1
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
2
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.
3
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
The classical information theory in system science is uncertainty based and has two
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
The classical information theory in system science is uncertainty based and has two
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
4
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.
6
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
7
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
8
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
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/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
necessary conditions for solvability of fuzzy relation equations is discussed. Linear,
nonlinear and multiobjective optimization models are designed subject to fuzzy relation
equations with different compositions as constraints and models are characterized for
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
9
fuzzy relational systems can be considered to have analogy with
binary fuzzy relation on finite sets can be encoded as a matrix). However here, the
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.
The work mainly explores the application of metaheuristics such as genetic algorithm,
neural network etc. and heuristics strategies such as algebraic method, covering method
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
binary fuzzy relation on finite sets can be encoded as a matrix). However here, the
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
The work mainly explores the application of metaheuristics such as genetic algorithm,
neural network etc. and heuristics strategies such as algebraic method, covering method
Methodologies used to
10
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
12
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].
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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.
17
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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