CHAPTER 1 Introduction - shodhganga.inflibnet.ac.inshodhganga.inflibnet.ac.in/bitstream/10603/14185/4/chapter1.pdf · knowledge or from vagueness, as the information about a particular

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CHAPTER 1

Introduction

______________________________________________________________________________

1.1 Why fuzzy?

“So far as the laws of mathematics refer to reality, they are not certain and so far as they

are certain they do not refer to reality.”

-Albert Einstein

‘‘Geometrie und Erfahrung’’, Lecture to Prussian Academy, 1921

Uncertainty can be thought of in an epistemological sense as being the inverse of

information. The uncertainty may arise because of complexity, from lack of information,

from chance, from various classes of randomness, from imprecision, from lack of

knowledge or from vagueness, as the information about a particular problem may be

incomplete, imprecise, fragmentary, unreliable, vague, contradictory, or deficient in some

other way [74].Practical systems throughout the world are too complicated and are

handled with some level of abstraction because of the inherent subjectivity. This

abstraction is the end result of compromise with the precision and accuracy of the

outcomes. Chinese Philosopeher LaoTsu (~600BC) has also said in his famous book Tao

Te Ching (1972) that:

“Knowing ignorance is strength,

Ignoring knowledge is sickness”

Also, Aristotle (384–322 BC) mentioned about precision and satisfaction that-

“It is the mark of an instructed mind to rest satisfied with that degree of precision which

the nature of the subject admits, and not to seek exactness where only an approximation

of the truth is possible.”

-Aristotle

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For many practical systems, there are two sources of information: human experts who

describe their knowledge about the system in natural languages (the subjective

information); the other is via measurements and mathematical models derived according

to physical laws (the objective information).

As we move into the information era, human knowledge becomes increasingly important

to meet the accuracy in results. Computational methods based on precise mathematical

formulae are incapable for handling the real life practical systems with much of

subjective information (imprecise and vague concepts). The only way out to handle them

is to incorporate this human knowledge in a systematic manner together with other

information derived from empirical methods.

But the key question is: “How to embed or transform this human knowledge into the

existing mathematical formulations?” So, special attention is paid to the uncertainty that

comes from imprecision and ambiguity in human affairs. The human behavior is

basically characterized by their capability to observe and analyze the world objects and

making inferences. The practice works in two steps: perception i.e. mental creation and

the interpretation. Perception i.e. constructions in the mind, which, only after being cast

in a linguistic form, become liable of analysis and logical tests (inference in form of

lingual representation). Generally, the content of perception is not identical to the

perceived entity or object and the inference come out to be an approximation with

imprecision. So, the whole thinking and inference process is a big source of vagueness

and imprecision that can also be understood as follows:

“There are differences between, what we think, what we want to say, what we think we

say, what we say, what they want to hear, what they hear, what they want to understand,

what they think they understand, and what they understand. That's why there are at least

nine reasons for people to misunderstand each other” [193].

-Yager

Figure 1.1 above shows the general human reasoning

been proposed for dealing with

probability theory [10,206

computing [212] and computing with words [

associated with an inherent limitation

uncertainty individually.

The classical information theory in system science is uncertainty based and has two

forms. Probability based information theory due to E.Shannon

information theory due to R.V.

governing the random phenomena

devoted to the handling of incomplete information. The name

was coined by Zadeh [209

Basically, possibility is associated with some fuzziness

set for which possibility is asserted.

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Figure 1.1: Human reasoning process

shows the general human reasoning process. Numerous

been proposed for dealing with uncertainty in an effective way in past. Some of these are

,206], fuzzy set theory [206], rough set theory [1

] and computing with words [211,213]. All these theories, however, are

associated with an inherent limitation and are insufficient to handle all facets of


forms. Probability based information theory due to E.Shannon [161] and possibility

information theory due to R.V. Hartley [48].The probability theory is the study of laws

governing the random phenomena while possibility theory is an uncertainty theory

devoted to the handling of incomplete information. The name “Theory of Possibility

09] in 1978 who interprets fuzzy sets as possibility distributions.

s associated with some fuzziness either in the background

set for which possibility is asserted.

