An Introduction to Analytical Fuzzy Plane Geometry [1st ed.] 978-3-030-15721-0;978-3-030-15722-7

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Studies in Fuzziness and Soft Computing

Debdas Ghosh Debjani Chakraborty

An Introduction to Analytical Fuzzy Plane Geometry

Studies in Fuzziness and Soft Computing Volume 381

Series Editor Janusz Kacprzyk, Polish Academy of Sciences, Systems Research Institute, Warsaw, Poland

The series “Studies in Fuzziness and Soft Computing” contains publications on various topics in the area of soft computing, which include fuzzy sets, rough sets, neural networks, evolutionary computation, probabilistic and evidential reasoning, multi-valued logic, and related fields. The publications within “Studies in Fuzziness and Soft Computing” are primarily monographs and edited volumes. They cover significant recent developments in the field, both of a foundational and applicable character. An important feature of the series is its short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. Contact the series editor by e-mail: [email protected] Indexed by ISI, DBLP and Ulrichs, SCOPUS, Zentralblatt Math, GeoRef, Current Mathematical Publications, IngentaConnect, MetaPress and Springerlink. The books of the series are submitted for indexing to Web of Science.

More information about this series at http://www.springer.com/series/2941

Debdas Ghosh Debjani Chakraborty •

An Introduction to Analytical Fuzzy Plane Geometry

123

Debdas Ghosh Department of Mathematical Sciences Indian Institute of Technology (BHU) Varanasi Varanasi, Uttar Pradesh, India

Debjani Chakraborty Department of Mathematics Indian Institute of Technology Kharagpur Kharagpur, West Bengal, India

ISSN 1434-9922 ISSN 1860-0808 (electronic) Studies in Fuzziness and Soft Computing ISBN 978-3-030-15721-0 ISBN 978-3-030-15722-7 (eBook) https://doi.org/10.1007/978-3-030-15722-7 Library of Congress Control Number: 2019934520 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To our Gurus, who teach us how to deal with the uncertainties in real life…

Preface

Understanding any optimization model involves a sequential process of obtaining a reasonable set of solutions. Instead of just achieving the final solution from a black-box algorithm, we wanted to look into the solution space of an optimization model which is defined in a fuzzy environment. Solution space of a fuzzy optimization model is hazy in nature and optimality lies on the edge of it. Thus our quest started by traversing the boundary of the fuzzy feasible region. Fuzzy Geometry is a study of shape, size, and nature of a region, surrounded by a hazy line or curve. In this book, the shape analysis has been done for obtaining the mathematical equations of hazy or fuzzy objects. The fuzzy objects have been visualized geometrically, and the construction procedure has been suggested to visualize it in the Euclidean space. Any entity in Euclidean space is the collection of points, restricted with some predefined constraints. The basic defining element in fuzzy geometry is a fuzzy point. This book starts with the concept of fuzzy point as a collection of different fuzzy numbers in the space. Though a fuzzy number is defined on the real line but to represent a fuzzy point in Euclidean space, there was a need to extend the concept of fuzzy number on any real line, excluding the real axis. The book mainly focuses on two-dimensional analytical fuzzy geometry. But the ideas may be extended to higher dimensions as well. The aim of the book is to synergize the different concepts of geometry and the algebra of real numbers in a methodical way. The joining of two fuzzy points as a fuzzy line segment and extending it bi-infinitely as fuzzy line are defined. The concepts of fuzzy triangle, fuzzy circle, and fuzzy parabola are illustrated. Proper care has been taken so that every concept coincides with the conventional definitions in classical geometry with zero uncertainty. The ideas on fuzzy geometry are applied to fuzzy multi-objective optimization problems. It is also shown that the proposed geometry can highly reduce the computational cost in obtaining the fuzzy decision feasible region.

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This book is suggested as a basic research monograph on fuzzy geometry. There is ample scope to extend the ideas further. Keywords Fuzzy point ∙ Same and inverse point ∙ Fuzzy distance ∙ Fuzzy line ∙ Different forms of fuzzy line ∙ Fuzzy angle ∙ Fuzzy triangle ∙ Fuzzy circle ∙ Fuzzy parabola ∙ Fuzzy multi-objective optimization problem ∙ Fuzzy non-dominance ∙ Construction of fuzzy feasible space Varanasi, India Kharagpur, India

Debdas Ghosh Debjani Chakraborty

Acknowledgements

As learners of fuzzy mathematics, first, we pay tribute to Prof. Lotfi A. Zadeh, the exponent of fuzzy-ism. It is also our privilege to read the seminal works on fuzzy set theory by Prof. Didier Dubois, Prof. Henri Prade, Prof. R. R. Yager, and Prof. H. J. Zimmermann. We are immensely indebted to the researchers who are actively participating in the investigation of source of fuzzy mathematics and its application. We would sincerely like to acknowledge the facilities provided by Department of Mathematics, Indian Institute of Technology Kharagpur, India, and by Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, India. Debdas Ghosh Debjani Chakraborty

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 A Step Towards Fuzzy Geometry . . . . . . . . . . . . . . . . . . . . 1.2 Introduction to Fuzzy Set Theory . . . . . . . . . . . . . . . . . . . . 1.3 Basic Set Theoretic Notions on Fuzzy Sets . . . . . . . . . . . . . 1.4 Alpha-Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Properties of Alpha-Cuts . . . . . . . . . . . . . . . . . . . . . 1.4.2 Decomposition Principle . . . . . . . . . . . . . . . . . . . . . 1.5 Extension Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Arithmetic Operations on Fuzzy Numbers . . . . . . . . . . . . . . 1.7.1 Same and Inverse Points with Respect to Two Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2 Addition, Multiplication, Subtraction and Division Operations on Two Fuzzy Numbers . . . . . . . . . . . . . 1.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basic Ideas on Fuzzy Plane Geometry . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Fuzzy Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Addition of Two Fuzzy Points . . . . . . . . . . . . . . . . . . . . . . 2.4.1 An Effective Way for Addition of Two Fuzzy Points . 2.5 Same and Inverse Points with Respect to Two Fuzzy Points . 2.6 Fuzzy Line Segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Fuzzy Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Fuzzy Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Construction of Fuzzy Line . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Fuzzy Line Passing Through nð  2Þ Collinear Fuzzy Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Point–Slope Form ( e L PS ) . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Slope–Intercept Form ( e L SI ) . . . . . . . . . . . . . . . . . . . 3.2.4 Intercept Form ( e LI ) . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Discussion and Comparison . . . . . . . . . . . . . . . . . . . . . . . . 3.4 General Form of a Fuzzy Line . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Fuzzy Triangle and Fuzzy Trigonometry 4.1 Introduction . . . . . . . . . . . . . . . . . . . 4.2 Fuzzy Triangle . . . . . . . . . . . . . . . . . 4.3 Similarity of Fuzzy Triangles . . . . . . 4.4 Fuzzy Trigonometry . . . . . . . . . . . . . 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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5 Fuzzy Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Fuzzy Numbers with a Pre-determined Fuzzy Distance from a Fuzzy Number . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 LR Fuzzy Numbers on the Real Line . . . . . . . . . . 5.2.2 LR Fuzzy Number Along a Line . . . . . . . . . . . . . 5.3 Construction of Fuzzy Circle . . . . . . . . . . . . . . . . . . . . . . e 1Þ . 5.3.1 Construction of the Membership Function lð:j C e 2Þ . 5.3.2 Construction of the Membership Function lð:j C 5.4 Discussion and Comparison . . . . . . . . . . . . . . . . . . . . . . 5.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Fuzzy Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Construction of Fuzzy Parabola . . . . . . . . . . . 6.2.1 Method 1 . . . . . . . . . . . . . . . . . . . . . 6.2.2 Method 2 . . . . . . . . . . . . . . . . . . . . . 6.2.3 Method 3: Symmetric Fuzzy Parabolas 6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Fuzzy Pareto Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Fuzzy Multi Objective Optimization Problem . . . . . . . . . . . . 7.2.1 Assumptions for the Problem . . . . . . . . . . . . . . . . . . 7.2.2 Construction of Fuzzy Feasible Space . . . . . . . . . . . . 7.2.3 Evaluation of Fuzzy Inequality Region by Extension Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Evaluation of Fuzzy Inequality Region by Inverse Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . e and X e0 . . . . . . . . . . . . . . . . . . 7.2.5 Relation Between X 7.2.6 Fuzzy Objective Function Space . . . . . . . . . . . . . . . . 7.2.7 Fuzzy Non-dominance . . . . . . . . . . . . . . . . . . . . . . . 7.3 A Direction Based Scalarization Technique to Generate e ð1Þ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y N eN . . . . . . . . . . . . . . . . . . . . . . . . 7.4 On Generation of Entire Y 7.4.1 Algorithmic Implementation of the Proposed Method 7.4.2 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . 7.5 Final Selection of Solution . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8 Concluding Remarks and Future Directions . . . . . . . . . . . . . . . . . . . 203 8.1 Chapter Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 203 8.2 Future Scopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

Chapter 1

Introduction

Mathematics that refers to reality is not certain and mathematics that is certain does not refer to reality. Albert Einstein

For realization of real-world decision problems, the inherent uncertainties are needed to be induced in the process. These uncertainties must be identified and hence, classified. Economists, mathematicians and management scientists classified these uncertainties in different manners, as found in literature. For the sake of model designing we need to understand the sources of uncertainties first and the genesis of these thereafter. We may ignore the uncertain factors involved in the decision process but this might lead us to an erroneous culmination, as we know uncertain components accumulate very fast in the sequence of arithmetic operations. On the other hand, considering all the uncertain factors may lead us to a large span of uncertain area within which the optimized decision lies. As a result, we generally get stranded to find a decision amongst a huge set of decision alternatives. In today’s world, machine intelligence is one of the most vibrant fields of research. We try to incorporate logic into the functioning of the machines, so that machines become more intelligent; as much as intelligent the human beings are. Hence we intend to inject our modus operandi with simple to more, and more complex thinking, in terms of logic within it. Artificial intelligence deals mainly with these issues. It is quite an obvious statement that, any real world decision problem involves human participation. But till date it is a Herculean task to consider these uncertainties, which is most pervasive in human language framing and communication. Human thought-process is a superimposition of frames of logical arguments. This process-flow is hampered if any logical frame is affected with uncertainty. As a result, confusion or indecisiveness is evolved. Human communication is the outcome of the human thought process. Thus, if we, the human beings, are full of confusion, then © Springer Nature Switzerland AG 2019 D. Ghosh and D. Chakraborty, An Introduction to Analytical Fuzzy Plane Geometry, Studies in Fuzziness and Soft Computing 381, https://doi.org/10.1007/978-3-030-15722-7_1

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1 Introduction

how can we embed our communication in the machine. How can we then inculcate our sense of uncertainties and its extent, into a mathematical model so that a machine can be fed? According to Weber’s argument—human is the only creator of all sorts of uncertainties (Özkan and Türk¸sen 2014). We may argue—is it due to our limitations to accumulate all the information available to us? Is it because of our knowledge, instinct, perception, cognition, and/or intelligence difference from one another? What if we argue—more intelligence or more knowledge may create more confusions? As only a so-called knowledgeable person can foresee more number of future options, which is again debatable whether the set is exhaustive or not, after all a human being is not omniscient and omnipotent like the God. On the contrary, ignoring inexact, incomplete and uncertain information might lead us to arrive at a point of unfavourable decision. Accumulating information is always valuable. While comprehending any real problem, we can undoubtedly accept the fact that gathering more information is always better than less information. On the other hand, accumulation of abundance of incomplete and contradictory information (which is quite obvious in explaining a realistic situation that requires considerable involvement of human perception/judgment), may lead us in some confusing situations. What we mean to say is that in a scale ranging from lack of information to huge non-specific information, our task should be to find a methodology that incorporates efficient and effective use of every bit of information. Let us assume certain situations to define a point—(i) we try to consider position of any point that appears randomly within a small region, (ii) we try to locate a tiny object which is moving centering a point in an open circle but the path is locally dense around center, (iii) there is a dark black spot in a space and we view it through a hazy glass, (iv) we try to locate a far distant star in the sky from earth, (v) we try to draw a dot on a bloating paper with fountain pen, (vi) we try to specify the body mass index of a tall young lady and so on. Now, one is asked to assign a tuple for locating the object for each measure with respect to suitable reference axes in a space. In order to assign possible numerals it is needed to accumulate and judge each bit of information given. Depending on our observing capability, we will try to capture as much information as we can. The nature of information varies—some are random in nature, in some cases there is lack or abundance of information, somewhere the information is perturbed with noise, elsewhere the information is perception dependent. There is no specific mathematical tool which can bring each and every situation in one platform (Özkan and Türk¸sen 2014). In literature, certain terms have been coined in this regard—fuzzy point (Buckley and Eslami 1997a), uncertain point (Caglioti 1994), indecisive point (Jørgensen et al. 2011), random point (Valtr 1995), etc. A common approach to start discussion on any geometric object (such as, line, circle, triangle etc.) is generally initiated with defining the most primary and smallest entity, i.e. the concept of a point within that genre of geometry. Before going into the detail in defining point in uncertain or imprecise or ill-defined or ambiguous situation let us refer an example, initially given by Zadeh (2009)—a spray-ink pen

1 Introduction

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is placed at a point on a paper. If we assume the paper as a Euclidean space R2 with an assumed origin, the position of the tip of the pen can be determined at (10, 12), say. Since the ink is spread out in the vicinity of (10, 12) we are unable to detect the position of the tip of the pen with a precise value. The portion of the ink in the paper can be expressed with digital geometry in image processing using optical intensity level. But if we restrict ourselves in R2 , in our opinion, it can only be expressed with a fuzzy point. A fuzzy point can be visualized as a set with collection or bunch or chunk of points or granules where belongingness of a point to the set is defined with a mathematical function, called membership function. The set of points covered with ink can be referred to as a granule as the very first definition of granule is defined as a clump of points drawn together by similarity, indistinguishability, physical adjacency and functionality (Zadeh 1996). Now, let us assume the clump of points with varied membership values, which are chosen together under a common criterion, are moving in a space. According to Zadeh (2009), the locus of this chunk or fuzzy point is defined as a fuzzy curve. Thus again the question arises—Are they moving together in the space maintaining the same shape and does there exist any pattern? If so, what is its mathematical representation? In earlier literature, though the number of citations is quite less than handful (Zadeh 2009), the presence of more or less similar sort of queries cannot be ruled out, there was no mention about the shape or possible width of the curve. The curve may be of varying width in the space and is named here as a fuzzy curve. Geometrically, a fuzzy curve may be defined with a collection of mathematical curves with different level of containment. The fuzzy curve may be composed of various linear or nonlinear curves. If it is linear, it corresponds to a fuzzy line, otherwise it generates fuzzy conic sections and so. In conventional Euclidean Geometry or in bivalent logic (Aristotelian logic) the writing instrument is a ballpoint pen, ruler, compass etc. In fuzzy logic the writing instrument is a spray pen with precisely known adjustable spray pattern (Bustince 2010). Then multiple issues may arise again from here—What should be the geometrical construction of these fuzzy objects? Is there any algebraic form of these? Can any combination of lines form a fuzzy line or any bunch of curves generate a fuzzy curve? It is quite challenging to find the impossible shapes of fuzzy objects or fuzzy locus. Our journey starts from this juncture. This book first introduces the concept of a fuzzy point. Then a fuzzy line, which is drawn from a fuzzy point to another fuzzy point, and this has been constructed mathematically and geometrically. A terminated fuzzy line can be further extended indefinitely with a pattern. A fuzzy circle has been formed with a fuzzy centre and fuzzy radius. The important question has also been answered here—whether a fuzzy circle can be constructed with any set of fuzzy centre and fuzzy radius. The kind of uncertainty as is dealt in fuzzy set theory poses a critical question on the concept of equality, like, equality or similarity of two fuzzy angles, fuzzy triangles, fuzzy circles, etc. This book consists of several axioms, propositions related to fuzzy shapes like fuzzy conic sections. Let us start with some preliminaries related to fuzzy set theory which are needed to describe the construction and hence its graphical representation and its mathematical expression of fuzzy shapes.

