end, discussion on measurable space is nec-

measurable set, a Borel algebra and also a Borel set, and measurable function are our

2.2.1 Distribution Functions

pre-dicted exactly, one of the ways is distribution functions For instance, throwing

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Studies in Systems, Decision and Control 254,

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9

coins, tossing dice, playing poker, stock pricing, marketing and market demand,lifetime and others.

As far as we know to obtain the distribution function of indeterminacy

second is our mental beliefs

Considering the previous observations factor, all observations can be shown bysome points on a function that is named distribution function

Suppose that we are talking about beautiful people and the question is, who isbeautiful? It is clear that we have a global set of people and a set of numbers belong

to closed interval between zero and one as belief (membership) degree of each

degree is one Otherwise it should be between zero and one It means, less beautiesless belief degrees The data for the examples like this, are concluded from thebeliefs Indeed, the behavior function in terms of belief degree is called a distri-

Any other example about personal characteristics and quality adjectives havealso a belief based distribution function In fact, these functions simulate thebehavior of an uncertain variable that will be explained later

In accordance with personal knowledge, the belief degree function represents thedegree of believes to the quantities and clearly in the case of changing the personalknowledge, the belief degree will change as well

Fig 2.1 A belief based

distribution function in

Gaussian form

Fig 2.2 A belief based

distribution function in S form

And the question is, is the belief degree true? I have to say that all of the beliefdegrees may wrong, but some of them are useful However without considering thetrueness or wrongness of all belief degrees, they are useful for decision making.

points out possibility or impossibility of the happening and the second one, points

case the uncertainty is presented by distributive function and in the second case it ispresented by membership function

frequency of distributive function and talks about the possibility or impossibility of

about the quality of the happening

correspond to uncertainties So, any calculation on these functions are the samecalculation on their uncertain variables

uncertain distribution function As we mentioned before, we are going to discussabout some other types of uncertain variables with different forms of distributionfunctions Some special uncertain distribution are listed as follows:

2.2.2 Measurable Space

Mathematically and essentially, the uncertainty theory is an alternative theory ofmeasure in a measurable space and this is why the uncertainty theory should discuss

on a measurable space In this section we are not going to discuss deeply about the

other books whom discuss about them completely

Definition 2.1 (Sigma-Algebra)

over nonempty set if the following conditions hold:

• M Contains the nonempty set

• Every member in the collection has its complement

• For containable members of M, it contains their union

If in the third condition the number of members is uncountable then an Algebra

is called a Sigma-Algebra

Definition 2.2 (Sigma-Algebra Space)

Spacety set M; (M; Nonempty setÞ is called a Sigma-Algebra space and anymember of nonempty set is called measurable set

Definition 2.3 (Borel Set)

The Borel-Algebra over the set of real numbers is defined as a smallestSigma-Algebra B; containing all open intervals and, any element in B is called aBorel set

Example 2.4 The intervals, open sets, closed sets, set of rational and irrationalnumbers are several types of Borel sets

Remark 2.5 The most important items to define an uncertain set with uncertainproperties are, the three mentioned above items In fact, they show that anyuncertain set has an uncertain property and any member of this set have a measure

by three above mentioned items

In an uncertain space, a measurable set is an event in uncertainty theory andevery number associated to happening of the event is actually the belief degreewhich we believe the event will happen Now it should be clear that the uncertainmeasure or belief degree depends on personal knowledge and it will be changed ifthe personal knowledge are changed.

1 Normality, the belief degree of an event is between 0 and 1 and it cannot exceed

1 and there is at least one event with belief degree of 1 (complete belief) For

2 Duality, the belief degree of a complementary event is equal to 1- belief degree

of the event

For instance, if a proposition is true with belief degree 0.3, then clearly it is false

at the same time with belief degree 0.7

It means that, how the belief degree of union of some events is generated by the

Exercises

Show that BD is an uncertain measure

infor-Definition 2.6 (Uncertainty Space)

Any space containing all triple members formed by, (a nonempty set, aSigma-Algebra on it, and belief degree (as an uncertain measure)), is calleduncertainty space

Definition 2.8 (Uncertain Variable)

An Uncertain Variable f as an event for any Borel set B, is a function that is

Remark 2.9 An event is a subset of real numbers and

Example 2.10 Any function from an uncertainty space (A nonempty set with powerset and with determined belief degree of each member) to the Borel set on realnumbers subject to the summation of belief degrees is one, is an uncertain variable