1.• Object

2.• Perception

3.• Mental representation

4.• Formal description

5.• Verbal description

6.• Interpretation

erous theories have

. Some of these are

], rough set theory [123], granular

All these theories, however, are

to handle all facets of


and possibility based

The probability theory is the study of laws

ossibility theory is an uncertainty theory

Theory of Possibility”

who interprets fuzzy sets as possibility distributions.

background or in the

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Generalized information theory also prevails containing both forms of uncertainty called

“imprecise probability” i.e. probability not known completely. Perhaps the first thorough

investigation of imprecise probabilities was made by Dempster [20], even though it was

preceded by a few earlier, but narrower investigations. Figure 1.2 shows the forms of

uncertainty that prevail in the information world.

Figure 1.2: Forms of uncertainty in the information world

Fuzzy set theory is designed to handle the particular kind of uncertainty namely

vagueness—which results when a property possessed by an object to varying degrees. In

other words, any notion is said to be vague when its meaning is not fixed by sharp

boundaries. According to the Oxford English Dictionary, the word “fuzzy” is defined as

“blurred, indistinct; imprecisely defined; confused, vague.” Fuzzy set theory acts as

machinery to transform the fuzziness (vagueness and imprecision) innate in human

thinking to the information that can be processed together with the classical mathematical

methods. The power of the paradigm is that it is capable of handling ambiguity that

appears in natural language and expressions.

Many times the fuzzy theory is misconceptualized to be a form of probability theory .But

the two theories are intrinsically different. Fuzziness describes the ambiguity of an event,

whereas randomness describes the uncertainty in the occurrence of the event. Also, the

fuzzy set paradigm is capable to deal with nonrandom objects where probability theory

Certain

Random

Fuzzy, imprecise

Uncertain

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fails to do so. Zadeh [210] has also claimed that “probability lacks sufficient

expressiveness to deal with uncertainty in natural language.”

Philosophical distinction between probability and fuzziness can be easily marked out. To

accomplish this, let the value of the membership function of A in x be equal to ,a i.e.

( ) ,A x a= and the probability that x belongs to A be equal to ,a i.e. { } ,P x A a∈ =

[0,1].a ∈ Upon observation of ,x the a priori probability { }P x A a∈ = becomes a

posteriori probability, i.e. either { } 1P x A∈ = or { } 0.P x A∈ = But ( ),A x a measure of

the extent to which x belongs to the given category, remains the same, in other words the

randomness disappears, but the fuzziness remains.

Modern mathematics has two pillars in the foundation: Set and relation. Set is the

collection of objects; the whole world is composed of. Relations are the way in which

two or more these objects are connected. As Goguen [32] writes:

“Science is, in a sense the discovery of relations between observables.”

So, the importance of studying relations is evident from the above statement. In fact, the

study of relations is equivalent to the general study of system. Hence, investigation of

relations is invaluable for understanding the general theory of systems. Basically, a

system is “a set or arrangement of things, so related or connected as to form a unity or an

organic whole.” Extracting the essence of this definition, we conclude that every system

consists of two components: a set of certain things and some relations among them. More

formally, S = (T, R), where symbols S, T, R denote a system, a set of things, and relation

among these things respectively. The components might be precise or imprecise as our

surroundings abound with the subjective information, information that is vague,

imprecise, uncertain, and ambiguous by nature. Moreover, it is natural when the

interaction amongst the different components results in vagueness and it becomes

difficult to neglect the subjectivity that usually appears in the relations.

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Fuzzy relations are the key mathematical tool to model systems having imprecise

relationships that pervade all over the world. Classical relations are based upon the idea

that whether two objects are related or nonrelated. The concept of a fuzzy relation instead

of dealing with related or non-related objects, considering objects that are related to some

degree, has thoroughly enriched the applicability of this fundamental concept. Being hard

the classical relations have the drawback that they are not efficient enough to model real

world situations. This forces us to dwell upon the world of fuzzy relations allowing

gradual relationships.

Applications of fuzzy relations are widespread, ranging from technical fields such as

control, signal and image processing, communications and networking to diagnostic,

medicine and finance, social networking, fuzzy modeling, psychology, economics and

sociology etc. An important application of fuzzy relations is fuzzy relation equations

(FRE), processing fuzzy information in relational structures especially in knowledge

based systems as they play the key role behind the fuzzy reasoning inference system. The

majority of fuzzy inference systems can be implemented by using the fuzzy relation

equations [169]. Fuzzy relation equations can also be used for processes of

compression/decompression of images and videos [50,51,91,109,110].