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1 Introduction

1.1 A Step Towards Fuzzy Geometry In spite of the introduction of fuzzy set theory in 1965 there have been a very few studies focusing on fuzzy geometrical ideas, the geometry of fuzzy sets. The ideas on fuzzy geometrical notions have been proposed by many researchers, only Buckley and Eslami (1997a, b) gave some ideas on construction and representation of the basic fuzzy geometrical entities in a mathematical framework. A brief review of studies on fuzzy geometry and topology of imprecise image subsets, including adjacency, separation and connectedness, is reported in Rosenfeld (1998). Prior to that Rosenfeld (1984) introduced concepts of the height, width and diameter of fuzzy sets using real integrals. Certain ideas about these are also investigated by Bogomolny (1987) using a projection of the fuzzy sets onto two mutually perpendicular directions. Bogomolny observed that the definitions introduced in the works (Rosenfeld 1984; Rosenfeld and Haber 1985) lack inner conformity when reduced to the corresponding customary definitions for crisp sets. To maintain this conformity, Bogomolny modified the definition given in Rosenfeld (1984). In the works of Rosenfeld (1984), Rosenfeld and Haber (1985) and Bogomolny (1987) measurements of the defined height, width, perimeter, etc. are considered as crisp numbers. However, Bogomolny (1987) dealt with the problem comparatively in more meaningful way. These should be fuzzy numbers and cannot be real numbers because if the region is itself ill-defined, then it is difficult to observe the measurement as precise one (Guha and Chakraborty 2010). In the literature, definition of a fuzzy triangle in the plane are given in four different ways—first, by intersection of three intersecting fuzzy half planes (Rosenfeld 1994a); second, by three fuzzy points as its vertices (Buckley and Eslami 1997b); third, by blurring the boundary of a crisp triangle (Li and Guo 2007) and last, by approximation of crisp triangle (Imran and Beg 2010, 2012). In Rosenfeld (1994b), fuzzy triangle is defined as a fuzzy sets whose α-cuts are similar triangles. Fuzzy triangle defined in Rosenfeld (1994b) cannot be a fuzzy triangle and it is a fuzzy point (according to Buckley and Eslami (1997b)) whose support is a triangular region. The fuzzy trigonometric functions and angles as is described by Rosenfeld (1994a) are crisp for a fuzzy triangle. These should be fuzzy and cannot be crisp in general, because the considered environment is itself not precisely defined. Whereas Buckley and Eslami (1997b) defined fuzzy triangle by three fuzzy points as its vertices. To form a fuzzy triangle, three intersecting fuzzy line segments are being adjoined. Fuzzy trigonometric functions have also been defined here for a right angled fuzzy triangle using ratio of fuzzy distances. However, if fuzzy trigonometric functions are tried to be generalized for any fuzzy angle, then fuzzy distance of a vertex to the opposite side of a fuzzy triangle has to be measured. But how to measure is not given. Liu and Coghill (2008) have defined fuzzy trigonometric functions using fuzzy unit circle, which has been named as fuzzy qualitative circle. Boundary of the crisp circle has been partitioned fuzzily and fuzzy qualitative angles are defined as 4-

1.1 A Step Towards Fuzzy Geometry

5

tuple trapezoidal fuzzy number. But it is very difficult to obtain the value of the trigonometric functions for arbitrary fuzzy angle in general. Imran and Beg (2010, 2012) studied fuzzy triangle or f -triangle as approximate triangle. It is reported that instead of drawing a triangle by ruler, any triangle drawn by free hand is a fuzzy triangle. Subsequently similarity of fuzzy triangles are also studied. But we note that core of this fuzzy triangle is not a crisp triangle. In Li and Guo (2007), fuzzy triangle is defined by blurring boundary of a crisp triangle using smooth unit step function and implicit functions. But in the obtained shape, its 1-level sets contain all the points which lie outside the considered crisp triangle instead of the points on the boundary. Though according to Zadeh (2009) the counterpart of a crisp triangle, C is a fuzzy triangle. The area and perimeter of fuzzy regions defined by Buckley and Eslami (1997b) are fuzzy numbers. Buckley and Eslami (1997a, b) initiated research on fuzzy plane geometry using the sup-min composition of fuzzy sets, which is similar to Zadeh’s extension principle. This work was further elucidated by Clark et al. (2008) and Yuan and Shen (2001) and since then has been extended by other researchers (Han et al. 2010; Qiu and Zhang 2006). In the application field of fuzzy geometry, Safi et al. (2007) used the geometrical concepts of Buckley and Eslami to solve fuzzy linear programming problems. Li and Guo (2007) applied the concepts in the modeling of fuzzy geometrical objects. Wang et al. (2011) used the concepts for location discovery in passive sensor networks. Bloch (1999) performed a study on different approaches to obtain fuzzy geometrical distances and their applications in image processing. Prior to the works of Buckley and Eslami, Chaudhuri (1991) defined some fuzzy geometrical shapes, but in general their cores do not correspond to the well-known shapes in classical geometry. For instance, Chaudhuri (1991) defined a fuzzy line as a fuzzy set for which any α-level set is either empty or a straight line for all α in (0, 1]. Pham (2001) studied the representation of fuzzy shapes and defined a fuzzy line as a collection of line segments with varied membership values. This fuzzy line is basically a fuzzy line segment. The notion of imprecise lines described by Löffler and Kreveld (2008) eventually becomes a set of lines. Geometric properties of sets of lines and fuzzy sets of lines have been investigated by Rosenfeld (1995). Gupta and Ray (1993) considered a fuzzy line as a fuzzy set consisting of collinear crisp points with varied membership values, i.e., a fuzzy set is a fuzzy line when its support set is a straight line. According to this view, α-cuts of a fuzzy line can be the union of disconnected line segments. Moreover, the fuzzy lines as described in Chaudhuri (1991) and Gupta and Ray (1993) can also have empty core. Thus, these methods of defining fuzzy lines may violate the corresponding definition of a crisp line. Ammar (1999) defined a fuzzy line through two fuzzy points as a fuzzy set whose membership function is a convex combination of membership values of those fuzzy points. Tang et al. (2006) defined a fuzzy cell complex structure for modeling fuzzy lines, which is a particular case of the approach used by Ammar (1999). Li and Guo (2007) modeled fuzzy lines using a smooth unit step function and implicit functions. To detect shapes in noisy data, Han and Kóczy (1994) proposed a fuzzy Hough transform and mentioned that a fuzzy straight line on R2 is a nor-

6

1 Introduction

mal fuzzy set whose core is a line in the classical sense. Rosenfeld (1995) defined geometric properties of a set of lines and a fuzzy set of lines and noted that a fuzzy subset of Hough space is a fuzzy set of lines. Guibas et al. (1989) studied epsilon-geometry (a type of perturbed geometry) as a framework for robust geometric algorithms for imprecise computations. They defined the epsilon-collinearity of three points using so-called epsilon-butterfly. The fuzzy line described by Buckley and Eslami (1997a) that passes through two fuzzy points whose supports are circles of equal radius has support like this epsilon-butterfly. Ferraro and Foster (1994) defined the fuzzy collinearity of three physical points as the orientation of a pair of points. They also studied the fuzzy straightness of a physical curve whose position may be imprecisely known. Muganda (1990) defined a fuzzy linear hull of two fuzzy points, but used fuzzy point therein may not be a fuzzy point since its core is not a crisp point. Very recently, Obradovi´c et al. (2013) proposed a few basic concepts for fuzzy plane geometry and basic spatial relations based on their previous work (Obradovi´c et al. 2011). They proposed a fuzzy linear combination on the linear fuzzy space R2 . The fuzzy line constructed by Obradovi´c et al. (2013) is similar to fuzzy line segment obtained by Buckley and Eslami (1997a). However, we note that their fuzzy line is a fuzzy line segment (Obradovi´c et al. 2013) and may not be a fuzzy line. Qiu and Zhang (2006) extended the Buckley–Eslami concept of fuzzy plane geometry to fuzzy space geometry. Schneider (2008) described some fundamental ideas for fuzzy lines. Apart from fuzzy geometry in the Euclidean case, Bloch (2000) presented some ideas on the geodesic case and the shortest path between two points. Fuzzy disk is defined in Rosenfeld and Haber (1985) where membership function depends on the distance from its center. It is a fuzzy point in the plane (Buckley and Eslami 1997a) and may not be a fuzzy disk. In the works of Liu and Coghill (2008), while constructive a fuzzy quantitative circle, the conventional unit circle has been modified by the introduction of quantity spaces for orientation and translation. The quantity space of every variable in the system is a finite and convex discretization of the real number line. Symmetric (fourtuple) trapezoidal fuzzy numbers are being placed on equispaced discrete points on the periphery of the unit crisp circle on the orientation space to form a fuzzy unit circle. Here support of the fuzzy unit circle is the unit crisp circle and its α-cuts are disconnected. The core of the constructed fuzzy unit circle is not a crisp unit circle as is described in Liu and Coghill (2008). Li and Guo (2007) defined fuzzy conics by blurring the boundaries of crisp conics using smooth unit step function and implicit functions. But in the obtained shapes, the core sets contain all the points which lie outside the conic instead of the points on the boundary of the desired conic, whereas in an earlier work Esogbue and Liu (1999) used fuzzy conics as prototype in fuzzy criterion clustering of finite number of crisp or fuzzy data. The definitions considered by Esogbue and Liu (1999) are appreciable but may not be taken as general definitions of fuzzy conics. Since, the center or foci of the fuzzy conics are taken as crisp points. Instead of crisp points, fuzzy points would have been more appropriate for general definition.

1.1 A Step Towards Fuzzy Geometry

7

Buckley and Eslami (1997b) defined fuzzy circle through fuzzy algebraic extension of the well-known classical equations of circles. The concepts for evaluating membership values of fuzzy lines, fuzzy circles, and fuzzy polygons in their studies are generalizations of the Theorem 1.1 in Goetschel and Voxman (1986). Except in Buckley and Eslami (1997b) and its explication (Clark et al. 2008), some ideas on fuzzy conics have been developed and applied in the works of different authors (El-Ghoul 1999; Harter 2008; Jooyandeh et al. 2009; Khatib et al. 2007; Kosko 1990, 1994; Li and Guo 2007; Qiu and Zhang 2006; Liu and Coghill 2008; Rosenfeld 1998, 1984; Rosenfeld and Haber 1985; Zadeh 2009). Zadeh (2009) has proposed that the counterpart of a crisp curve C in the Euclidean geometry is a fuzzy curve. A fuzzy shape may be formed by a fuzzy transform of C, where C acts as a prototype for the fuzzy shape. It is helpful to visualize a fuzzy line, a fuzzy circle or a fuzzy conic section as a fuzzy transform of C drawn by a spray pen (Zadeh 2009). Jian et al. (2010) noted that a fuzzy line can be viewed as a locus of a fuzzy point along a fixed direction. Therefore, a fuzzy line can also be visualized as an infinitely long hazy band consisting of a bunch of crisp lines with varied membership grades. Its core must contain a crisp line and its membership function must smoothly decrease from the core to the neighboring points. It is proposed in this study that the spread of the support of a fuzzy line cannot suddenly be widened like epsilon-butterfly shape (Guibas et al. 1989) as described by Buckley and Eslami (1997a). Furthermore, the slope of a fuzzy line cannot have more than one number in its core. Otherwise, it lacks inner conformity in deriving the customary definition of a crisp line. It is also to be noted that due to formation of such epsilon-butterfly shape of Buckley-Eslami’s fuzzy line, it cannot be observed as an infinite extension of a fuzzy line segment. It may also be noted form the existing literature that only Buckley and Eslami (1997b) studied some concepts for the construction of fuzzy circles through the extension of the algebraic equations of a circle. However, it is observed that fuzzy circle defined by Buckley and Eslami (1997b) cannot be observed as locus of a fuzzy point with a fixed fuzzy distance from a fixed fuzzy point. Thus, the defined fuzzy circle in Buckley and Eslami (1997b) may be violating the customary classical definition of a circle. For the other geometrical ideas proposed by Buckley and Eslami (1997a, b), similar observations can be made. Fuzzy regression line or fuzzy line fitting corresponding to a given data set of imprecise locations is a topic of extensive application of fuzzy line. Kao and Chyu (2002) mentioned that the existing ideas on fuzzy line fitting give the fuzzy responses (dependent variable) with wider and wider spreads as the independent variable increases its magnitude. But this is not suitable in general. Ge and Wang (2007) also mentioned that at the situation with lack of a significant amount of data to determine the regression model, fuzzy linear regression provides a suitable way for finding vague relationships between the dependent and independent variables. Due to the successful applications of fuzzy regression line in various fuzzy systems as reported in (Chen and Dang 2008), substantially a large number of research work focused on fuzzy linear regression corresponding to crisp data or fuzzy data (Shakouri and Nadimi 2013; Wu 2011; Ge and Wang 2007; Coppi et al. 2006; Hong and Hwang 2004).

8

1 Introduction

Let us start with some preliminaries related to fuzzy set theory which are needed to describe the construction of fuzzy shapes and hence its graphical representations and its mathematical expressions as well.

1.2 Introduction to Fuzzy Set Theory Definition 1.2.1 (Fuzzy set) (Zadeh 1965). Let X be a classical set of elements which should be evaluated with regard to a fuzzy statement. Then the set of order pairs  = {(x, μ(x| A))  : x ∈ X }, where μ : X → [0, 1], A  is called the membership is called a fuzzy set on X . The evaluation function μ(x| A)  function or the grade of membership of x in A.  Capital or small letters with a tilde bar ( A, B, …and a,  b, …) denote fuzzy subsets of Rn .

1.3 Basic Set Theoretic Notions on Fuzzy Sets In doing operations on fuzzy sets the membership function plays the important role. Fuzzy operations are defined via membership functions which are discussed below.  of the intersection C = A ∩  • The membership functionμ(x|C) B is pointwise  = min μ(x| A),  μ(x|  defined by μ(x|C) B)}.  of the union C = A ∪  • The membership function μ(x|C) B is pointwise defined     by μ(x|C) = max μ(x| A), μ(x| B)}.  is also pointwise • The membership function of the complement of a fuzzy set A c   and defined by μ(x| A ) = 1 − μ(x| A).  and   is said to be a subset of  • Let A B be two fuzzy sets on X . The fuzzy set A B,     presented by A ⊆ B, if μ(x| A) ≤ μ(x| B) for all x in X .  and  =   = • Two fuzzy sets A B on X are said to be equal, presented by A B, if μ(x| A)  μ(x| B) for all x in X .

1.4 Alpha-Cuts  Definition 1.4.1 (α-cut of a fuzzy set) (Dubois and Prade 1980). For a fuzzy set A,  its α-cut is denoted by A(α) and is defined by:   ≥ α} {x : μ(x| A) if 0 < α ≤ 1  A(α) =  closur e{x : μ(x| A) > 0} if α = 0.

1.4 Alpha-Cuts

9

 > 0} is called support of the fuzzy set A.  The set {x : μ(x| A)  = {x : μ(x| A)  = 1} is said to be core of the fuzzy set A.  The set A(1)  is the supremum of the membership values of A.  Height of a fuzzy set A  in X is said to be normal if there exists an x0 in X such that Fuzzy set A  = 1. μ(x0 | A)  in X is said to be convex if • A fuzzy set A      μ(x2 | A)  , x1 , x2 ∈ X , λ ∈ [0, 1].  ≥ min μ(x1 | A), μ λx1 + (1 − λ)x2  A • • • •

1.4.1 Properties of Alpha-Cuts  is a fuzzy set, then α-level set of A  is defined as: A(α)   ≥ α}. If A = {x : μ(x | A The following properties hold for all α ∈ [0, 1]. (i) (ii) (iii) (iv) (v)

  ). ⊇ A(α If α > α, then A(α)   ∪   ∪ If A and B are two fuzzy sets, then ( A B)(α) = A(α) B(α).  and  ∩   ∩ If A B are two fuzzy sets, then ( A B)(α) = A(α) B(α). ⊆   A B if and only if A(α) ⊆ B(α). =   A B if and only if A(α) = B(α).

1.4.2 Decomposition Principle Decomposition principle signifies that every fuzzy set can be uniquely represented by the family of its alpha cuts. Theorem 1.4.1 (Decomposition principle) (Wang et al. 2009).  can be represented by Any fuzzy set A

=  A α A(α), α∈[0,1]

 where the set α A(α) is defined by:   μ(x|α A(α)) =

 α if x ∈ A(α) 0 otherwise.

 = sup{α : x ∈ From the ‘decomposition principle’, it can be seen that μ(x| A)   A(α)}. We denote sup{α : x ∈ A(α)} by the notation α.  x∈ A(α)

10

1 Introduction

 To represent of membership function of a fuzzy set A, the  the construction   = notation x : x ∈ A(0) is frequently used, which also means μ(x| A)  sup{α : x ∈ A(α)}.