It means

i¼1

Thus it is clear that,

Definition 2.11 (Sign of Uncertain Variable)

An uncertain variable is said to be positive (negative) if the belief degree ofnegative (positive) uncertain variable is zero respectively And this is true fornonnegative and non positive one

Definition 2.12 (Equality of Uncertain Variables)

Two uncertain variables from the same uncertainty space are equal if for almostall c from the universal set they are equal

And it is obvious that,

Fig 2.4 An uncertain variable

Definition 2.13 For n uncertain variables ff gi n

mono-tone real valued function f then

f cð Þ ¼ f fð 1ð Þ; fc 2ð Þ; ; fc nð ÞcÞ; 8c 2 universal set

Is an uncertain variable It means, the summation, subtraction, multiplication anddivision of a countable number of uncertain variables are an uncertain variable as

Remark 2.14 In the division the denominator should not contains zero

2.2.4 Uncertainty Distribution Functions

In this section we are going to cover a discussion on uncertainty distribution that is

Definition 2.15 (Uncertainty Distribution)

For any real number x if f  x then the set of all belief degrees of f is defined as

an uncertainty distribution Indeed the domain of the function is an uncertain

0; 1

It is clear that if x increases then the belief degree increases Thus the function is

Fig 2.5 An uncertainty distribution function

Example 2.16 Suppose the nonempty set is A ¼ af 1; a2g and

the uncertain variable

and Lebesgue measure on it Then

Remark 2.18 In general, if f and u are uncertain variables and continuous tainty distributions respectively, and g is a invertible strictly monotone functionthen for any real x,

uncer-w1ð Þ ¼ u gx 1ð Þx

and w2ð Þ ¼ 1  u gx 1ð Þx

if g is strictly decreasing

Remark 2.19 It can be said that, two uncertain variables have the same distribution

or they are identically distributed

of uncertainty distributions as follows

Definition 2.20 (Linear Uncertain Variable)

If the uncertainty distribution of an uncertain variable is linear then the uncertainvariable is called linear Generally,

It is clear that the real numbers a; b can be chosen from the set of real numbers.The following example is as the form of Zigzag uncertain variable Suppose that

we are talking about an old person, any person with age less than 40 is not old, butthe aged person between 40 and 60 is a little old and the age between 60 and 70 is

Definition 2.21 (Experimental Uncertain Variable)

An uncertain variable f is called experimental if its uncertain distribution

where 1 i  n  1; 0\h  1:

Example 2.22 An uncertain variable f is called experimental if its uncertain

Fig 2.6 A linear uncertainty distribution of a linear uncertain variable

Fig 2.7 A zigzag uncertainty distribution of a zigzag uncertain variable

following uncertainty distribution:



Borel Algebra and BD is a measure (here it is Lebesgue measure) Show that the

Fig 2.8 An experimental uncertainty distribution of an experimental uncertain variable

Fig 2.9 A normal uncertainty distribution of a Normal uncertain variable

3 Suppose that f is an uncertain variable with uncertainty distribution function /.

w xð Þ ¼ / ln xð ð ÞÞ; x [ 0

4 Suppose that f is an uncertain variable with uncertainty distribution / Now

and properties For instance, the set of all real numbers greater than 2 In this case

exactly and there is an ambiguity on it For instance, set of old, tall or young men It

with probability measure

the uncertain sets with the name of fuzzy sets And in the fuzzy set, the property isuncertain and the measure that is used in the fuzzy set is probability measure In thefollowing, we will see that a member to belong to an uncertain set should have amembership degree (it is exactly the belief degree or uncertainty measure) and all ofthe degrees form a function that is called a membership function In the uncertainsets these membership functions have the same meaning

Definition 2.23 (Uncertain Set)

A set with uncertain event or belief degree is a function f from an uncertaintyspace to the set of Borel on real numbers and is called an uncertain set

membership degree and depends on individual belief degree In an exact view, the

Example 2.24 Suppose that f cð Þ is an uncertain function that is defined from anuncertainty space with a nonempty set using a Borel and measures of its members.

members

Definition 2.25 (Operations on Uncertain Sets)

Let ff gi n

have the following properties,

then their union, intersection and complements are an uncertain sets as well

f cð Þ ¼ f ðf1ð Þ; fc 2ð Þ; ; fc nð ÞÞ; 8c 2 Universal setc

Remark 2.28 Ameasurable function like f can be explained as follow,

Y; R1

measurable, if:

Remark 2.29 Based on the 2.28, any summation and multiplication of twouncertain set is an uncertain set It should be noted that the empty uncertain setannihilates any other uncertain set but it still is an uncertain set

uncertain set of old men, then we have many types of old men uncertain sets andtheir summation is another uncertain set of old men but it is not equal to multiplyingone of them by scalar 2

Fig 2.10 A measurable

function

Example 2.30 Consider Example2.26,

If the degree is 0, the member does not belong to the set and if the degree is 1, themember belong to the set completely

This is why any crisp or real set is a special case of the uncertain set and have thecharacteristic degree and function In this case the range of change of the mem-

f x; l xð ð ÞÞjl xð Þ 2 0; 1½ ; x 2 fg

degree

In this concept, if x belongs to the uncertain set with the membership degree

degree

f cð Þ ¼ c; c½ ; 8c 2 0; 1½ And its membership function is as follow,

p

; 1  x  1



Remark 2.35 The membership function of a trapezoidal uncertain set

Generally, if the uncertain set is, f cð Þ ¼ f½ lð Þ; fc uð Þc  where

Remark 2.36 If consider R cð Þ ¼ L cð Þ ¼ c then the membership function is calledtrapezoidal and

flð Þ ¼ a þ b  ac ð Þc;

Remark 2.37 The definition of trapezoidal uncertain set can be shown in anothersimilar way as follow,

Remark 2.38 The graph of the membership function is depended on the L and Rcompletely In some cases L and R can be considered as decreasing maps g :

For instance, g xð Þ ¼ max 0; 1  xð Þ.

In this case we will have,

flð Þ ¼ b  b  ac ð ÞL1ð Þ;c

fuð Þ ¼ c þ d  cc ð ÞR1ð Þ; c 2 0; 1c ½ The membership function is,

the trapezoidal uncertain two core points are the same then we have a triple ordered

flð Þ ¼ a þ b  ac ð ÞL1ð Þ;c

fuð Þ ¼ c  c  bc ð ÞR1ð Þ; c 2 0; 1c ½ The membership function is,

Remark 2.40 The membership function of a triangular uncertain set

in another similar way as follow,

flð Þ ¼ b  b  ac ð ÞL1ð Þ;c

fuð Þ ¼ b þ c  bc ð ÞR1ð Þ; c 2 0; 1c ½ The membership function is,

In mentioned above cases the membership function is linear because of the

l xð Þ ¼

3 ¼ 1

1 þ 4x 3

1 þ 2 x4 2

“young”, “old”, “beautiful”, “tall”, “short”, “good”, “not bad”, “warm”, “cold”,

“most”, “high”, “low”, “medium”, “strong”, “weak”, “more”, “less”, “salty”,

“handsome”, “ugly” and any other words

measure to evaluate the people is percentage and the membership function of the

Fig 2.13 A piece wise linear membership function

Fig 2.14 The membership

function

Note 1: The totally ordered uncertain set defined on a continuous uncertainty spacealways have membership function What totally ordered means is, for two arbitrary

uncertain set and conversely

, we know that it is a real function for any

uncertainty space, so the following function is as corresponding uncertain set of

f cð Þ ¼ ð ffiffiffiffiffiffiffiffiffiffi

lnc

p

; þpffiffiffiffiffiffiffiffiffiffilncÞ; 0  c  1:

opera-tions on the membership funcopera-tions To this end, one of the ways is using thefollowing level wise membership functions

2.2.7 Level Wise Membership Function or Interval Form

Level wise membership function is indeed an inverse function of membershipfunction that propose an interval valued function In fact, any cut or level on thevertical axes gives us an interval on the horizontal axes We should know that any

It means that there is one to one map between two functions, membership

uncertain set has a membership degree and also any degree corresponds to amember This is why, we claim that for any interval there is a degree and vice versa

The union of this levels corresponds to all intervals who contain the interval [a,b] Actually an interval driven from an uncertain set is a restriction on the functionfor easy using in the calculations Then to have computations on the distributive

Remark 2.44 A triangular uncertain set in level wise form

f r½  ¼ ½flð Þ; fr uð Þ ¼ ½b þ b  ar ð Þ r  1ð Þ; b þ c  bð Þ 1  rð Þ

Fig 2.15 An interval level form of an uncertain set