The importance of the theory of fuzzy relational equations is best described by Zadeh in

the preface of the monograph by Di Nola et al [23]:

“Human knowledge may be viewed as a collection of facts and rules, each of which may

be represented as the assignment of a fuzzy relation to the unconditional or conditional

possibility distribution of a variable. What this implies is that the knowledge may be

viewed as a system of fuzzy relational equations. In this perspective, then, inference from

a body of knowledge reduces to the solution of a system of fuzzy relational equations.”

Fuzzy relation equations provide a rich framework within which many complicated

problems that cannot be solved using linear equations can be solved. These problems can

be solved by solving corresponding fuzzy relation equations. This makes the exposition

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of different mathematical characteristics of fuzzy relations and fuzzy relation equations

an appealing subject of research.

The domain of problems that arise from the area of FRE has two branches: fuzzy

identification problems and fuzzy inverse problems. The problem of fuzzy identification

arises when the system itself is to be identified with the observed available output. The

resolution of inverse problem is to determine the entire solution set of fuzzy relation

equations. For this firstly, the solvability of the system is examined i.e. whether the

system has an exact solution or not. If the system is solvable then in general the solution

set of FRE comprises of unique maximum solution and possibly finite number of

minimal solutions (or dually, by a unique minimum solution and finitely many maximal

solutions) [4,49,155,156]. The maximum solution can easily be computed but finding the

entire set of minimal solutions is not a trivial task and is considered as an NP hard

problem [87,95]. If the system is unsolvable, then approximate solutions are determined.

When these problems form the feasible domain of some optimization problem, the

problem becomes a fuzzy relational optimization problem. More precisely, fuzzy

relational optimization is a branch of fuzzy optimization dealing with the optimization

problems with one or more objective functions subject to fuzzy relation equations

constraints based on certain algebraic compositions. The area of problems is important

from application perspective as decision and optimization theory of real world events

have concern with uncertain information.

The decision space in this case in general is non-convex, so the conventional optimization

techniques cannot be applied directly in their original form. Hence, the exploration of

efficient methods to solve these problems, offering lesser computational complexity is

always in demand. In the same area the application of the metaheuristics is useful to

handle such optimization problems.

1.2 Objectives and methodology

The foremost objective behind the work lies in the exposition of

characteristics of fuzzy relations and

nonlinear and multiobjective optimization problems with fuz

constraints. The two major types of fuzzy relation equations are with

and inf -t

Θ composition, where

(implication) respectively. The optimization models considered are with fuzzy relation

equations subject to sup-

different resolution strategies for determining the

necessary conditions for solvability of fuzzy relation equations is discussed. Linear,

nonlinear and multiobjective optimization models are designed subject to fuzzy

equations with different compositions as constraints and models are characterized for

obtaining optimal solutions/

programming problems with fuzzy relation equations having no unique solution, not

of approximate solutions is given.

compression and decompression/

outline of the objectives of the research work

Figure 1.

Fuzzy Identification problems

Decision problems

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and methodology

objective behind the work lies in the exposition of different mathematical

fuzzy relations and fuzzy relation equations and study of fuzzy linear,

nonlinear and multiobjective optimization problems with fuzzy relation equations as

constraints. The two major types of fuzzy relation equations are with sup-

composition, where t and tΘ denote a t-norm and a resid

respectively. The optimization models considered are with fuzzy relation

sup-t composition. Characterization of the feasible domain,

different resolution strategies for determining the complete solution set and establishing


nonlinear and multiobjective optimization models are designed subject to fuzzy


obtaining optimal solutions/approximate solutions. In case of linear/nonlinear

programming problems with fuzzy relation equations having no unique solution, not

of approximate solutions is given. Applications of fuzzy relations are discussed in image

compression and decompression/reconstruction, diagnosis etc. Figure 1.

of the objectives of the research work:

Figure 1.3: Objectives of the research work

Fundamentals of

Fuzzy Sets and Systems

Analysis of fuzzy relation equations

(FRE)