1.5 Extension Principle Suppose ϕ is a real-valued function of n variables x1 , x2 , . . . , xn . The extension prinx2 , . . . ,  xn ), ciple, as stated by Zadeh (1975), allows us to extend this function to ϕ ( x1 ,  which is a fuzzy set,  y say, with membership function: ⎧ ⎨ sup min μ(xi | xi ) if ϕ−1 (y) = ∅ i=1,2,...,n y=ϕ(x ,x ,...,x ) 1 2 n μ(y| y) = ⎩ 0 if ϕ−1 (y) = ∅. Let ⊕ and  be the extended addition and multiplication, respectively. For two fuzzy numbers a and  b, according to the extension principle, the membership function  of  a ⊗ b (× represents + or .) is defined by  a ), μ(y| b) . μ(z| a ⊗ b) = sup min μ(x| x×y=z

1.6 Fuzzy Numbers Definition 1.6.1 (Fuzzy number) (Buckley and Eslami 1997a).  on R is called a fuzzy number if its membership function μ has the A fuzzy set A following properties:  is upper semi-continuous, (i) μ(x| A)  (ii) μ(x| A) = 0 outside some interval [a, d], and  is increasing (iii) there exist real numbers b and c so that a ≤ b ≤ c ≤ d and μ(x| A)  on [a, b] and decreasing on [c, d], and μ(x| A) = 1 for each x in [b, c].  the notation (a/b/c) is used here. Set of all fuzzy To represent a fuzzy number A  is upper semi-continuous for a numbers in R is represented by F(R). Since μ(x| A)  the set {x : μ(x| A)  ≥ α} is closed for all α in [0, 1]. Thus the α-cuts fuzzy number A, are closed and bounded intervals. For the fuzzy numbers A  and of a fuzzy number A a,

1.6 Fuzzy Numbers

11

   l   lα , A rα and  a (α) =  aα ,  we write A(α) = A aαr , respectively. If the membership  is continuous, the fuzzy number A  is called a continuous fuzzy function μ(x| A) number. Definition 1.6.2 (L R-type fuzzy number) (Zimmermann 2001). A function L : [0, +∞) → [0, 1] which is non-increasing and satisfies either (i) L(0) = 1 and L(1) = 0, or (ii) L(x) > 0 for x in [0, +∞) and L(+∞) = 0 is called as a reference function of a fuzzy number.  is called an L R-type fuzzy number if there exist two reference A fuzzy number A  can be functions L and R, and two numbers α > 0 and β > 0 such that μ(x| A) written as:  L( m−x ) if x ≤ m α  = μ(x| A) ) if x ≥ m. R( x−m β Here, we use the notation (m − α/m/m + β) L R to represent an L R-type fuzzy  is number. In particular, if L(x) = R(x) = max{0, 1 − |x|}, the fuzzy number A called a triangular fuzzy number. A triangular fuzzy number is denoted by (m − α/m/m + β). The following reference functions are commonly used in practical applications: (i) (ii) (iii) (iv)

L(x) = max{0, 1 − |x| p }, p > 0, L(x) = exp(−|x| p ), p > 0, 1 L(x) = 1+|x| p , p > 0, and L(x) = 1, x ∈ [−1, 1] and 0 otherwise.

1.7 Arithmetic Operations on Fuzzy Numbers For an increasing continuous function ϕ : Rn → R, Dubois and Prade (1979) proved the following two lemmas. Lemma 1.7.1 Let ϕ : Rn → R be an increasing continuous function. Let f 1 , f 2 , …, f n be n convex continuous functions; we suppose that f i is strictly increasing on (−∞, bi ], strictly decreasing on [bi , ∞) and f i (R) = [0, 1] for each i = 1, 2, . . . , n. Let (x1 , x2 , …, xn ) be an element of Rn satisfying ϕ(x1 , x2 , . . . , xn ) ≤ ϕ(b1 , b2 , . . . , bn ). Then there exist x1∗ , x2∗ , . . . , xn∗ such that

12

1 Introduction

(i) (ii) (iii) (iv)

xi∗ < bi for each i, f 1 (x1∗ ) = f 2 (x2∗ ) = · · · = f n (xn∗ ), ∗ ∗ ∗ ϕ(x  1 , x2 , . . . , xn ) = ϕ(x1 , x2 , . . . , xn ), and ( f 1 , f 2 , . . . , f n ) = f 1 (x1∗ ) = · · · = f n (xn∗ ). μ ϕ(x1 , x2 , . . . , xn )ϕ

Lemma 1.7.2 Let μ(xi | m i ) be the membership function of the continuous fuzzy m i ) is increasing on [ai , bi ] and decreasing number m i , i = 1, 2, . . . , n; each μ(xi | on [bi , ci ] for all i, possibly ai = −∞, bi = ±∞, ci = +∞. Let xi , i = 1, 2, . . . , n m i ) = ω ∈ [0, 1] for all i. Then, for be such that xi ≤ bi and μ(x  an increasing  i | ( m1, m 2 , . . . , m n ) = ω. function ϕ of n variables, μ ϕ(x1 , x2 , . . . , xn )ϕ

1.7.1 Same and Inverse Points with Respect to Two Fuzzy Numbers Let ϕ be an increasing function and m 1 , m 2 , . . . , m n be n continuous fuzzy numbers. 2 , . . . , m n ) by the extension principle, we take a number To evaluate  y=ϕ ( m1, m y ∈ y(0). From Lemmas 1.7.1 and 1.7.2 and Theorem 1.7.1, it is clear that in the following two situations the combinations (x1 , x2 , . . . , xn ) with y = ϕ(x1 , x2 , . . . , xn ) are redundant: (i) if there exist x j , xk such that μ(x j | m j ) = μ(xk | m k ), j, k ∈ {1, 2, . . . , n}, or (ii) if there exist x j , xk such that x j > m j and xk < m k , j, k ∈ {1, 2, . . . , n}. This redundancy leads us to define same points and inverse points with respect to continuous fuzzy numbers in R and continuous fuzzy points in the plane R2 . Definition 1.7.1 (Same points with respect to continuous fuzzy numbers (Ghosh and Chakraborty 2012)) Let x, y be two numbers belonging to the supports of the continuous fuzzy numbers  a and  b, respectively. The numbers x and y are said to be same points with respect to  a and  b if: (i) μ(x| a ) = μ(y| b), and (ii) x ≤ a and y ≤ b, or x ≥ a and y ≥ b, where a and b are midpoints of  a (1) and  b(1), respectively.

Example 1.1 Consider  a = 2 = (1/2/3) and  b = 6 = (5/6/7). For each particular α ∈ [0, 1], the pairs 1 + α, 5 + α and 3 − α, 7 − α are same points.

1.7 Arithmetic Operations on Fuzzy Numbers

13

Example 1.2 Let  a = (1/2/3) and  b be defined as: ⎧ 2 ⎪ ⎨(x − 4) if 4 ≤ x ≤ 5 8−x  μ(x|b) = if 5 ≤ x ≤ 8 3 ⎪ ⎩ 0 elsewhere. The pairs of numbers 45 ,  b(0).

9 2

and 83 , 7 are same points, where 45 , 83 ∈  a (0) and 29 , 7 ∈

Definition 1.7.2 (Inverse points with respect to continuous fuzzy numbers) Let x, y be two numbers belonging to the supports of the continuous fuzzy numbers  a and  b, respectively. The numbers x and y are said to be inverse points with respect to  a and  b if x, −y are same points with respect to  a and   −b, where −b is scalar multiplication of  b by −1.

Example 1.3 Consider  a = 2,  b = 6 in the illustration of Example 1.1. For each particular α ∈ [0, 1], the pairs 1 + α, 7 − α and 5 − α, 7 + α are inverse points. Example 1.4 Let  a and  b be the two fuzzy numbers considered in Example 1.2. The 71 22 pairs of numbers 54 , 29 and , are inverse points with respect to  a and  b, where 4 25 5 5 71 29 22  , ∈ a (0) and 4 , 5 ∈ b(0). 4 25

1.7.2 Addition, Multiplication, Subtraction and Division Operations on Two Fuzzy Numbers Definition 1.7.3 (Arithmetic operations on L R-type fuzzy numbers) (Zimmermann 2001). b = (b − γ/b/b + δ) L R be two L R-type fuzzy Let  a = (a − λ/a/a + β) L R and  numbers. Then     (i)  a ⊕ b = (a + b) − (λ + γ) (a + b) (a + b) + (β + δ) L R ,     (ii)  a  b = (a − b) − (λ + δ) (a − b) (a − b) + (β + γ) L R when L = R, and (iii) for some constant k ∈ R \ {0}, the scalar multiplication of the fuzzy number  a a ). We can see that is given by the membership function μ(x|k a ) = μ( xk | (−1)  a = − a = (−a − β/ − a/ − a + λ) R L .

14

1 Introduction

 l r  l r b(α) =  bα ,  Let  a (α) =  aα , aα and  bα . For each α ∈ [0, 1], the α-cuts of their addition and scalar multiplication can be obtained as:   l bαl ,  aαr +  bαr . ( a ⊕ b)(α) =  aα +    λ a l , λ if λ ≥ 0, aαr  (λ a )(α) =  αr if λ < 0. aαl λ aα , λ The following theorem allows easy application of the extended n-ary operation on continuous fuzzy numbers. 2 , . . . , m n be continuous fuzzy numbers Theorem 1.7.1 (Hong 1997) Let m 1 , m whose membership functions are surjective and whose supports are bounded. Let ϕ 2 , . . . , m n ) is be a continuous increasing n-ary operation. Then the extension ϕ ( m1, m a continuous fuzzy number whose membership function is continuous and surjective from R to [0, 1]. This fuzzy number can be constructed by applying Lemma 1.7.2 to m i ) separately. This decomposition is the increasing and decreasing parts of μ(xi | authorized by Lemma 1.7.1. Example 1.5 For two triangular fuzzy numbers  2 and  6 with  2 = (1/2/3),  6= (5/6/7), (i) the membership functions of  2 and  6 are continuous and surjective,   (ii) the supports of 2 and 6 are bounded, and (iii) μ(x1 | 2) is increasing on [1, 2] and decreasing on [2, 3]; a similar property 6). applies to μ(x2 | Take ϕ to be the normal algebraic addition +, which is a continuous increasing operator. (a) Determination of redundant combinations: Take 1.8 ∈  2(0) and 6.7 ∈  6(0). 2 Here, (1.8, 6.7) is an element of R and ϕ(1.8, 6.7) = 8.5 > 8 = ϕ(2, 6). In the following, we show that although (1.8, 6.7) is an element of R2 and ϕ(1.8, 6.7) = 8.5, this is a redundant combination for obtaining the value of μ(8.5| 2 ⊕ 6) because ∗ ∗ Lemma 1.7.1 implies that there exist x1 , x2 such that: (i) they lie on the right-hand side of  2(1) and  6(1), respectively (x1∗ > 2 and x2∗ > 6), 2) = μ(x2∗ | 6)), (ii) they have same membership value (μ(x1∗ | ∗ ∗ (iii) x1 + x2 = 8.5, and 2) = μ(x2∗ | 6). (iv) μ(8.5| 2 ⊕ 6) = μ(x1∗ | We can find x1∗ = 2.25 and x2∗ = 6.25. Except for (x1∗ , x2∗ ), there are pairs of combinations, such as (1.5, 7), (1.9, 6.6) and (1.8, 6.7), such that ϕ for each of them is 8.5. However, these do not offer the supremum of the computation μ(8.5| 2 ⊕ 6) =   sup min{μ(x1 |2), μ(x2 |6)} and do not satisfy the condition given in Lemma x1 +x2 =8.5

1.7.1. Thus, such combinations may be discarded when computing μ(8.5| 2 ⊕ 6). We

1.7 Arithmetic Operations on Fuzzy Numbers

15

call these combinations redundant or irrelevant because without considering them we can still evaluate μ(8.5| 2 ⊕ 6). By contrast, we call the combination (2.25, 6.25) as perfect or effective combination (or a combination of same points). In addition, (2.25, 6.25) satisfies all the conditions of Lemma 1.7.1. (b) Evaluation of μ(.| 2 ⊕ 6): Apparently, ( 2 ⊕ 6)(0) = [6, 10] and μ(8| 2 ⊕ 6) = 1. By applying the extension principle and evaluating the membership value for each point in [6, 10], the membership function of  2 ⊕ 6 can be obtained as: μ(x | 2 ⊕ 6) =

⎧ ⎪ x−6 ⎨ 2

10−x ⎪ 2

⎩

0

if 6 ≤ x ≤ 8 if 8 ≤ x ≤ 10 elsewhere.

This analytical form of μ(x | 2 ⊕ 6) can be obtained very easily by combining perfect combinations or same points as follows. Let α ∈ [0, 1]. The numbers in  2(0) with membership value α are 1 + α, 3 − α. The numbers in  6(0) with membership value α are 5 + α, 7 − α. According to Lemma 1.7.2, as 1 + α ≤ 2, 5 + α ≤ 6 and they have membership value α, the membership value of (1 + α) + (5 + α) in  2 ⊕ 6   2⊕ will be α, i.e., μ((1 + α) + (5 + α)|2 ⊕ 6) = α. Similarly, μ((3 − α) + (7 − α)|  6) = α. Therefore, μ(6 + 2α| 2 ⊕ 6) = α and μ(10 − 2α| 2 ⊕ 6) = α. Clearly, these two functional equations will provide the analytical form of the membership function of  2 ⊕ 6, which is identical to that written above. Therefore, we conclude that in computing μ(x| 2 ⊕ 6) there are infinitely many   pairs (x1 , x2 ) with x1 ∈ 2(0), x2 ∈ 6(0) and x1 + x2 = x. However, consideration of the combinations for which either (i) μ(x1 | 2) = μ(x2 | 6) or (ii) x1 < 2, x2 > 6 or x1 > 2, x2 < 6 is unnecessary. This is why these types of combinations may be called redundant or irrelevant. (c) Evaluation of μ(.| 2  6): Using extension principle, the membership function   of 2  6 can be obtained as: ⎧ √ ⎪ ⎨-3+√ 4 + x if 5 ≤ x ≤ 12 μ(x | 2  6) = 5− 4 + x if 12 ≤ x ≤ 21 ⎪ ⎩ 0 elsewhere. Here, in  2  6 for (1 + α)(5 + α) and (3 − α)(7 − α) the membership values are α. Thus, μ((1 + α)(5 + α)| 2  6) = α and μ((3 − α)(7 − α)| 2  6) = α. In order to obtain μ(x | 2  6) there are infinitely many pairs (x1 , x2 ) with x1 ∈  2(0), x2 ∈  6(0) and x1 .x2 = x. For example, let us consider three pairs (1.5, 6.5), (1.68, 5.803) and (1.708, 5.708) for which x = 9.75. For (1.5, 6.5), we get {μ(1.5| 2), μ(6.5| 6)} = {.5, .5}. Similarly, we have {μ(1.68| 2), μ(5.803| 6)} = {.68, .803} and {μ(1.708| 2), μ(5.708| 6)} = {.708, .708}. Therefore,

16

1 Introduction

 μ(z = 9.75| 2  6) = sup min μ(x| 2), μ(y| 6) = .708. x.y=z

Again we can re-confirm that consideration of the same point combinations for which either (i) μ(x1 | 2) = μ(x2 | 6) or (ii) x1 < 2, x2 > 6 or x1 > 2, x2 < 6 is unnecessary. This is why these types of combinations may be called redundant or irrelevant. (d) Evaluation of μ(·| 6  2): Subtraction of two fuzzy numbers can be viewed as   6 ⊕ (−1)  2. Applying extension principle the membership value of  6  2 is as follows: ⎧ x−2 ⎪ ⎨ 2 if 2 ≤ x ≤ 4 μ(x | 6  2) = 6−x if 4 ≤ x ≤ 6 2 ⎪ ⎩ 0 elsewhere. Here the points 1 + α ≤ 2, 7 − α ≥ 6 or 3 − α ≥ 2, 5 + α ≤ 6 are the perfect combinations according to Lemma 1.7.2. These combinations are the inverse combinations according to Definition 1.7.2 and are responsible to get the functional equations of membership function. This analytical form is again same with the function written above. (e) Evaluation of μ(·| 6  2): In fuzzy geometry this operation works within the inverse combinations of  6 and  2. thus it is worthy to mention that μ((5 + α)/(3 − α)| 6  2) = α and μ((7 − α)/(1 + α)| 6  2) = α. We get the same result if extension principle is used. Applying extension principle we get the membership function as follows: ⎧ 3x−5 ⎪ if 53 ≤ x ≤ 3 ⎪ ⎨ x+1 μ(x | 6  2) = 7−x if 3 ≤ x ≤ 7 x+1 ⎪ ⎪ ⎩ 0 elsewhere.

1.8 Conclusion A few basic ideas on fuzzy set theory have been summarised with a revisit to fuzzy number definition.Two basic concepts—same point and inverse point are defined here. The combinations of these points only used to define different operations among fuzzy numbers. If the fuzzy number is the combinations of many crisp numbers, though with varying membership values, then it is needed to graphically explain the position of same point and inverse point within it. In the subsequent chapters further details have been discussed graphically and with more mathematical rigor.

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Chapter 2

Basic Ideas on Fuzzy Plane Geometry

2.1 Introduction Euclidean geometry employs Cartesian coordinate system in which reference axes are perpendicular to each other. In this system every point is represented by a unique tuple which is crisp. In fuzzy set theory it is considered that the universe is non-fuzzy, and the way we perceive any object is fuzzy or imprecise. This hypothesis guided us to assume a different reference frame, which act as yardstick to measure in fuzzy plane geometry.

