CHAPTER 1 Introduction

Transcription

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 ( 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 1

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 2

3 Object Perception Mental representation Formal description Verbal description Interpretation Figure 1.1: Human reasoning process Figure 1.1 above shows the general human reasoning process. Numerous theories have been proposed for dealing with uncertainty in an effective way in past. Some of these are probability theory [10,206], fuzzy set theory [206], rough set theory [123], granular computing [212] and computing with words [211,213]. All these theories, however, are associated with an inherent limitation and are insufficient to handle all facets of uncertainty individually. 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 based 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 was coined by Zadeh [209] in 1978 who interprets fuzzy sets as possibility distributions. Basically, possibility is associated with some fuzziness either in the background or in the set for which possibility is asserted. 3

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. Random Uncertain Certain Fuzzy, imprecise 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 4

5 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, a [0,1]. Upon observation of x, the a priori probability P{ x A} = a becomes a posteriori probability, i.e. either P{ x A} = 1 or P{ x A} = 0. 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. 5

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 6

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. 7

8 1.2 Objectives and methodology The foremost objective behind the work lies in the exposition of different mathematical characteristics of 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-t composition and inf -Θ t composition, where t and Θ t denote a t-norm and a residuation operation (implication) respectively. The optimization models considered are with fuzzy relation equations subject to 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 relation 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, notion of approximate solutions is given. Applications of fuzzy relations are discussed in image compression and decompression/reconstruction, diagnosis etc. Figure 1.3 presents the outline of the objectives of the research work: Fundamentals of Fuzzy Sets and Systems Analysis of fuzzy relation equations (FRE) Analysis of Fuzzy Relations and Compositions Fuzzy Identification problems Decision problems Fuzzy Inverse problem Optimization problems Figure 1.3: Objectives of the research work 8

9 The fuzzy relational systems can be considered to have analogy with linear algebra (a binary fuzzy relation on finite sets can be encoded as a matrix). However here, the underlying algebra is exotic, and non non-linear linear in the traditional sense, generally. Basically, the underlying algebra is latticized with lattice operations and some other operations from 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 other o fields under various names such as: min min-max algebras. Because of the nonlinearity onlinearity and the unusual algebraic framework the classical mathematical tools and techniques cannot be applied directly to deal with of the fuzzy systems systems. So, special methods are developed based upon some heuristics and metaheuristics 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 optimization problems studied. Methodologies used us to carry out the work can be summarized in the figure 1.4 as follows: Fuzzy relational calculus Modified classical Algebraic methods methods Covering method Evolutionary techniques Soft computing techniques Genetic Algorithm, Memetic Algorithm etc Artificial intelligence Hybrid methods Algebraic methods+soft computing techniques Covering method+soft computing techniques Figure 1.4: Methodologies to carry out research 9

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 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 wellknown 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 10

11 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 11

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. Offsprings New generation Initialization Decode strings Genetic operators Parents Evaluation Mate Selection Reproduction 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 [ ]. Fuzzy relations and the concepts of similarity and fuzzy orderings were first introduced by Zadeh [ ]. Binary fuzzy relations were further investigated by Rosenfeld [147] and Yager [194]. Boixader, Jacas, Recasens [9] considered fuzzy relations on a single set 12

13 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]. 13

14 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 maxmin 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, ]. Yuan and Klir [73,204] also studied approximate solutions of fuzzy 14

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]. 15

16 Yang and Cao [ ] investigated a special nonlinear programming with geometric objective function both with lattice operators and the algebraic functions and maxproduct 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- Θ fuzzy relation equations where t denotes a t-norm and t Θt 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. 16

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 maxmin 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. ********** 17