Fuzzy Identification

Decision problems

Fuzzy Inverse problem

Optimization problems

Analysis of Fuzzy Relations and Compositions

different mathematical

fuzzy relation equations and study of fuzzy linear,

zy relation equations as

sup-t composition

norm and a residuation operation

respectively. The optimization models considered are with fuzzy relation

composition. Characterization of the feasible domain,

complete solution set and establishing


nonlinear and multiobjective optimization models are designed subject to fuzzy relation


solutions. In case of linear/nonlinear

programming problems with fuzzy relation equations having no unique solution, notion

discussed in image

Figure 1.3 presents the

Analysis of Fuzzy Relations and Compositions

The fuzzy relational systems can be

binary fuzzy relation on finite sets can be encoded as a matrix). However here, the

underlying algebra is exotic, and non

the underlying algebra is latticized with lattice operations

fuzzy set theory. This kind of structure where maximum plays the role of addition and

some fuzzy conjunction plays the role of product has been studied in o

various names such as: min

algebraic framework the classical mathematical

directly to deal with of the fuzzy systems

some heuristics and metaheuristics

The work mainly explores the application of metaheuristics such as genetic algorithm,

neural network etc. and heuristics strategies such as algebraic method, covering method

for resolution of the decision and

carry out the work can be summarized in the figure

Figure

Modified classical methods

Soft computing techniques

Hybrid methods

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fuzzy relational systems can be considered to have analogy with


underlying algebra is exotic, and non-linear in the traditional sense, generally.

the underlying algebra is latticized with lattice operations and some other operations from

This kind of structure where maximum plays the role of addition and

some fuzzy conjunction plays the role of product has been studied in o

such as: min-max algebras. Because of the nonlinearity

the classical mathematical tools and techniques cannot be applied

to deal with of the fuzzy systems. So, special methods are developed based upon

some heuristics and metaheuristics.



decision and optimization problems studied. Methodologies us

can be summarized in the figure 1.4 as follows:

Figure 1.4: Methodologies to carry out research

• Fuzzy relational calculus

• Algebraic methods

• Covering method

Modified classical methods

• Evolutionary techniques

• Genetic Algorithm, Memetic Algorithm etc

• Artificial intelligence

Soft computing techniques

• Algebraic methods+Soft computing techniques

• Covering method+Soft computing techniques

Hybrid methods

linear algebra (a


linear in the traditional sense, generally. Basically,

and some other operations from

This kind of structure where maximum plays the role of addition and

some fuzzy conjunction plays the role of product has been studied in other fields under

onlinearity and the unusual

techniques cannot be applied

are developed based upon



Methodologies used to

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In many cases there is more to be gained from cooperation than from arguments over

which methodology is best. A case in point is the concept of soft-computing. Soft

computing is not a methodology, it is a partnership of methodologies that function

effectively in an environment of imprecision and/or uncertainty and is aimed at exploiting

the tolerance for imprecision, uncertainty, and partial truth to achieve tractability,

robustness and low solution costs.

1.2.1 An introduction to genetic algorithm

Genetic algorithm (GA) is a class of evolutionary algorithms that mimics the metaphor of

natural biological evolution. Today they have set up as a type of general problem solvers

that can be successfully applied to many difficult optimization problems even when the

problem-specific knowledge is absent. Though GA does not guarantee convergence nor

of the optimal solution but do provide, on average, a “good” solution.

Genetic algorithm was originally developed by John Holland [53] and his co-workers in

the University of Michigan in the early 60’s. Although genetic algorithms were not well-

known at the beginning, after the publication of Goldberg's book [33] followed by Deb’s

book [18] they have been established as an effective and powerful global optimization

algorithm providing robust search in multimodal and nonlinear complex search spaces,

for any combinatorial optimization problems, performing well even for problems with

discrete optimization parameters, non-differentiable and/or discontinuous objective

functions.