2.2 Coordinate System In the study on conventional geometry, the underlying space is evidently the regular Euclidean two-dimensional space R2 . With the help of two perpendicular directions or directed axes O X 1 and O X 2 say, the R2 plane is algebraically described by the collection of two-tuple points (x1 , x2 ). The primary intension in mathematical study on geometry is to describe the geometrical curves or boundary of a planer region through algebraic equations. Towards this end, we try to interrelate the variables x1 and x2 in the general point (x1 , x2 ) by some equation f (x1 , x2 ) = 0. For instance, f (x1 , x2 ) ≡ x12 + x22 − 1 = 0 describes a circle, f (x1 , x2 ) ≡ x12 − x22 − 1 = 0 describes a hyperbola, etc. The axes O X 1 and O X 2 are the fundamental in describing and hence identifying the concerned geometrical curves. We note that the axes O X 1 and O X 2 are just two real number lines being placed perpendicular to each other. These perpendicular axes essentially constitute a reference frame to geometrically describe an equation f (x1 , x1 ) = 0. The geometrical description of a given equation may vary according to differently chosen reference frame. For instance, the equation x12 + x22 = 1 describes a circle of unit radius with

© Springer Nature Switzerland AG 2019 D. Ghosh and D. Chakraborty, An Introduction to Analytical Fuzzy Plane Geometry, Studies in Fuzziness and Soft Computing 381, https://doi.org/10.1007/978-3-030-15722-7_2

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respect to the O X√ 1 -O X 2 reference frame. However, the same equation describes a circle of radius 2 in OY1 -OY2 reference frame where OY1 and OY2 are the perpendicular lines x1 − x2 = 0 and x1 + x2 = 0. More surprisingly, in O Z 1 -O Z 2 reference frame, with z 1 = x12 and z 2 = x22 , the equation x12 + x22 = 1 determines a straight line. Thus the chosen reference frame or coordinate system in describing a geometrical curve or an equation is of supreme importance. It is worth noting that the common idea in all the reference frame to describe a planner curve is that the axes in a reference frame are just two real number lines, being placed perpendicular to each other. Thus the axes are essentially comprised of real numbers. In research on fuzzy plane geometry, we intend to mathematically characterize the fuzzy geometrical elements through the membership functions. Much similar to the primary intention of the study on conventional geometry, we want to algebraically identify the fuzzy geometrical objects as a collection of two-dimensional points with varied membership values. For instance, to describe a so-called fuzzy point ‘around (a, b)’ one can think it as a collection of neighboring points to (a, b) with varied membership values. Obviously, in order to give a profound mathematical description, the distribution of the membership values cannot be arbitrary and they should follow some pattern. Different distribution of membership values will determine different fuzzy points at (a, b). As the underlying space in determining planer fuzzy geometrical objects is considered as the usual Euclidean space, we have an inherent reference frame, namely, the O X 1 -O X 2 frame. We call it as the first reference frame. One can also think of one other reference frame to visualize the fuzzy geometrical plane. Towards this end, unlike the first reference frame, we think the axes are also fuzzy and comprised of fuzzy numbers (x − α/x/x + β) L R instead of just the real numbers x. Mathematically, the axes are given by the following union 

(x − α/x/x + β) L R .

x∈R

In particular, the left-spread α and the right-spreads β of these fuzzy number which lie on the fuzzy axes can be considered as equal. We call this reference as the second reference frame. Accumulating all, in the first reference frame, the axes are real and the membership functions of fuzzy points on the plane R2 are realized as surfaces in R3 . In the second reference frame, the axes are comprised of fuzzy numbers. Figure 2.1 gives a detailed visualization of this reference frame for α = β = 21 . For the clarity of presentation, in Fig. 2.1, although the fuzzy numbers on the axes are depicted in a non-overlapping manner. If all ∪x∈R (x − α/x/x + β) L R ’s for all

2.2 Coordinate System

23

Fig. 2.1 Second reference frame in the fuzzy geometrical plane

x’s across R are depicted, this overlapping scenario of the underlying fuzzy numbers in the fuzzy axes will be easily observed.   In this reference frame any fuzzy location can be represented by a, ˜ b˜ and its location can be easily obtained through moving a˜ = (a − α/a/a + β) L R unit along the fuzzy O X 1 axis and then b˜ = (b −α/b/b  + β) L R unit along O X 2 . We note that ˜ the support set of the fuzzy location a, ˜ b is the rectangular region [a − α, a + β] × [b − α, b + β]. However, this reference frame has the following drawbacks: (i) this reference frame cannot offer a position to the fuzzy location with non rectangular support set. Nonetheless, it cannot offer a position to fuzzy locations with arbitrary rectangular support set. For instance, it is difficult to identify the ˜ 1) ˜ where 1˜ = (1 − α/1/1 + (β + 1)). location of the fuzzy coordinate (1, ˜ (ii) We note that in this reference frame   the addition of two locations (1, 0) and ˜ 0) which is given by 1˜ ⊕ 2, ˜ 0 is not identical to (3, ˜ 0) which is already (2, underlying on the fuzzy axes. It is also difficult to find the locations for the linear combinations of a pair of fuzzy coordinates. For example, consider the fuzzy ˜ 3). ˜ Apparently, the second reference frame cannot ˜ 2) ˜ and (1, coordinates (1, ˜ 2) ˜ ⊕ c2 (1, ˜ 3) ˜ for any c1 and c2 in R. For instance, offer a location to c1 (1, ˜ 2) ˜ ⊕ c2 (1, ˜ 3) ˜ = (1˜ ⊕ 1, ˜ 2˜ ⊕ 3), ˜ being letting c1 = c2 = 1, we observe that c1 (1, a fuzzy coordinate with support set [2 − 2α, 2 + 2β] × [5 − 2α, 5 + 2β], does not have any location in the second reference frame. Thus, the second reference frame leads to a restricted environment. In this book, we retain ourselves in the first reference frame to formulate fuzzy geometrical ideas.

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2 Basic Ideas on Fuzzy Plane Geometry

2.3 Fuzzy Point Definition 2.1 (Fuzzy point) (Buckley and Eslami 1997).  b), is defined by its membership function A fuzzy point at (a, b) in R2 , written as P(a, which satisfies the following conditions: (i) (ii) (iii)

 b)) is upper semi-continuous, μ((x, y)| P(a,  b)) = 1 if and only if (x, y) = (a, b), and μ((x, y)| P(a,  P(a, b)(α) is a compact and convex subset of R2 , for each α in [0, 1].

1 (a, b), P 2 (a, b), P 3 (a, b), … and P 1 , P 2 , P 3 , … are used to represent The notations P fuzzy points. Example 2.1 (Fuzzy point with elliptical base) Let (a, b) be a point in R2 . Consider a right elliptical cone with elliptical base   y−b 2 x−a 2 (x, y) : ( p ) + ( q ) ≤ 1 and vertex (a, b). This right elliptical cone can be  b). The mathematical form of taken as membership function of a fuzzy point P(a,  μ(.| P(a, b)) is: 



 b) = μ (x, y)| P(a,



1 − ( x−a )2 + ( y−b )2 if ( x−a )2 + ( y−b )2 ≤ 1 p q p q 0

elsewhere.

  b)(α) = (x, y) : ( x−a )2 + ( x−b )2 The α-cut of this fuzzy point is P(a, p q ≤ (1 − α)2 , 0 ≤ α ≤ 1. Example 2.2 (Fuzzy point)  Let (a, b) be a point in R2 . Consider a right cone with base (x, y) : | x−a |3 + | y−b |3 p q ≤ 1} and vertex (a, b). This right cone can be taken as membership function of a  b). The mathematical form of μ(.| P(a,  b)) is: fuzzy point P(a,  

y−b 3 3  if | x−a | + | | |3 + | y−b |3 ≤ 1 1 − | x−a p q p q  b) = μ (x, y)| P(a, 0 elsewhere.    b)(α) = (x, y) : | x−a |3 + | y−b |3 ≤ (1 − α) , The α-cut of this fuzzy point is P(a, p q 0 ≤ α ≤ 1. Example 2.3 (Fuzzy point with rectangular base) Let (a, b) be a point in R2 . Consider a right elliptical cone with elliptical base   y−b |, | |} ≤ 1 and vertex (a, b). This right elliptical cone can be (x, y) : max{| x−a p q  b). The mathematical form of taken as membership function of a fuzzy point P(a,  b)) is: μ(.| P(a,

2.3 Fuzzy Point

25

Fig. 2.2 Fuzzy points with different bases



3

 1 − max{| x−a |, | y−b |} if max{| x−a |, | y−b |} ≤ 1 p q p q  b) = μ (x, y) P(a, 0 elsewhere.   b)(α) = (x, y) : max{| x−a |, | y−b |} The α-cut of this fuzzy point is P(a, p q 3 ≤ (1 − α) , 0 ≤ α ≤ 1 (Fig. 2.2).

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2 Basic Ideas on Fuzzy Plane Geometry

Example 2.4 (Fuzzy point with rhomboidal base) Let (a, b) be a point in R2. Consider a right elliptical cone with elliptical base  | + | y−b | ≤ 1 and vertex (a, b). This right elliptical cone can be (x, y) : | x−a p q  b). The mathematical form taken as membership function of a fuzzy point P(a,  of μ(.| P(a, b)) is:

 1 − | x−a | + | y−b | if | x−a | + | y−b |≤1 p q p q  μ (x, y)| P(a, b) = 0 elsewhere.    b)(α) = (x, y) : | x−a | + | y−b | ≤ 1 − α , The α-cut of this fuzzy point is P(a, p q 0 ≤ α ≤ 1. Example 2.5 (Fuzzy point with core at the periphery of the boundary of the support) Let a > 1 be any number. Consider a right cone with base right loop of the lemniscate of Bernoulli (x, y) : (x 2 + y 2 ) ≤ a 2 (x 2 − y 2 ) and vertex (0, 0). This cone can be  0). The mathematical form of taken as membership function of a fuzzy point P(0,  0)) is: μ(.| P(0, 



 0) = μ (x, y)| P(0,



1−

x 2 +y 2 a(x 2 −y 2 )

0

if x 2 + y 2 ≤ a(x 2 − y 2 ), y ≥ 0 elsewhere.

 This 2.3. The α-cut  fuzzy 2point2is depicted in Fig. of this fuzzy point is P(0, 0)(α) = 2 2 (x, y) : x + y ≤ a(1 − α)(x − y ), y ≥ 0 , 0 ≤ α ≤ 1. It is to notice that in polar coordinate the membership function of the fuzzy point  0) can be described by P(0,

 1−  μ (r, θ)| P(0, 0) = 0

r a cos 2θ

if r ≤ a cos 2θ, θ ∈ [0, π4 ] ∪ [3 π4 , π] elsewhere.

Fig. 2.3 Fuzzy point with core at the periphery of the support-boundary

2.3 Fuzzy Point

27

According to the definition of fuzzy points, a fuzzy point can be viewed in two different ways. These two ways are described below. Representation 1. A fuzzy point is a collection of crisp points with different membership values.  b) is the collection of points For instance, in Example 2.1, the fuzzy point P(a, x−b 2 2 ) + ( ) = 1, associated with membership value which are inside the ellipse ( x−a p q  1 − ( x−a )2 + ( x−b )2 . If the membership values are depicted by grey-level optical p q density (the higher grey level implies the higher membership value), fuzzy points can be observed as the figures those are depicted in Fig. 2.2.  b) as a Representation 2. In this method, we observe a fuzzy point P(a, collection of fuzzy sets whose (i) bases are the line segments those are obtained by intersection of the base  b) and the lines those are passing through (a, b), and of P(a, (ii) membership functions by the intersecting curves of the sur are obtained   b) and the planes those are perpendicular to face of z = μ (x, y) P(a, the x y-plane and pass through (a, b). For a geometrical visualization of Representation 2, let us look at Fig. 2.4. In this  b) is a given fuzzy point. The leftmost grey-leveled figure gives Reprefigure, P(a,  b). If we depict membership value along z-axis, then membership sentation 1 of P(a,  function of P(a, b) can be observed as a right cone as depicted in the next figure. Let us consider a plane Pθ , say which is perpendicular to the x y-plane and passes through a line L θ in the x y-plane that passes through (a, b); θ being the angle of pθ be the fuzzy set whose elevation of the line L θ and the positive x-axis. Let  membership function is the intersecting curve of the plane Pθ and the surface of

  b) . We note that the base of the fuzzy set  z = μ (x, y) P(a, pθ is the line segment Aθ Bθ . Varying θ in [0, π], different  pθ ’s will be obtained along different L θ ’s. The  b) can be observed as union of all such  complete fuzzy point P(a, pθ ’s. Thus,  b) = P(a,



 pθ .

θ∈[0,π]

Hence, a fuzzy point can be observed as a collection of fuzzy sets along different lines (L θ ’s) passing through the core of the fuzzy point. Definition 2.2 (Fuzzy number along a line) In defining a fuzzy number, conventionally the real line (R) is taken as the universal

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2 Basic Ideas on Fuzzy Plane Geometry

Fig. 2.4 Geometric visualization of Representation 2

set. Accordingly a fuzzy number p, ˜ say, is defined as a collection of real numbers those are neighboring to the real number p with different membership values. Let us take the x-axis of the x y-plane as the real number line. Let  p be a fuzzy number on the real line. We now consider to define the fuzzy number  p on the x-axis through associating membership value μ(x| p ) to the point (x, 0) over the x-axis. We call this fuzzy set, defined on the x-axis, as fuzzy number  p on the x-axis and denote p X can be written as it by  p X . Explicitly, membership function of 

μ ((x, y)| pX ) =

μ(x| p ) if y = 0 0 elsewhere.

In the fuzzy number  p X , the x-axis is regarded as the universal set. Instead of x-axis, consider any line ax + by = c on the x y-plane as the universal set. Note that the bijective transformation T : R2 → R2 given by T (x, y) = ((x − h) cos θ + (y − k) sin θ, −(x − h) sin θ + (y − k) cos θ) transforms x-axis to the line ax + by = c, where  (i) the point (h, k) = a 2ac , bc being point of intersection for ax + by = c and +b2 a 2 +b2 its perpendicular line through origin bx − ay = 0, and (ii) θ being the angle of elevation of ax + by = c with the positive x-axis. For a complete geometrical visualization we refer to Fig. 2.5.

2.3 Fuzzy Point

29

Fig. 2.5 Fuzzy number along the line ax + by = c

Let (u, v) = T (x, y). Now on the uv-space, regard the axis of U as the universal set. Note that the line ax + by = c in the x y-plane is the u-axis in the uv-plane. Considering u-axis as the universal set, we now consider to define the fuzzy number p X ) to the point  p X on the u-axis through associating a membership value μ ((x, y)| (u, v), where (u, v) = T (x, y). We call this fuzzy set, defined on the u-axis, as fuzzy number  p on the line ax + by = c. Explicitly this fuzzy number  pU is defined by its membership function as

μ((u, v)| pU ) =

μ((x, y)| p X ) if v = 0, where T (x, y) = (u, v) 0 elsewhere.

Example 2.6 (Fuzzy number along a line) Let  3 be a fuzzy number with membership function: ⎧ ⎪ ⎨x − 2 if 2 ≤ x ≤ 3 x−5 2 μ(x| 3) = if 3 ≤ x ≤ 5 2 ⎪ ⎩ 0 elsewhere. Here, the support of  3 is {x : 2 ≤ x ≤ 5}. This  3 can also be placed on the line x − y = 0 as follows. In R2 , the x-axis can be imagined as real line. Considering the x-axis as the universal set, the fuzzy number  3 can be expressed as: ⎧ ⎪ ⎨x − 2 if 2 ≤ x ≤ 3, y = 0 x−5 2 μ((x, y)| 3X ) = if 3 ≤ x ≤ 5, y = 0 2 ⎪ ⎩ 0 elsewhere. Here, the support of 3 is {(x, y) : 2 ≤ x ≤ 5, y = 0}. A transformation involving only 45◦ rotation of the axes transforms the x-axis to x − y = 0. This transformation is 3 on x − y = 0 can be expressed as: T (x, y) = ( √x2 + √y2 , √x2 − √y2 ). Now 

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2 Basic Ideas on Fuzzy Plane Geometry

⎧√ √ ⎪ 2u − 2 if 2 ≤ u ≤ √32 , v = 0 ⎪ ⎨ √ 2 2u−5 μ((u, v)| 3U ) = if √32 ≤ u ≤ √52 , v = 0 2 ⎪ ⎪ ⎩ 0 elsewhere. This fuzzy number is said to be fuzzy number √ three on the line x − y = 0. Here, the support of  3U is the line segment {(x, y) : 2 ≤ x ≤ √52 , x − y = 0}. Note 2.1 The definition has been titled ‘fuzzy number along a line’ because for each and every fuzzy number  p on line ax + by = c there always exists a unique fuzzy number on the real line and the converse is also true (this converse fuzzy number on the real line can be obtained by mapping T −1 ). However, the title ‘(normal, convex) fuzzy set (or fuzzy point) along a line’ may be more appropriate. In our discussion, the two terminologies are used interchangeably.