A genetic algorithm for a particular problem must have the following five components:

(i) a genetic representation for potential solutions to the problem i.e. encoding (ii) a way

to create an initial population of potential solutions i.e. initialization (iii) an evaluation

function that plays the role of the environment, rating solutions in terms of their fitness

i.e. fitness function (iv) genetic operators that alter the composition of children (v) values

for various parameters that the genetic algorithm uses (population size, probabilities of

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applying genetic operators, etc.). The fundamental procedure of genetic algorithms can be

summarized as follows:

At first, the solution needs to be defined within the genetic algorithm. The genetic

representation of the solution is called as the chromosome. Each individual or

chromosome in the population represents a potential solution to the problem under

consideration and a point in search space. The fitness function is possibly the most

important component of GA. Since each chromosome represents a potential solution, the

evaluation of the fitness function quantifies the quality of that chromosome, i.e. how

close the solution is to the optimal solution.

The three genetic operations are selection (or reproduction), crossover, and mutation are

the core of the algorithm and the unique cooperation in the three is the key factor

responsible for the efficient functioning of GA. The crossover operator is the main search

tool. It mates chromosomes in the mating pool by pairs and generates candidate offspring

by crossing over the mated pairs with probability. A thorough investigation on selection

and the two genetic operators in genetic algorithm is presented in [5,18,54,63,100].

The fitness is the link between genetic algorithms and the problem to be solved. The

fitness function should include all criteria to be optimized. In addition to optimization

criteria, the fitness function can also reflect the constraints of the problem through

penalization of those individuals that violate constraints. Through three main genetic

operators together with fitness, the population at a generation evolves to form the next

population. After some number of generations, the algorithm converges to the best string

which hopefully represents the optimal or approximate optimal solution to the

optimization problem. The whole cycle of genetic algorithm is shown in figure 1.5.

The proper settings of parameters in GA play important role in its convergence and

efficient functioning. Mainly, there are three parameters the crossover probability,

mutation probability and the size of the initial population. The probability to perform

crossover operation is chosen in a way so that recombination of potential strings (highly

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fitted individuals) increases without disruption. Generally, the crossover rate lies between

0.6 and 0.9. Since mutation occurs occasionally, it is clear that the probability of

performing mutation operation will be quite low. Typically, the mutation rate lies

between 0.01 and 0.1.

Figure 1.5: Cycle of genetic algorithm

1.3 Survey of the literature

The revolutionary concept of fuzzy sets was the brainchild of pioneer researcher Zadeh

[206]. Since then the invention has established as a device to handle the vague systems

prevailing throughout the world. The various theories to handle different forms of

uncertainty have been described in [48,123,161,206,211,213].The mathematical

foundation of fuzzy logic has been discussed by Gottwald [37, 40], Hájek [45] ,Novok

[111] and Belohlavek [7-8]. General discussion on fuzzy logic can be found in Wang et

al. [184], Zadeh [211-214].

Fuzzy relations and the concepts of similarity and fuzzy orderings were first introduced

by Zadeh [206-207]. Binary fuzzy relations were further investigated by Rosenfeld [147]

and Yager [194]. Boixader, Jacas, Recasens [9] considered fuzzy relations on a single set

Selection

Evaluation

Initialization

Genetic

operators

Offsprings

Mate

New generation

Parents

Reproduction

Decode strings

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and described the indistinguishability operators for them and used it as a tool to relate

different ways to generate such operators when the given t-norm is Archimedean.

A variety of literature is available on the applications of fuzzy relations. Rotshtein and

Rakityanskaya [149] considered the use of backward logical inference in expert

diagnostic systems. A genetic algorithm (GA) based approach was used to find the

solutions of fuzzy logic equation formed. Rotshtein and Rakityanskaya and Hanna [150]

discussed a fault diagnosis problem based on a cause and effect analysis which is

formally described by fuzzy relations and proposed a genetic algorithm as a tool to solve

the problem. Noburah, Hirota, Pedrycz and Sessa [110] studied a decomposition problem

of a fuzzy relation and discussed image decomposition as an application of fuzzy

relations. Vigier and Terceno [179] discussed the model of diseases of firms. The core

idea was to determine a matrix of economic and financial knowledge stating the fuzzy

relations between symptoms and causes that generate anomalies in the firms. Some other

applications of fuzzy relations can be viewed in [1,75,91,106,196].