The following example illustrates how to find the fuzzy number on the real line corresponding to a given fuzzy number on a line ax + by = c. Example 2.7 (Fuzzy number along a line) Let L π be the line y − x = 1; A ≡ (1, 2), B ≡ (2, 3) and C ≡ (3, 4) being three 4 p π be a given fuzzy number, defined on the line segment joining points on L π . Let  4 4 A to C, which is described by the membership function:  ⎧   2 + (y − 3)2 ⎪ √1 L if (x, y) ∈ AB (x − 2) ⎪  ⎨  2   1 μ (x, y)| p π = R √ (x − 2)2 + (y − 3)2 if (x, y) ∈ BC ⎪ 2 4 ⎪ ⎩ 0 elsewhere, where L and R are two reference functions with L(0) = R(0) = 1 and L(1) = R(1) = 0. Let θ be the angle of elevation of the line L π with the positive x-axis, and (h, k) 4 be he point of intersection for L π and its perpendicular line through origin, i. e., 4  y + x = 0. Then, θ = π4 and (h, k) = − 21 , 21 . Note that the transformation (u, v) = T (x, y) = ((x − h) cos θ + (y − k) sin θ, −(x − h) sin θ + (y − k) cos θ)   √ , y−x √ − √1 = x+y 2 2 2 transforms the x y-plane to the uv-plane where the axes of u and v are the lines y = x + 1 and y + x = 0, respectively.

2.3 Fuzzy Point

31

In the reverse way, the transformation (u, v) = T −1 (x, y) = (u cos θ − v sin θ + h, u sin θ + v cos θ + k)   √ − 1 , u+v √ + 1 = u−v 2 2 2 2 transforms the x y-plane to uv-plane. In the uv-plane, the fuzzy number  p π can be expressed by the membership func4 tion  ⎧  5 u ⎪ √ if √32 ≤ u ≤ √52 , v = 0 L − ⎪ 2 ⎨ 2 μ ((u, v)| pU ) = R √u − 5 if √5 ≤ u ≤ √7 , v = 0 2 ⎪ 2 2 2 ⎪ ⎩ 0 elsewhere. p π on the line L π can be The fuzzy number  p X on x-axis corresponding to  4 4 obtained through the interrelation of the axes of u and x: x = u cos θ − v sin θ + h with v = 0 = Therefore,

√u 2

− 21 .

⎧ ⎪ ⎨ L (2 − x) if 1 ≤ x ≤ 2, y = 0 μ ((x, y)| p X ) = R (x − 2) if 2 ≤ x ≤ 3, y = 0 ⎪ ⎩ 0 elsewhere.

Figure 2.6 gives a visualization of this example.

Fig. 2.6 Fuzzy number along the line y = x + 1 in Example 2.7

32

2 Basic Ideas on Fuzzy Plane Geometry

2.4 Addition of Two Fuzzy Points  and Q  be two fuzzy points. Due to Representation 2 of a fuzzy point, we have Let P = P



=  pθ and Q

θ∈[0,π]



 qθ .

θ∈[0,π]

According to the extension principle, ⊕ Q = P



 pθ ⊕

θ∈[0,π]

=





 qθ

θ∈[0,π]

( pθ ⊕  qφ ).

θ,φ∈[0,π]

That is, for all possible values of θ and φ, the fuzzy sets  pθ and  qφ have to be added  ⊕ Q.  by the extension principle to obtain the complete P Lemma 2.1 Refer to Fig. 2.7. Let p˜ θ be a fuzzy number along the line y = tan θx + c p and q˜θ be a fuzzy number along the line y = tan θx + cq . Then (i) the support of  pθ ⊕  qθ is a line segment in the line y = tan θx + (c p + cq ) qθ is a fuzzy number along the line y = tan θx + (c p + cq ). (ii)  pθ ⊕ 

Proof (i) Let (x, y) ∈  pθ ⊕  qθ . Then there exist (x p , x p ) ∈  pθ and (xq , xq ) ∈  qθ such that (x, y) = (x p , y p ) + (xq , yq ). pθ , y p = tan θx p + c p . Similarly, yq = tan θxq + cq . ThereSince (x p , y p ) ∈  fore, (x, y) = (x p , tan θx p + c p ) + (xq , tan θxq + cq )  = x p + xq , tan θ(x p + xq ) + (c p + cq ) ⇒ y = tan θx + (c p + cq ). (ii) This part is similar to the addition of two regular fuzzy numbers on the real line. Note 2.2 Lemma 2.1 informs that  pθ ⊕  qφ is a fuzzy number along a line for θ = φ. qφ is not a fuzzy set along a line, as its support is not a However, for θ = φ,  pθ ⊕  line segment.

2.4 Addition of Two Fuzzy Points

33

Fig. 2.7 Geometric visualization of Lemma 2.1

 and Q  as a collection of fuzzy numbers along In order to perform addition of P  ⊕ Q,  we define addition of P  pθ ⊕  qφ ) as P lines, instead of taking ∪θ,φ∈[0,π] (  and Q in a fuzzy geometrical plane as + Q  := P



( pθ ⊕  qθ ).

θ∈[0,π]

+ Q  is a fuzzy point when P  and Q  are two continuous Theorem 2.2 proves that P fuzzy points.

2.4.1 An Effective Way for Addition of Two Fuzzy Points  b) and Q(c,  d) be two fuzzy points and Let P(a,  b) = P(a,

 θ∈[0,π]

 d) =  pθ and Q(c,

 θ∈[0,π]

 qθ .

34

2 Basic Ideas on Fuzzy Plane Geometry

The membership function of  pθ can be written as:

μ pθ (x, y) := μ((x, y)| pθ ) =

 if (x, y) ∈ l pθ μ((x, y)| P) 0 otherwise,

where l pθ is a line passing through (a, b) (Fig. 2.8) with angle θ to the line, l say, joining (a, b) and (c, d). Let us look at the membership function of p˜ θ . Here,  gradually increases as x increases, and for for (x, y) ∈ l pθ and x ≤ a, μ((x, y)| P)  gradually decreases as x increases. The same (x, y) ∈ l pθ and x ≥ a, μ((x, y)| P) qθ ). reasoning is applied to y for y ≤ b and y ≥ b and we write μqθ (x, y) := μ((x, y)|  b) and Q(c,  d): In computing the addition of the fuzzy points P(a,  b) + Q(c,  d) = P(a,



( pθ ⊕  qθ ),

θ∈[0,π]

as per the definition of extension principle to compute  pθ ⊕  qθ , its base consists of pθ and (x2 , y2 ) ∈  qθ }. Thus to compute  pθ ⊕  qθ one {(x1 , y1 ) + (x2 , y2 ) : (x1 , y1 ) ∈  pθ and has to take the union of all the points those are addition of some (x1 , y1 ) ∈   b) + Q(c,  d) there qθ . It is evident that for many (x3 , y3 ) in P(a, some (x2 , y2 ) ∈  pθ , (x2 , y2 ) ∈  qθ such that (x3 , y3 ) = are infinitely many combinations of (x1 , y1 ) ∈  (x1 , y1 ) + (x2 , y2 ). Theorem 2.1 helps to find some redundant combinations and separates out the qθ much efficiently than the regular effective combinations in order to compute  pθ ⊕  extended addition process. First, we provide the following lemma, which is needed to prove the theorem. In Lemma 2.2 and Theorem 2.1, the lines l, l pθ and lqθ , the pθ and  qθ bear the same meaning as functions μ pθ and μqθ , and the fuzzy numbers  defined above.  b) and Q(c,  d) be two fuzzy points. If (x1 , y1 ) and Lemma 2.2 Let P(a,  and lqθ ∩ Q(0),  respectively, with x1 ≤ a, (x2 , y2 ) are two points in l pθ ∩ P(0)  = μ((x2 , y2 )| Q)  = ω, then x2 ≤ c and μ((x1 , y1 )| P)  + Q)  = ω. μ((x1 + x2 , y1 + y2 )| P

  Proof Let (x1  , y1  ) ∈ l pθ ∩ P(0) and (x2  , y2  ) ∈ lqθ ∩ Q(0), where (x1  , y1  ) = (x1 , y1 ) and (x2  , y2  ) = (x2 , y2 ) but (x1  + x2  , y1  + y2  ) = (x1 + x2 , y1 + y2 ). Four cases may arise: Case 1. x1  > x1 and y1  ≤ y1 , i.e., x2  ≤ x2 and y2  > y2 . Case 2. x1  > x1 and y1  > y1 , i.e., x2  ≤ x2 and y2  ≤ y2 .

2.4 Addition of Two Fuzzy Points

35

Fig. 2.8 Addition of two fuzzy points for the increasing and decreasing  along parts of μ((x, y)| P) l pθ

Case 3. x1  ≤ x1 and y1  < y1 . Case 4. x1  ≤ x1 and y1  > y1 .  In all four cases, either x1  ≤ x1 or x2  ≤ x2 . Therefore, min{μ((x1  , y1  )| P),      μ((x2 , y2 )| Q)} ≤ ω since μ((x, y)| P) and μ((x, y)| Q) are increasing with respect to the first variable x along l pθ for x ≤ a and along lqθ for x ≤ c, respectively.  μ((x2  , y2  )| Q))  ≤ ω and the maxiThus, in any situation, min(μ((x1  , y1  )| P), mum is attained for (x1 , y1 ) and (x2 , y2 ), which yields the result.  b) and Q(c,  d) be two continuous fuzzy points. If Theorem 2.1 Let P(a,   are two points such that x1 + (x1 , y1 ) ∈ l pθ ∩ P(0) and (x2 , y2 ) ∈ lqθ ∩ Q(0) ∗ ∗  and (x2∗ , y2∗ ) ∈ lqθ ∩ Q(0)  x2 ≤ a + c, then there exist (x1 , y1 ) ∈ l pθ ∩ P(0) such that: (i) x1∗ ≤ a, x2∗ ≤ c,  = μ((x2∗ , y2∗ )| Q),  (ii) μ((x1∗ , y1∗ )| P)

36

2 Basic Ideas on Fuzzy Plane Geometry

(iii) x1 + x2 = x1∗ + x2∗ , y1 + y2 = y1∗ + y2∗ , and  + Q)  = μ((x1∗ , y1∗ )| P)  = μ((x2∗ , y2∗ )| Q).  (iv) μ((x1 + x2 , y1 + y2 )| P ˜  b) and q˜θ = lqθ ∩ Q(c,  d). Since P(a, Proof Let p˜ θ = l pθ ∩ P(a, b) is a continuous fuzzy point, the membership function of p˜ θ , μ pθ , increases as x increases for x ≤ a.  d), along lqθ , the function μqθ is increasing with respect to x for Similarly, for Q(c, x ≤ c. Here two cases may arise. Case 1. In this case, we consider that μ pθ and μqθ are strictly increasing for x ≤ a −1 and x ≤ c, respectively. Then, both μ pθ and μqθ are bijective and hence μ−1 pθ and μqθ exist and they are continuous and strictly increasing on [0, 1]. −1 Consider the function g = μ−1 pθ + μqθ . Then, obviously, g is strictly increasing and continuous on [0, 1]. Let (X, Y ) = (x1 + x2 , y1 + y2 ) and ω be the value of g −1 (X, Y ). For this ω, we −1 ∗ ∗ consider the points (x1∗ , y1∗ ) = μ−1 pθ (ω) and (x 2 , y2 ) = μqθ (ω). ∗ ∗ ∗ ∗ Addition of these two points is (x1 + x2 , y1 + y2 ) = g(ω) = (X, Y ). Moreover, −1 x1∗ ≤ a since μ−1 pθ is strictly increasing in [0, 1] and μ pθ (1) = (a, b). Similarly, ∗ x2 ≤ c. Since (x1∗ , y1∗ ) and (x2∗ , y2∗ ) are two points on l pθ and lqθ , respectively, with x1∗ ≤ a  + Q)  = and x2∗ ≤ c. According to Lemma 2.2, we obtain μ((x1 + x2 , y1 + y2 )| P ∗ ∗  ∗ ∗  μ((x1 , y1 )| P) = μ((x2 , y2 )| Q) = ω, which proves the theorem in this case. Case 2. Consider another case in which μ pθ and μqθ are not strictly increasing for x ≤ a and x ≤ c, respectively. In other words, there exist two intervals [a1 , a2 ] and [c1 , c2 ] (possibly a1 = a2 and c1 = c2 ) such that μ pθ and μqθ are constant, ω say, for  + Q)  x in [a1 , a2 ] and [c1 , c2 ], respectively. In this case, we claim that μ((x, y)| P     = ω for all (x, y) ∈ ( P + Q)(0) ∩ l with x ∈ [a1 + c1 , a2 + c2 ], where l is the line with angle θ to l and passing though (a + c, b + d).  ∩ l pθ and (x2 , y2 ) ∈ Q(0)  ∩ lqθ be two points with x1 ∈ Let (x1 , y1 ) ∈ P(0)    x2 ∈ [c1 , c2 ]. Then min μ((x1 , y1 )| P), μ((x2 , y2 )| Q) [a1 , a2 ] and = min{ω, ω} = ω. 1 (0) ∩ l pθ and (x2 , y2 ) ∈ P 2 (0) ∩ lqθ be such that x1 + x2 Let (x1 , y1 ) ∈ P    = a1 + c1 . If x1 ≤ a1 , then μ((x1 , y1 )| P) ≤ ω, and hence min μ((x1 , y1 )| P),  ≤ ω. If x1 > a1 , then x2 ≤ c1 . Thus, μ((x2 , y2 )| Q)  ≤ ω. This imμ((x2 , y2 )| Q)   μ((x2 , y2 )| Q)  ≤ ω and equality occurs for x1 = a1 , plies that min μ((x1 , y1 )| P), and x2 = c1 . 1 (0) ∩ l pθ and (x2 , y2 ) ∈ The same result holds true for two points (x1 , y1 ) ∈ P 2 (0) ∩ lqθ with x1 + x2 = a2 + c2 . P  + Q)(0)  Hence, for (x, y) ∈ ( P ∩ l  with x ∈ [a1 + c1 , a2 + c2 ],

2.4 Addition of Two Fuzzy Points

 + Q)  = μ((x, y)| P =

37

sup

(x1 ,y1 )+(x2 ,y2 )=(x,y)

 μ((x2 , y2 )| Q)}  min{μ((x1 , y1 )| P),

 μ((x2 , y2 )| Q)}  sup min{μ((x1 , y1 )| P),

x1 +x2 =x

= ω. Thus, our claim is proved and this result assures the existence of many (x1∗ , y1∗ ) and (x2∗ , y2∗ ) that satisfy properties (i)–(iv) in the theorem. Hence, the theorem is proved. Example 2.8 (Example supporting to Theorem 2.1)  Let P(−1, 0) be a fuzzy point whose membership function is

 1 − (x + 1)2 + y 2  μ (x, y)| P(−1, 0) = 0 



if (x + 1)2 + y 2 ≤ 1 elsewhere.

 3) be another fuzzy point whose membership function is Let P(2,

  1 − 4(x − 2)2 + 16(y − 3)2  3) = μ (x, y)| P(2, 0

if 4(x − 2)2 + 16(y − 3)2 ≤ 1 elsewhere.

˜ At first we find P˜ + Q.  For each α ∈ [0, 1], the α-cut of the fuzzy point P(−1, 0) is {(x, y) : (x + 1)2 + y 2 ≤ (1 − α)2 }.  3) is The α-cut of the fuzzy point P(2, {(x, y) : 4(x − 2)2 + 16(y − 3)2 ≤ (1 − α)2 }. Let l pθ :

x+1 cos θ

=

y sin θ

and lqθ :

x−2 cos θ

=

y−3 . sin θ

 The intersection of the boundary of P(−1, 0)(α) and the line l pθ is the point (−1 + (1 − α) cos θ, (1 − α) sin θ).  is the union of all the points (−1 + (1 − α) cos θ, Note that the fuzzy point P (1 − α) sin θ) associated with the membership values α, i.e.,  P(−1, 0) =



(−1 + (1 − α) cos θ, (1 − α) sin θ) .

α∈[0,1]

Similarly, for each  α ∈ [0, 1], the intersection of the boundary of the α-cut of  3) with lqθ is 2 + 1 cos θ, 3 + 1 (1 − α) sin θ . The fuzzy point Q(2, 3) can Q(2, 2 4 be expressed by

38

2 Basic Ideas on Fuzzy Plane Geometry

  2 + 21 cos θ, 3 + 41 (1 − α) sin θ .