The notion of fuzzy relation equations was first proposed and investigated by Sanchez

[155]. Nola and Sessa [21] discussed the theory of fuzzy relation equations under lower

and upper semicontinuous t-norms. The basic theory of resolution of finite fuzzy relation

equations can be found in Higashi and Klir [49] and Bourke and Fisher [11]. Baets [4]

studied the analytical behavior of fuzzy relation equations and proposed analytical

methods for determining complete solution set of system of polynomial lattice equations

in distributive lattices. Stamou and Tzafestas [170] gave the concept of mean solution in

the solution set for the fuzzy relation equations and proved its existence. Fuzzy relations

equations over continuous t-norms have been studied by Shieh [162,164]. The

fundamental results for fuzzy relation equations with max-product composition are

credited to Pedrycz [126,132].Perfilieva and Nosková [140] studied fuzzy relation

equations with dual compositions. Infinite fuzzy relation equations in a complete

Brouwerian lattices are discussed by Shieh[163] ,Wang [183] and Xiong and Wang

[191]. More work on fuzzy relation equations over Brouwerian lattices can be found in

[47,143,192].

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Chen and Wang [14-15] developed a new method and algorithm to solve a system of

fuzzy relation equations and asserted that finding all minimal solutions for a general

system of fuzzy relation equations is an NP -hard problem in terms of computational

complexity. Luoh et al. [93] considered the problem of solving fuzzy relation equations

with max-min or max-product composition. A computer based algorithm was proposed to

solve the problem which operates systematically and graphically on a matrix pattern to

get all the solutions of the problem.

Markovskii [95] considered max-product fuzzy relation equations and showed that

solving these equations is closely related with the covering problem, which belongs to the

category of -NP hard problems. Lin [87] considered the problem of solving fuzzy relation

equations with Archimedean t-norms and provided a one-to-one correspondence between

the minimal solutions of the equations and the irredundant coverings, as previously

discovered by Markovskii [95] for fuzzy relation equations with max-product

composition. Peeva [135] proposed a universal algorithm and software for solving max-

min and min-max fuzzy relation equations.

Molai and Khorram [103] studied the problem of solving a max-∗ composite finite fuzzy

relation equations, where ∗ is a special class of pseudo t-norms. Some necessary

conditions of solvability were presented for the minimal solutions. Yeh [203] investigated

the minimal solutions of sup-t fuzzy relation equations with max-min composition and

gave an algorithm for computing all minimal solutions. Peeva[135] and Peeva and

Kyosev [136] presented a quasi-characteristic matrix to detect the minimal solutions of

the system. Recent monograph of Li and Fang [86] presents a detailed analysis of fuzzy

relation equations and its types.

In case of system not having a unique solution the notion of approximate solutions of

FRE is addressed. Approximate solutions of fuzzy relation equations were first studied by

Pedrycz [124] and Gottwald [34,38,39]. Gottwald and Pedrycz [35,36] studied the

solvability indices of fuzzy relation equations. More literature on this issue can be found

in [125,128-131]. Yuan and Klir [73,204] also studied approximate solutions of fuzzy

15

relation equations. In the same area, different neural network approaches have been

suggested to find the approximate solutions of the system [79,153,181]. The use of

genetic algorithms for solving fuzzy relation equations was suggested by Sanchez [157].

More literature in this regard can be found in [94,108]. Other theoretical discussions over

fuzzy relational composition and fuzzy relation equations are present in

[2,3,7,46,111,203].

Fuzzy optimization problems with different kinds of fuzzy relation equations as

constraints are an important area of research. The problem of minimizing a linear

objective function subject to a system of max-min fuzzy relation equations was first

investigated by Fang and Li [27] and later by Wu et al. [186] and Wu and Guu [187].

Optimization problem with max-product composition was further considered by

Loetamonphong and Fang [89]. Pandey [117] studied the optimization of fuzzy relation

equations with continuous t-norms and with linear objective function. Pandey and

Srivastava [115] gave efficient procedure for optimization of linear objective function

subject to fuzzy relation equations as constraints. More work in this regard can be found

in Pandey [116,118,121] and Pandey and Srivastava [119]. Wu [188] and Khorram and

Ghodousian [66] studied a linear optimization problem with max-average fuzzy relation

equations. Thapar, Pandey and Gaur [174] discussed a linear optimization model subject

to max-Archimedean fuzzy relation equations. Some more literature by this triad on this

topic can be viewed in[173,175]. More work on study of optimization problems with

fuzzy relation equations as constraints is available in [30,31, 43,67,82,137,144,190].