 3) = P(2,

α∈[0,1]

+ Q  is the union of all the points Therefore, P  (−1 + (1 − α) cos θ, (1 − α) sin θ) + 2 + 21 cos θ, 3 + 14 (1 − α) sin θ  = 1 + 23 cos θ, 3 + 45 (1 − α) sin θ associated with membership value α. Mathematically,   1 + 23 cos θ, 3 + 54 (1 − α) sin θ .

+ Q = P

α∈[0,1]

+ Q  is the set Therefore, boundary of the α-cut of P ⎧ ⎨ ⎩

(x, y) :

(x−1)2  2 3 2

+

(y−3)2  2 5 4

≤ (1 − α)2

⎫ ⎬ ⎭

.

+ Q  can be expressed by its membership function as Hence, P  + Q  = μ (x, y)| P

 ⎧ 2 ⎪ ⎨1 − (x−1)  2 + 3 ⎪ ⎩

2

0

(y−3)2  2 5 4

if

(x−1)2  2 3 2

+

(y−3)2  2 5 4

≤1

elsewhere.

We now consider θ = 0 and hence the lines l pθ and lqθ will reduce to the lines y = 0 and y = 3.   and (x2 , y2 ) = Accordingly, consider two points (x1 , y1 ) = (− 47 , 0) ∈ l p0 P 5 3  Here (x1 , y1 ) + (x2 , y2 ) = ( , 3). The membership value of ( 3 , 3) ( 2 , 3) ∈ lq0 ∩ Q. 4 4 1 + Q  is . on P 6  and Q  are respectively (a, b) = (−1, 0) and (c, d) = (2, 3). Here The cores of P 3 x1 + x2 = 4 < 1 = a + c and also note that there are several pair of points, indeed  and Q  whose addition is ( 3 , 3). For instance, infinitely many, on the supports of P 4 (−1.85, 0), (2.6, 3); (−1.9, 0), (2.65, 3); (−1.5, 0), (2.25, 3), etc.  and However, we note that we have the pair of points (x1∗ , y1∗ ) = (− 76 , 0) ∈ P 23 ∗  which satisfy (x2 , y2∗ ) = ( 12 , 3) ∈ Q (i) x1∗ ≤ −1, x2∗ ≤ 2,  = μ((x2∗ , y2∗ )| Q)  = 1, (ii) μ((x1∗ , y1∗ )| P) 6 (iii) x1 + x2 = x1∗ + x2∗ = 43 , y1 + y2 = y1∗ + y2∗ = 3, and  + Q)  = μ((x1∗ , y1∗ )| P)  = μ((x2∗ , y2∗ )| Q)  = 1. (iv) μ((x1 + x2 , y1 + y2 )| P 6

2.4 Addition of Two Fuzzy Points

39

It is worthy to note that the points (x1∗ , y1∗ ) and (x2∗ , y2∗ ) are intersecting points of   3)( 1 ), respectively. 0)( 16 ) and P(2, l p0 and lq0 with P(−1, 6  + Q,  to obtain the memNote 2.3 Theorem 2.1 informs that in computing P bership value of any point (x1 + x2 , y1 + y2 ), there are many (x1  , y1  ) ∈ l pθ ∩   and (x2  , y2  ) ∈ lqθ ∩ Q(0) such that (x1  + x2  , y1  + y2  ) = (x1 + x2 , P(0) y1 + y2 ). However, consideration of the combinations for which either  = μ((x2  , y2  )| Q)  or (i) μ((x1  , y1  )| P) (ii) x1  < a, x2  > c or x1  > a, x2  < c is irrelevant. This is why these types of combinations of points can be called redundant combinations, and on the other hand we call the combinations of the points (x1∗ , y1∗ ) and (x2∗ , y2∗ ) in the theorem as effective combination. Note 2.4 It should be mentioned that in applying a binary increasing operator for the sup-min composition on continuous fuzzy sets, while taking the minimum of two membership values of two different elements, the lower membership value always dominates the higher one. Thus, it is reasonable to take only combinations of elements with the same membership values (effective combinations) because otherwise higher membership values would not have any effect on the membership value. In fact, Theorems 1.8.1 and 2.1 suggest that this composition should be applied. Note 2.5 Observe that combinations besides effective combinations are redundant + Q  can be combined. We call these combinasince only effective combinations P tions as combinations of same points. A formal definition of same points with respect to two continuous fuzzy points is given in the next section. Accumulating all the discussions in this section, it is evident to write that + Q = P

 {(x1∗ , y1∗ ) + (x2∗ , y2∗ ) : where (x1∗ , y1∗ ) ∈ p˜ θ , (x2∗ , y2∗ ) ∈ q˜θ are the points in Theorem 2.1}.

It is note worthy to mention that the symbols ⊕ and + carry different meanings. 

⊕ Q = P

( pθ ⊕  qφ ),

θ,φ∈[0,π]

however,

+ Q = P



( pθ ⊕  qθ ).

θ∈[0,π]

40

2 Basic Ideas on Fuzzy Plane Geometry

2.5 Same and Inverse Points with Respect to Two Fuzzy Points This section introduces the concepts of same and inverse points. These concepts are then used to define various fuzzy geometrical ideas. Definition 2.3 (Same points with respect to fuzzy points)  b), Let (x1 , y1 ), (x2 , y2 ) be two points on the supports of the fuzzy points P(a,  P(c, d), respectively, and let L 1 be a line joining (x1 , y1 ) and (a, b).  b) is a fuzzy point, along L 1 , a fuzzy number,  r1 say, is situated on the As P(a,  b). The membership function of this fuzzy number support of P(a, r1 can be written  b)) for (x, y) in L 1 , and 0 otherwise. as: μ((x, y)| r1 ) = μ((x, y)| P(a, r2 say, Similarly, along a line (L 2 ) joining (x2 , y2 ) and (c, d), a fuzzy number,   d). will be obtained on the support of P(c,  b) Now the points (x1 , y1 ), (x2 , y2 ) are said to be same points with respect to P(a,  and P(c, d) if: (i) (x1 , y1 ) and (x2 , y2 ) are same points with respect to  r1 ,  r2 , and (ii) L 1 , L 2 make the same angle with the line joining (a, b) and (c, d). In other words, the points (x1 , y1 ) and (x2 , y2 ) are same points with respect to  b) and P(c,  d) if P(a,  b)) = μ((x2 , y2 )| P(c,  d)), (i) μ((x1 , y1 )| P(a, (ii) (x1 , y1 ) and (x2 , y2 ) lie on the same side of the line joining (a, b) and (c, d), and (iii) the lines L 1 and L 2 are parallel. Example 2.9 (Same points with respect to fuzzy points)  2) be a fuzzy point whose membership function is a right circular cone with Let P(2,  6) be another base {(x, y) : (x − 2)2 + (y − 2)2 ≤ 2} and vertex (2, 2). Let P(5, fuzzy point whose membership function is a right elliptical cone with base {(x, y) : 2 (x−5)2 + (y−6) ≤ 1} and vertex (5, 6). 5/3 5/2 The bases of these fuzzy points are depicted in Fig. 2.9 by the circle centered at Q 1 (2, 2) and the ellipse centered at Q 2 (5, 6). Consider the points P1 ≡ (1.5, 2.5) and P2 ≡ (4.5, 6.5) from the supports of  2) and P(5,  6), respectively. The line joining P1 and Q 1 is L 1 : x + y = 4. P(2,  2). r1 say, on the support of P(2, Along L 1 , there exists a triangular fuzzy number,  The base of  r1 is the set {(x, y) : (x − 2)2 + (y − 2)2 ≤ 2, x + y = 4}. Visualizing  2) as a surface in R3 , the membership function the membership function of P(2, of  r1 may be perceived as the union of the straight line segments from (1, 3, 0) to (2, 2, 1) and from (2, 2, 1) to (3, 1, 0).

2.5 Same and Inverse Points with Respect to Two Fuzzy Points

41

Similarly, along the line joining P2 and Q 2 , L 2 : x + y = 11, there exists a trian 6). The support set of  r2 is the gular fuzzy number,  r2 say, on the support of P(5, (y−6)2 (x−5)2 line segment {(x, y) : 5/3 + 5/2 ≤ 1, x + y = 11}. The membership function of  r2 is the union of the straight line segments from (4, 7, 0) to (5, 6, 1) and from (5, 6, 1) to (6, 5, 0). In Fig. 2.9, A1 ≡ (1, 3), B1 ≡ (3, 1), A2 ≡ (4, 7), B2 ≡ (6, 5). Apparently, r2 (0) = A2 B2 .  r1 (0) = A1 B1 and  Note that (i) with respect to  r1 and  r2 , the points P1 ≡ (1.5, 2.5) and P2 ≡ (4.5, 6.5) are same points.  2)) = μ((1.5, 2.5)| (ii) μ((1.5, 2.5)| P(2, r1 ) = 0.29,  μ((4.5, 6.5)| P(5, 6)) =μ((4.5, 6.5)| r2 ) = 0.29. (iii) The line joining (2, 2) and (5, 6) is 4y − 3x = 2. Both the lines L 1 : x + y = 4 and L 2 : x + y = 11 make the same angle θ = tan−1 (−7) with 4y − 3x = 2.

 1 ) and P(Q  2) Fig. 2.9 Same and inverse points for two continuous fuzzy points P(Q

42

2 Basic Ideas on Fuzzy Plane Geometry

Therefore, the points (1.5, 2.5) and (4.5, 6.5) are same points with respect to the  2) and P(5,  6). fuzzy points P(2, Definition 2.4 (Addition of two fuzzy points) 2 is denoted by P 1 + P 2 and its membership 1 and P Addition of the fuzzy points P 2 ) = sup{α : (x, y) = λ(x1 , y1 ) + (x2 , y2 ), 1 + P function is defined by μ((x, y)| P 1 (0) and (x2 , y2 ) ∈ P 2 (0) are same points with membership where (x1 , y1 ) ∈ P value α}. Definition 2.5 (Scalar multiplication of a fuzzy point)  b) by λ is written as λ P(a,  b) Let λ ∈ R. Scalar multiplication of a fuzzy point P(a, and its membership function is defined by:

⎧   x y  ⎪ μ ( P(a, b) if λ = 0 , )

⎪ ⎪ λ λ



⎨ 

 b) if λ = 0, (x, y) = (0, 0)  b) = sup μ (u, v) P(a, μ (x, y) λ P(a, ⎪ (u,v)∈R2 ⎪ ⎪ ⎩ 0 if λ = 0, (x, y) = (0, 0). Definition 2.6 (Convex and linear combination of fuzzy points) 1 and P 2 is denoted by λ P 1 + (1 − λ) P 2 The linear combination of the fuzzy points P   and its membership function is defined by μ((x, y)|λ P1 + (1 − λ) P2 ) = sup{α : 1 (0) and (x2 , y2 ) ∈ P 2 (0) (x, y) = λ(x1 , y1 ) + (1 − λ)(x2 , y2 ), where (x1 , y1 ) ∈ P are same points with membership value α}. 2 1 + (1 − λ) P If λ is taken in the interval [0, 1], then a linear combination λ P 1 and P 2 . becomes a convex combination of P 1 , P 2 are two continuous fuzzy points, then Theorem 2.2 If P (i) (ii) (iii)

1 is a fuzzy point for all λ ∈ R, λP 2 is a fuzzy point, and  P1 + P 1 + λ2 P 2 , λ1 , λ2 ∈ R is also a fuzzy point. the linear combination λ1 P

Proof (i) This part is obviously followed from Definition  2.5.  1 + P 2 ≥ α is (ii) Let α ∈ [0, 1]. First, we argue that the set A(α) := z : μ z P convex and bounded. Let z 1 , z 2 ∈ A(α). Thus, there exist same points x1 , y1 and 1 ), μ(x2 | P 1 ), x2 , y2 such that z 1 = x1 + y1 , z 2 = x2 + y2 with each of μ(x1 | P 2 ) and μ(y2 | P 2 ) is greater than or equal to α. Let λ ∈ R. Note that μ(y1 | P λz 1 + (1 − λ)z 2 can be expressed as (λx1 + (1 − λ)x2 ) + (λy1 + (1 − λ)y2 ). 1 (α) and P 1 (α) is convex, λx1 + (1 − λ)x2 ∈ P 1 (α). SimiSince x1 , x2 ∈ P 2 (α). Thus, whether or not λx1 + (1 − λ)x2 and larly, λy1 + (1 − λ)y2 ∈ P

 2 is at least α. 1 + P λy1 + (1 − λ)y2 are same points, μ λz 1 + (1 − λ)z 2 P 1 (α), Therefore, λz 1 + (1 − λ)z 2 ∈ A(α), and hence A(α) is convex. As P 2  P2 (α) are both compact subsets of R , they are bounded. Any z in A(α) can 1 (α) and be obtained by taking a combination of same points that belong to P

2.5 Same and Inverse Points with Respect to Two Fuzzy Points

43

2 (α). Thus, A(α) is bounded trivially. P Now we prove that A(α) is closed. If the set of all limit points of A(α) is empty, then this part is obviously true. If the set is not

empty, then let z 0 be a limit point

P  = β. Thus, β < α. Now +P of A(α). If possible, let z ∈ / A(α). Let μ z 0     0 1 2 1 + P 2 ) ≥ α ⊂ z : μ z P 1 + P 2 ≥ β . Let be the distance z0 ∈ / z : μ z P between z 0 and A(α). It is easily perceived that > 0. Now A(α) and the open ball B(z 0 , ) have empty intersection and hence z 0 cannot be a limit point of A(α), which is a contradiction. Thus, z 0 ∈ A(α). Since z 0 is arbitrarily taken, A(α) is closed.   2 ≥ t is closed. Thus, the mem1 + P Obviously, ∀ t ∈ R the set z : μ z P 1 + P 2 ) is upper semi-continuous. bership function μ(z| P Since A(α) is closed and bounded, A(α) is a compact subset of R2 .   Let

points at the points (a, b) and (c, d), respectively. Then  P1 , P2 be fuzzy 1 + P 2 = 1. Thus, P 2 is a fuzzy point. 1 + P μ (a + c, b + d) P (iii) This part is an application of the previous two parts, and the proof is omitted. Definition 2.7 (Fuzzy point dividing a fuzzy line segment in m:n ratio) The fuzzy point internally dividing the fuzzy line segment in a given ratio m : n is n 1 + m P 2 . The midpoint of the two fuzzy points can be the fuzzy point m+n P m+n obtained by taking m = 1, n = 1. Example 2.10 We consider the fuzzy points taken in Example 2.9. The fuzzy point that internally divides the line segment joining those two fuzzy points in the ratio  2 : 3 is P(3.2, 3.6), whose membership function is a right circular cone with base  2 2 (x, y) : (x−3.2) + (y−3.6) ≤ 1 and vertex (3.2, 3.6). 1.362 1.482 Definition 2.8 (Inverse points with respect to fuzzy points) Let (x1 , y1 ) and (x2 , y2 ) be two points belonging to the supports of two different  b) and P(c,  d), respectively. The points (x1 , y1 ), (x2 , y2 ) fuzzy points P(a,  b) and P(c,  d) if (x1 , y1 ), are said to be inverse points with respect to P(a,   d), where (−x2 , −y2 ) are same points with respect to P(a, b) and − P(c,   − P(c, d) is λ P(c, d), with λ = −1. Figure 2.9 explains same and inverse points for two continuous fuzzy points  1 ) and P(Q  2 ). The outer circle centered at Q 1 and the outer ellipse centered P(Q  1 ) and the at Q 2 are their respective supports. The inner circle is the α-cut of P(Q  inner ellipse is the α-cut of P(Q 2 ). M L is a line joining Q 1 and Q 2 . A1 B1 and A2 B2 are lines passing through Q 1 and Q 2 , respectively. Both A1 B1 and A2 B2 make the same angle with M L, ∠A1 Q 1 L = ∠A2 Q 2 L = θ (say). The points A1 , A2 ; P1 , P2 ; R1 , R2 ; … are pairs of same points and P1 , R2 ; A1 , B2 ; … are pairs of inverse points.

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2 Basic Ideas on Fuzzy Plane Geometry

Example 2.11 (Inverse points with respect to fuzzy points)  2) and P(5,  6) in Example 2.9. The points P1 ≡ Consider the fuzzy points P(2,  2) and P(5,  6). (1.5, 2.5), R2 ≡ (5.5, 5.5) are inverse points with respect to P(2, 1 and P 2 say, can be Note 2.6 Note that subtraction of two fuzzy points/numbers, P done by taking the supremum over the combination of inverse points in the sup-min 2 = P 1 ⊕ (− P 2 ). 1  P composition of the fuzzy sets, because it can be proved that P The same observation can be applied in evaluating a fuzzy distance.

2.6 Fuzzy Line Segment The usual procedure to obtain a crisp line segment joining two crisp points is to consider the union of all possible convex combinations of the points. Similarly, we 2 as the union of all 1 and P define the fuzzy line segment joining two fuzzy points P 2 . If we denote the fuzzy line segment by 1 and P possible convex combinations of P  L P1 P2 , then   1 + (1 − λ) P 2 ). L P1 P2 = (λ P λ∈[0,1]

2 , only the additions of 1 and P Theorem 2.1 implies that for two fuzzy points P ∗ ∗ ∗ ∗   1 + P 2 . the same points (x1 , y1 ) ∈ P1 and (x2 , y2 ) ∈ P2 are sufficient to evaluate P Similarly, it can be proved that the convex combinations of the same points are also 2 for any λ ∈ [0, 1]. 2 + (1 − λ) P sufficient to evaluate the convex combination λ P Therefore, only the union of the line segments whose extremities the same points 2 will construct the complete fuzzy line segment  1 and P L P1 P2 . Thus, a fuzzy of P line segment can be defined through the following two ways. Definition 2.9 (Fuzzy line segment (Method 1)) 2 can be defined by 1 and P The fuzzy line segment joining two fuzzy points P  L P1 P2 =



1 + (1 − λ) P 2 ). (λ P

λ∈[0,1]

Basically, in this method we observe the fuzzy line segment as the union of all the 2 , λ ∈ [0, 1]. 1 + (1 − λ) P fuzzy points λ P Definition 2.10 (Fuzzy line segment (Method 2)) 1 and P 2 can be defined by The fuzzy line segment  L P1 P2 joining the fuzzy points P  L P1 P2 =



1 and (x2 , y2 ) ∈ P 2 are same points . λ(x, y) + (1 − λ)(x2 , y2 ) : (x1 , y1 ) ∈ P

Method 2 essentially defines the fuzzy line segment  L P1 P2 through its membership function as

2.6 Fuzzy Line Segment

45

Fig. 2.10 Fuzzy line segment  L P1 P2 in Method 2

  μ (x, y)| L P1 P2 = sup{α : where (x, y) lies on the line segment joining same points 1 and (x2 , y2 ) ∈ P 2 with membership value α}. (x1 , y1 ) ∈ P

We now obtain an equation of the fuzzy line segment  L P1 P2 , in Method 2, joining 2 (c, d). For a visualization of the procedure, we follow Fig. 2.10. 1 (a, b) and P P   1 or (x, y) ∈ 2 , q = max x : (x, y) ∈ P 1 or (x, y) ∈ P Let p = min x : (x, y) ∈ P  P2 ,   1 or (x, y) ∈ P 2 and s = max y : (x, y) ∈ P 1 or (x, y) ∈ r = min y : (x, y) ∈ P 2 . P L , there always exist two It is worth noting that for the fuzzy line segment  P1 P2

curves f (x, y) = 0 and g(x, y) = 0 in [ p, q] × [r, s] that are the boundaries of  L P1 P2 (0) on either side of  L P1 P2 (1). If we consider a line, l say, perpendicular to  L P1 P2 (1), then the cross-section of  L P1 P2 (0) on the vertical plane passing through l must be an L R-type fuzzy number along l. Considering different l, we obtain different fuzzy numbers along l whose reference functions Land R are  identical. Thus, we can write the equation of the fuzzy line segment as f (x, y) (y − b) −   d−b (x − a) g(x, y) = 0, where the membership function μ(.| L ) gradually c−a

LR

P1 P2

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2 Basic Ideas on Fuzzy Plane Geometry

increases from 0 to 1 on either side of (y − b) − d−b (x − a) = 0; L and R are c−a suitable reference functions.     (x − a) g(x, y) = 0 means that The equation f (x, y) (y − b) − d−b c−a LR

along any line perpendicular to (y − b) − d−b (x − a) = 0 there exists an L R-type c−a fuzzy number. In addition, this equation does not mean that there exists (x, y) ∈ R2 for which f (x, y), (y − b) − d−b (x − a) and g(x, y) vanish together. c−a The following question may arise: how can the proposed method be extended to obtain the fuzzy line segment determined by two fuzzy points when another fuzzy point with a core collinear to those of the other two points? More precisely, if same points of the three fuzzy points are not collinear, how should the points be used? The answer to this question is as follows. 1 and P 2 . Let P 3 be an additional L P1 P2 is the line segment joining P Suppose that    2 (1) and P 3 (1) fuzzy point to be added to L P1 P2 to extend it. It is given that P1 (1), P are collinear. According to our suggested method, one of the following can be done. 1 be situated on the L represents the required extended form of  L P1 P2 and let P Let     left-hand side of P2 in L P1 P2 . Now L can be obtained as follows.  1 and P 2 , then  3 lies in between P L = L P1 P3  L P3 P2 . (i) If P      (ii) If P3 lies on the left-hand side of P1 , then L = L P3 P1  LP P .  1 2 3 lies on the right-hand side of P 2 , then  (iii) If P L = L P1 P2  L P2 P3 . Details of this account is given in Chap. 3 when we construct a fuzzy line joining more than two fuzzy points.  Definition 2.11 (Containment of point on a fuzzy line segment L)   a fuzzy   d−b  be a fuzzy point and L ≡ f (x, y) (y − b) − Let P (x − a) g(x, y) =0 c−a LR  ∈  must be fuzzily ∀ (x, y) ∈ [ p, q] × [r, s]. If P(1) L(1), then the fuzzy point P  contained in L with some membership value, β say. This β may be obtained as: ⎧ ⎪ 1 ⎪ ⎪ ⎪ ⎨β 1 β= ⎪ β ⎪ 2 ⎪ ⎪  ⎩ min β1 , β2 where β1 =

sup

(x,y): f (x,y)=0

⊆   ⊂ if P L or P(0) L(0)   if P(0) exceeds L(0) on the side of f  exceeds  if P(0) L(0) on the side of g

 exceeds  if P(0) L(0) on the either sides of  L(1),

 and β2 = μ((x, y)| P)

sup

(x,y):g(x,y)=0

 μ((x, y)| P).

 ∈  cannot be fuzzily contained in  If P(1) / L(1), then P L and we define β = 0 in this situation.  is contained in   ∩ Note 2.7 If a fuzzy point P L P1 P2 , then for any point (x, y) ∈ P(0)

   ≥ β. L(0), μ (x, y) P

2.6 Fuzzy Line Segment

47

Fig. 2.11 Visualization of Example 2.12

1 (1, 1), P 2 (5, 5), P 3 (2, 2), Example 2.12 Consider the following five fuzzy points P 5 (4, 5) whose membership functions are all right circular cone with 4 (3, 3), and P P the following  mentioned bases. 1 (1, 1) = (x, y) : (x − 1)2 + (y − 1)2 ≤ 1 , P  2 (5, 5) = (x, y) : (x − 5)2 + (y − 5)2 ≤ 1 , P   2 3 (2, 2) = (x, y) : (x−2)2 + (y−2) ≤ 1 , P 4 4   4 (3, 3) = (x, y) : (x−3)2 + (y − 3)2 ≤ 1 , and P 4  5 (4, 5) = (x, y) : 4(x − 4)2 + 4(y − 5)2 ≤ 1 . P 3 on the fuzzy line segment We now find the containment of the fuzzy points P  L P1 P2 . We refer to Fig. 2.11 for this example. 3 (2, 2) ⊂  3 on  4 Since P L P1 P2 , the containment of P L P1 P2 is 1. Containment of P  on L P1 P2 is given by min{β1 , β2 } where     4 : y − x − 1 = 0 and β2 = sup μ (x, y)| P 4 : y − x + 1 = 0 . β1 = sup μ (x, y)| P

5 on  5 (1) = Thus β = min{0.5, 0.5} = 0.5 Containment of P L P1 P2 is 0, since P  (4, 5) ∈ / L P1 P2 .

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2 Basic Ideas on Fuzzy Plane Geometry

2.7 Fuzzy Distance Definition 2.12 (Fuzzy distance between two fuzzy points) 2 may be defined  between two continuous fuzzy points P 1 and P The fuzzy distance D by its membership function:  2 (0) are inverse points,  = sup α : where d = d(u, v), u ∈ P 1 (0) and v ∈ P μ(d| D) 2 ) = α . 1 ) = μ(v| P μ(u| P Here, d(, ) is the usual Euclidean distance metric. 1 and P 2 , Theorem 2.3 For two continuous fuzzy points P   1 (α), v ∈ P 2 (α) are inverse points (i) D(α) = d : d = d(u, v), where u ∈ P for all α in [0, 1].  is a fuzzy number in R. (ii) D  2 (α) are in1 (α) and v ∈ P Proof (i) Let A(α) = d : d = d(u, v), where u ∈ P  verse points . We prove that A(α) = D(α) for 0 < α ≤ 1. If this result is true  for 0 < α ≤ 1, then obviously D(0) = A(0), since support of a fuzzy number is the union of all of its α-cuts.   To prove that D(α) is a subset of A(α) for any α ∈ (0, 1], let d ∈ D(α) and  μ(d| D) = β, say. Then β ≥ α. If β > α, then there exists γ ∈ R with α < γ ≤ β such that d ∈ A(γ). As  A(γ) ⊆ A(α), so d ∈ A(α). Hence, in this case D(α) is a subset of A(α).  For the case when β = α, observe that μ(d| D) = sup {t : d = d(u, v), where 2 (0) are inverse points, μ(u| P 1 ) = μ(v| P 2 ) = t} = β = α. 1 (0) and v ∈ P u∈P 1 ) = Obviously, there exist sequences of inverse points {u n }, {vn } with μ(u n | P  μ(vn | P2 ) = δn and d = d(u n , vn ) such that {δn } is a nondecreasing sequence that converges to α. Therefore, for any > 0, there exists K ∈ N such that β − < δn for all n ≥ K . Here, d ∈ A(δn ) for any n and A(δn ) ⊆ A(β − ) for all n ≥ K . This implies that d ∈ A(β) because > 0 is arbitrarily taken. Therefore, in this  case D(α) is also a subset of A(α).  Thus, D(α) is a subset of A(α) for any α ∈ (0, 1].  Consider d ∈ A(α), where α ∈ (0, 1]. From the definition of A(α) and μ(d| D),   we obtain μ(d| D) ≥ α. Thus, d belongs to D(α) and therefore A(α) is a subset  of D(α).  Thus, D(α) = A(α)∀ α ∈ (0, 1], and hence for all α ∈ [0, 1]. 2 (α) are closed and bounded subsets of R2 , A(α) is a closed  (ii) Since P1 (α) and P  and bounded interval of R for all α ∈ [0, 1], and therefore so is D(α). Let    D(α) = [a(α), c(α)] and D(0) = [a, c]. Thus, μ(d| D) = 0 for all d not in [a, c].  It is obvious from the definition of D(α) that for 0 ≤ α ≤ β ≤ 1, [a(β), c(β)] =   D(β) ⊆ D(α) = [a(α), c(α)]. Therefore, as α increases, a(α) increases and c(α) decreases.

2.7 Fuzzy Distance

49

 ≥ t} is closed and bounded. Therefore, Now, for all t ∈ R, the set {d : μ(d| D)  is upper semi-continuous. the membership function of D 2 (1) = ( p, q). Now, D(1)  1 (1) = (a, b) and P = A(1) = d((a, b), Let P ( p, q)) = a(1) = c(1).  is a fuzzy number. Hence, D 1 and P 2 be two fuzzy points at (1, 0) and (2, 0), respectively. Example 2.13 Let P 1 (0) = {(x, y) : (x − 1)2 + 1 is a right circular cone with base P The shape of P 1 2 y ≤ 4 } and vertex (1, 0). 2 is a right circular cone with base P 2 (0) = {(x, y) : (x − 2)2 + The shape of P 1 2 y ≤ 4 } and vertex (2, 0). 1 and P 2 with memFor each α ∈ [0, 1], the inverse points with respect to P 1 1 bership value α are P : (1 + 2 (1 − α) cos θ, 2 (1 − α) sin θ) and Q : (2 − 21 (1 − α) cos θ, − 21 (1 − α) sin θ), respectively (θ ∈ [0,√ 2π]). {1 + (1 − α)2 − 2(1 − α) cos θ}. The distance between P and Q is d(P, Q) = √ 2 Now, inf d(P, Q) = {1 + (1 − α) − 2(1 − α)} = α, and θ∈[0,2π] √ sup d(P, Q) = {1 + (1 − α)2 + 2(1 − α)} = 2 − α. θ∈[0,2π]

1 and P 2 is D,  say, then for any d ∈ [α, 2 − α], Clearly, if the distance between P   μ(d| D) ≥ α. Therefore, D(α) = [α, 2 − α].  is the fuzzy number defined by the following membership function: Thus, D ⎧ ⎪ if 0 ≤ d ≤ 1 ⎨d  μ(d| D) = 2 − d if 1 ≤ d ≤ 2 ⎪ ⎩ 0 elsewhere.

Definition 2.13 (Coincidence of two fuzzy points) 1 and P 2 may be defined The degree of fuzzy coincidence (κ) of two fuzzy points P as: ⎧ 1 (1) = P 2 (1) 0 if P ⎪ ⎪ ⎨   if P1 = P2 κ= 1









⎪

⎪ 1 (1) = P   2 (1) but P 1 = P 2 . ⎩1 − sup μ (x, y) P1 − μ (x, y) P2 if P (x,y)∈R2

Note 2.8 If two fuzzy points coincide fuzzily (i.e., κ > 0), then their fuzzy distance  is a fuzzy number with μ(0| D)  = 1, i.e., D  is a fuzzy number  D 0. 1 and P 2 be two fuzzy Example 2.14 (Coincidence of two fuzzy points) Let P 1 (0) = points whose membership functions are right circular cones with bases P 2 2 2 2  {(x, y) : x + y ≤ 1} and P2 (0) = {(x, y) : (x − 1) + y ≤ 1} and vertices (0, 0) 2 is ‘zero’ because 1 and P and (1, 0), respectively. The degree of coincidence of P   P1 (1) = (0, 0) = (1, 0) = P2 (1).

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2 Basic Ideas on Fuzzy Plane Geometry

1 and P 2 be two fuzzy points Example 2.15 (Coincidence of two fuzzy points) Let P 1 (0) = {(x, y) : whose membership functions are right circular cones with bases P 2 (0) = {(x, y) : x 2 + y 2 ≤ 2}, both of which have vertex (0, 0). x 2 + y 2 ≤ 1} and P 1 and P 2 is 1 − sup |μ((x, y)| P 1 ) − In this example, the degree of coincidence of P (x,y)∈R2

2 )| = √1 . μ((x, y)| P 2 2 are said to be collinear if P 1 (1) = 1 and P Note 2.9 The two fuzzy points P    P2 (1) but P1 = P2 . Definition 2.7.1 (Angle between two fuzzy line segments) 1 , P 2 and P 3 be three continuous fuzzy points and  Let P L P1 P2 ,  L P2 P3 be fuzzy line 1 , P 2 and P 2 , P 3 respectively. The angle between  segments joining P L P1 P2 and  L P2 P3  and is defined by: is denoted by Θ  = sup{α : θ is the angle between the line segments L uv and L vw , μ(θ|Θ) where u, v and v, w are same points with membership value α; 1 (0), v ∈ P 2 (0), w ∈ P 3 (0)}. u∈P 1 , P 2 and P 3 , Theorem 2.7.1 For three continuous fuzzy points P  (1) Θ(α) = {θ : θ is the angle between line segments L uv and L vw , where 2 (α) and w ∈ P 3 (α), and u, v and v, w are same points} 1 (α), v ∈ P u∈P ∀α ∈ [0, 1].  is a fuzzy number in R. (2) Θ

Proof The proof is similar to that of Theorem 2.3 and is omitted. Example 2.16 (Fuzzy angle)  6), P(1,  2) and P(14,  Let P(7, 15) be three fuzzy points.  6) is the right elliptical cone with The shape of the membership function of P(7, (y−6)2 (x−7)2 base {(x, y) : 4 + 9 ≤ 1} and vertex (7, 6).  2) is the right circular cone with base {(x, y) : The membership function of P(1, (x − 1)2 + (y − 2)2 ≤ 1} and vertex (1, 2).  The shape of the membership function of P(14, 15) is the right elliptical cone (y−15)2 2 with base {(x, y) : (x − 14) + 4 ≤ 1} and vertex (14, 15). Here the angle between the fuzzy line segments   joining the first two and last  with support π − tan−1 131 , π − tan−1 47 ; two fuzzy points is a fuzzy number Θ 4 134 4 126



    = 0 = μ π − tan−1 131 Θ  and the core of Θ  is π − tan−1 2 . μ π − tan−1 131 Θ 4

134

4

134

4

3

2.8 Conclusion

51

2.8 Conclusion In this chapter the concept of fuzzy point has been discussed with respect to defined reference frame. In order to establish geometrical relationship between fuzzy points the existence of same and inverse point play the important role. The mathematical theory of same and inverse point has been elaborated along with the graphical illustrations. The distance between two fuzzy points which are connected with a fuzzy line segment is also measured. Fuzzy line segment contains different fuzzy points fully or partially, hence a measure of belongingness has been computed here. The angle between two fuzzy line segments is also proposed here.