The nonlinear optimization problem with fuzzy relation equations as constraints was first

studied by Lu and Fang [92]. Li, Fang & Zhang [83] considered a problem of minimizing

a nonlinear objective function with system of max-min fuzzy relation equations and

reduced it to a 0-1 mixed integer programming problem and solved it using an existing

solver. Nonlinear optimization with max-average fuzzy relation equations has been

discussed by Khorram and Hassanzadeh [68]. More literature on nonlinear fuzzy

relational optimization can be viewed in [120,172,176].

16

Yang and Cao [197-201] investigated a special nonlinear programming with geometric

objective function both with lattice operators and the algebraic functions and max-

product fuzzy relation equation as constraints. Further, Wu [189] discussed the problem

of minimizing a geometric objective function with single term exponents subjected to

fuzzy relation equations specified in max-min composition. Zhou and Ahat [217]

considered a geometric programming problem with max-product fuzzy relational

constraints and gave the min-max method to find optimal solution.

Wang [182] extended the study to multiobjective mathematical programming problem

with fuzzy relation equations as constraints. Loetamonphong, Fang, and Young [90]

studied max-min composition with multiple objective functions. Recently, Thapar et al.

[177] considered a multiobjective optimization problem subjected to a system of fuzzy

relational equations based upon the max-product composition and applied a problem

specific nondominated sorting genetic algorithm for solving the same.

1.4 Organization of the thesis

Work is divided into eight chapters and two appendices A and B. Chapter 2 presents a

brief description of some introductory and fundamental concepts of fuzzy set theory and

fuzzy logic theory.

Chapter 3 discusses exposition of different mathematical characteristics of fuzzy relations

and fuzzy relation equations such as the algebraic and analytic behavior of fuzzy

relations, a concise description of the sup- ,t inf- tΘ fuzzy relation equations where t

denotes a t-norm and tΘ denotes the residuation operator (implication), logical operators,

basic operations of fuzzy relational calculus etc. At the end some applications of fuzzy

relations and resolution problem are presented. Chapter 4 presents a nonlinear

optimization problem with max-Archimedean t-norm fuzzy relation equations. A two

step procedure based on covering method and genetic algorithm is adopted to solve the

problem.

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Chapter 5 considers a nonlinear optimization problem subject to fuzzy relation equation

when the system has no unique solution. Two nonlinear optimization problems are

discussed and two different solution procedures are designed to solve them respectively.

Chapter 6 considers a posynomial geometric optimization problem with a system of max-

min fuzzy relation equations constraints. A hybrid strategy with algebraic method and

genetic algorithm is applied to solve the problem.

Chapter 7 discusses two geometric optimization problems with special discrete form of

geometric objective function subjected to the system of fuzzy relational constraints. For

the optimization problem with system of max-product fuzzy relational system of

equations as constraints, a reduction procedure is employed to solve the problem. For the

geometric optimization problem with max-Archimedean composition a binary coded

genetic algorithm is employed to solve the problem. Chapter 8 presents a multiobjective

optimization problem subjected to a system of fuzzy relation equations with max-

Archimedean t-norm based composition. Concept of utility function is used to solve the

problem. A hybridized genetic algorithm is applied to solve the transformed optimization

problem. Three local search strategies have been tested and their efficiency comparison

has been discussed. Further the original NSGA-II is modified according to the problem so

as to result an efficient set of Pareto solutions. The results obtained with the proposed

method are compared with the results obtained with the help of the modified NSGA-II

algorithm.

Appendix A discusses some applications of fuzzy relations. Appendix B at the end

describes the modified NSGA-II procedure that has been used in the chapter 8.

The reference list is given at the end of the thesis comprising mainly the works cited in

the text and notes, and covers relevant books and significant papers on the work

undertaken.

**********

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CHAPTER 1 Introduction - shodhganga.inflibnet.ac.inshodhganga.inflibnet.ac.in/bitstream/10603/14185/4/chapter1.pdf · knowledge or from vagueness, as the information about a particular