Reference Buckley, J.J., Eslami, E.: Fuzzy plane geometry I: points and lines. Fuzzy Sets Syst. 86, 179–187 (1997)

Chapter 3

Fuzzy Line

3.1 Introduction In conventional Euclidean geometry, a straight line is obtained by extending a line segment bi-infinitely (second postulate of Euclid). Thus, a line is the locus of a point along a fixed direction. The slope of the line determines its direction and it is the inclination angle of the tangent to the positive x-axis. Any linear equation of two variables, ax + by = c with (a, b, c) = (0, 0, 0), always represents a (straight) line. However, in a fuzzy environment, often there may not exist any pair ( x,  y) that satisfies the fuzzy linear equation  a x + b y = c (Buckley and Qu 1991). For example, if  a = (1/2/3),  b = (0/0/0) and  c = (2/4/5), then there does not exist y = (y − α y /y/y + β y ) with αx , α y , βx , β y ≥ 0 any  x = (x − αx /x/x + βx ) and  such that  a x + b y = c, since fitting all the quantities in  a x + b y = c, we obtain βx as negative. Thus, compared to classical geometry, a fuzzy line may not be expressed by a fuzzy linear equation. Although there may not exist any fuzzy linear equation to express a fuzzy line, this chapter shows that a fuzzy line may be expressed mathematically if at least two of its attributes—two points, point-slope, slope-intercept or two intercepts—are known (Chakraborty and Ghosh 2014). In the following subsections, we investigate mathematical forms for a fuzzy line when information about two of its attributes is available. This chapter mainly addresses the questions: What is a fuzzy line? How to construct a fuzzy line? And what is mathematical form of fuzzy line?

3.2 Construction of Fuzzy Line When two (or more) fuzzy points are given for a fuzzy line, analogous to classical plane geometry, a fuzzy line may be obtained by extending the fuzzy line segment joining the two (or more) fuzzy points bi-infinitely. Now we define a fuzzy line passing through several given fuzzy points. © Springer Nature Switzerland AG 2019 D. Ghosh and D. Chakraborty, An Introduction to Analytical Fuzzy Plane Geometry, Studies in Fuzziness and Soft Computing 381, https://doi.org/10.1007/978-3-030-15722-7_3

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3.2.1 Fuzzy Line Passing Through n(≥2) Collinear Fuzzy Points 1 (a1 , b1 ), P 2 (a2 , b2 ), …, P n (an , bn ) are n fuzzy points whose cores Suppose that P are collinear and they lie on the line l, say. Let the points (ai , bi ) be arranged from left to right for i = 1, 2, . . . , n. To construct a fuzzy line  L passing through all of L say, joining these n fuzzy points, we first need to construct a fuzzy line segment,      P1 , P2 , …, Pn . Then we can extend L to infinity on both sides to obtain the fuzzy L can be obtained by joining the fuzzy line line  L. Note that the fuzzy line segment     segments L P1 P2 , L P2 P3 , . . . , L Pn−1 Pn successively. Thus,  L = L P1 P2



 L P2 P3



···



 L Pn−1 Pn .

Here it is obvious that the core of  L must be the line l on which all the core points L bi-infinitely, we consider two hypo(a1 , b1 ), (a2 , b2 ), …, (an , bn ) lie. To extend  thetical fuzzy points with a core on l: one of them is an infinite distance away on 1 (a1 , b1 ) and the other is also an infinite distance away on the right the left of P 1∞ and P n∞ , respectively. Let   L 1∞ and of Pn (an , bn ). Let these two points be P 1∞ and P 1 , and P n and P n∞ ,  L n∞ be the semi-infinite fuzzy line segments joining P respectively. Then the fuzzy line  L can be defined as  L 1∞ L =



 L P1 P2



···



 L Pn−1 Pn



 L n∞ .

1∞ , P n∞ and what does the phrase ‘infinite How can we find the fuzzy points P distance away’ actually mean? Answers to these questions are presented in Propo1∞ sition 3.2.1, which states that for construction of  L, the shape and position of P    and Pn∞ may not be of due importance. The fuzzy points P1∞ and Pn∞ are two hypothetical fuzzy points. To construct  L, only the following information is needed: n∞ must be compact sets and their core must lie on l. 1∞ and P the supports of P L 1∞ and  L n∞ , we need to take the union of all the line segments joining To obtain  1∞ , and P n and P n∞ , respectively. However, according to 1 and P same points of P Proposition 3.2.1, these line segments are always parallel to the line l. Thus, the L 1∞ and  L n∞ on  L are two bunches of half-lines semi-infinite fuzzy line segments  with varied membership values and the half-lines must be parallel to the line l. Proposition 3.2.1 To consider a fuzzy line  L as a collection of fuzzy points, the slope   of all the half-line segments in L 1∞ (0) and L n∞ (0) must have the same slope as that of l ≡  L(1).

3.2 Construction of Fuzzy Line

55

Proof On contrary, let there exists a half-line, l  say, in  L 1∞ (0) whose slope is not equal to the slope of l. L 1∞ (0). According to the formulation Here l  is a (semi-infinite) line segment in   of L 1∞ (0), it is union of the line segments joining same points of the fuzzy points 1∞ (0). Therefore, there must exist two same points (x1 , y1 ) ∈ P 1 (0) and 1 (0) and P P 1∞ (0) that are two extremities of l  . (x1∞ , y1∞ ) ∈ P L 1∞ (1) or the distance between the Since l  is not parallel to l, either l  intersects  L 1∞ (1) must be infinitely large. The former case point (x1∞ , y1∞ ) and the half-line  1 (0) and (x1∞ , y1∞ ) ∈ P 1∞ (0) are not same points, because implies that (x1 , y1 ) ∈ P in this case they lie on two different sides of  L 1∞ (1). The latter case implies that the 1∞ cannot be a fuzzy point. Thus, both 1∞ is unbounded and hence P support of P cases are impossible. A contradiction arises. L 1∞ (1) or, equivalently, l  is parallel to l. Therefore, Hence, l  must be parallel to  L 1∞ (0) must be the same as the slope of the slope of all the half-line segments in  l. In a similar manner, we can show that the slope of all the half-line segments in  L n∞ (0) must be equal to the slope of l. Hence, the result follows. To obtain  L, we first consider the line segment  L joining same points (Definition    2.10) of P1 , P2 , . . . , Pn and then the semi-infinite fuzzy line segments  L 1∞ and  L n∞ are adjoined on either side of  L. Here  L 1∞ and  L n∞ can be evaluated using the following formulations:  L 1∞ =

 1 (0) and l1∞ (x, y) is a semi-infinite line l1∞ (x, y) : where (x, y) ∈ P 1 segment parallel to l with membership value (x, y) on P



and  L n∞ =

 n (0) and ln∞ (x, y) is a semi-infinite line ln∞ (x, y) : where(x, y) ∈ P  n . segment parallel to l with membership value (x, y) on P

Figure 3.1 shows the method to construct  L through five fuzzy points, where the 1 , P 2 , P 3 , ellipse, rhombus, ellipse, circle and triangle are the supports of these points P 5 , respectively. The variation of the membership grades for the fuzzy points 4 and P P is indicated by the shading intensity. Darker shading indicates a higher membership value. The region between the upper and lower extended lines/curves is the support 2 , P 3 , 1 , P of the fuzzy line. The two dashed lines join consecutive same points of P 5 , and the fuzzy line  4 and P L is the union of these dashed lines. In the following, P a particular case of this construction, called the two-point form of a fuzzy line, is proposed.

56

3 Fuzzy Line

Fig. 3.1 Fuzzy line passing through five fuzzy points whose cores are collinear

1 and P 2 be two fuzzy points. Definition 3.2.1 (Two-point form ( L 2P )). Let P   1 and P 2 is the union  L 1∞  L P1 P2  L 2∞ , The fuzzy line  L 2P passing through P   where L 1∞ and L 2∞ are two semi-infinite fuzzy line segments as defined above. Thus,  L 2P can be defined by its membership function as: ⎧ 1 ) ⎪ μ((u, v)| P sup  ⎪ ⎪ ⎪ 1 (0) (u,v)∈l(x,y) P ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ 1 1 ⎪sup α : yy−y = xx−x , where (x1 , y1 ) ⎨ 2 −y1 2 −x 1

⎪    2 (0) are same μ (x, y) L 2P = in P1 (0), (x2 , y2 ) in P  ⎪ ⎪ ⎪ ⎪ points with membership value α ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ) μ((u, v)| P sup  ⎪ ⎩ (u,v)∈l(x,y)

2 (0) P

if (x, y) ∈  L 1∞ (0)

if (x, y) ∈  L P1 P2 (0) if (x, y) ∈  L 2∞ (0),

where l(x, y) is the line passing through (x, y) and parallel to the core line of  L 2P .

3.2 Construction of Fuzzy Line

57

In this definition, a fuzzy line is considered as a collection points with  of crisp L 2P explicitly varied membership values. However, the expression for μ (x, y)  shows that a fuzzy line is the union of all line segments whose extremities are same 1 and P 2 , inside the points. The membership value of a line segment l between P region  L 2P (0), can be defined as   L 2P = min μ((x, y)| L 2P ). μ l| (x,y)∈l

The following theorem shows how to obtain the membership value of the line L 2P using same points. segment l in  Theorem 3.2.1 Suppose that l is a line in  L 2P and there exist two same points 1 and (x2 , y2 ) ∈ P 2 with μ((x1 , y1 )| L 2P ) = μ((x2 , y2 )| L 2P ) = α (x1 , y1 ) ∈ P y−y1 y2 −y1 = . Then, such that l ≡ x−x x2 −x1 1 μ(l| L 2P ) = α.

Proof We argue that (i) μ(l| L 2P ) < α and (ii) μ(l| L 2P ) > α. L 2P ) < α. Thus, there exists (x3 , y3 ) ∈  L 2P (0) such that (i) On contrary, let μ(l| L 2P ) < α. Suppose that μ((x3 , y3 )| L 2P ) = β. As (x3 , y3 ) (x3 , y3 ) ∈ l and μ((x3 , y3 )| ∈ l and l is the line segment joining two same points with membership value L 2P ) = sup{γ : (x, y) lies on the line joining same points with α, μ((x3 , y3 )| membership value γ} ≥ α. However, β < α, so a contradiction arises. Therefore, L 2P ) < α. μ(l| L 2P ) = min{α : (x, y) lies on l and (ii) This part is obvious since μ(l| μ((x, y)| L 2P ) = α} and both (x1 , y1 ) and (x2 , y2 ) lie on the line l. L 2P ) = α. Hence, μ(l| Note that on  L 2P , if a line perpendicular to  L 2P (1) is considered, then along that line there exists a fuzzy number on  L 2P (0). For example, if the line C D is considered in Fig. 3.1, then along C D, a fuzzy number of L R type, (F/G/H ) L R , exists. Thus, the whole fuzzy line can be visualized as a three-dimensional figure (basically a L 2P (1) is a fuzzy number such as subset of R2 × [0, 1]) whose cross-section across  (F/G/H ) L R . 1 and P 2 , the surfaces of z = μ((x, y)| P 1 ) and Although  L 2P is constructed by P 2 ) always lie in the interior of z = μ((x, y)| L 2P ) and a portion of z = μ((x, y)| P 2 ) touches the surface of z = 1 ) and z = μ((x, y)| P both the surfaces z = μ((x, y)| P 1 (0) (or P 2 (0)) whose membership μ((x, y)| L 2P ). Hence, there exist some points in P 2 ) is less than the membership grade at  1 (or P L 2P . For instance, in Fig. 3.1, grade at P the points on the periphery of the left most elliptic region have a membership value

58

3 Fuzzy Line

1 , whereas all those points except where P 1 (0) touches of ‘zero’ for the fuzzy point P  L(0) have positive a membership value for  L 2P . 1 (0, 0) and P 2 (1, 1) be two Example 3.1 (Two-point form of a fuzzy line). Let P fuzzy points whose membership functions are right circular cones. The bases of 2 (1, 1) are {(x, y) : x 2 + y 2 ≤ ( 1 )2 } and {(x, y) : (x − 1)2 + (y − 1 (0, 0) and P P 3 1 and 1)2 ≤ ( 13 )2 }, respectively. The membership function of  L 2P passing through P    P2 can be generated by moving the membership function of P1 or P2 along the straight line y = x. Here,  L 2P (1) is the straight line y = x and √ √   2 2  ≥ 0, x − y − ≤0 . L 2P (0) = (x, y) : x − y + 3 3 Note 3.1 This is the Example 6 in Buckley and Eslami (1997), according to which 1 (0) and P 2 (0), but widens as the graph of z = μ((x, y)| L 2P ) is ‘thinï£ between P we move along y = x for x > 1 or for x < 0. More precisely, the spread of  L 2P (0) across the line y = x is 2/3 when 0 < x < 1, but suddenly increases when x < 0 or x > 1. By contrast, according to Definition 3.2.1, the spread of  L 2P (0) across the line y = x is uniform and is 2/3 for any x. 1 (1, 2) be a fuzzy point whose Example 3.2 (Two-point form of a fuzzy line). Let P  2 membership function is a right circular cone with base (x, y) : (x−1) + (y − 2)2 ≤ 22  2 (5, 7) be another fuzzy point whose membership function 1 and vertex (1, 2). Let P   is a right circular cone with base (x, y) : (x − 5)2 + (y − 7)2 ≤ 22 and vertex (5, 7). For a particular α ∈ [0, 1], same points with membership  value α ∈ [0, 1] 2(1−α) cos θ 2(1−α) sin θ) 1 (0) and Q : 2 (5, 7) are P : 1 + √ 1 (1, 2) and P √ ∈P , 2 + on P 2 2 1+3 sin θ

1+3 sin θ

2 (0), where θ ∈ [0, 2π]. The fuzzy line (5 + 2(1 − α) cos θ, 7 + 2(1 − α) sin θ) ∈ P    L 2P passing through P1 and P2 is the union of the two semi-infinite fuzzy line L 1∞ and  L 2∞ , with the collection of the line segments joining P and Q segments    L 1∞  L P1 P2  L 2∞ . Here, for different values of α and θ, that is,  L 2P =   L P1 P2 (0) =



 

α∈[0,1] θ∈[0,2π]

(x, y) :

y − (7 + 2(1 − α) sin θ) x − (5 + 2(1 − α) cos θ) =

5 + 2(1 − α) sin θ[1 − √ 4 + 2(1 − α) cos θ[1 −

1 ] 1+3 sin2 θ , √ 1 ] 1+3 sin2 θ

 p≤x ≤q ,

where p (= −1) and q (= 7) are the minimum and maximum of the sets {x : 2 (0)}, respectively, and  1 (0)} and {x : (x, y) ∈ P L 2P (1) = {(x, y) : 5x − (x, y) ∈ P 4y + 3 = 0}.

3.2 Construction of Fuzzy Line

3.2.1.1

59

2 P ) Construction of the Membership Function µ(.| L

The mathematical form of μ(.| L 2P ) is not always simple to evaluate because the membership value at a particular point is the supremum of a set of real numbers obtained by solving nonlinear  equations. We note that Definition 3.2.1 and Theorem  3.2.1 imply μ (x, y)  L 2P =   x − x1 y − y1 1 (0), (x2 , y2 ) ∈ P 2 (0) are same points . = , (x1 , y1 ) ∈ P sup α : y2 − y1 x2 − x1   Thus, to obtain μ (x, y)  L 2P , we must first take two generic same points with membership value α ∈ [0, 1]. Then all possible α values for which (x, y) lies on the line joining two same points with membership value α are identified. We may have to solve nonlinear equations  The supremum of all of these α values is  to get an α. the membership value of μ (x, y)  L 2P . The following example illustrates the procedure to find the membership value of a point on a fuzzy line. Example 3.3 (Finding the membership value in a two-point form). Consider the fuzzy line in Example 3.2. Suppose the membership value at (2, 4) has to be identified. Here the x-coordinate of the point (2, 4) lie between −1 and 7. Hence the membership 1 (0) and L P1 P2 . Let P ∈ P grade of (2, 4) on  L 2P must be exactly equal to that on  2 (0) be same points with membership grade α ∈ [0, 1]. The equation for the Q∈P line segment P Q is 2 + 2k(1 − α) sin θ − (7 + 2(1 − α) sin θ) 1 y − (7 + 2(1 − α) sin θ) = , where k =  . x − (5 + 2(1 − α) cos θ) 1 + 2k(1 − α) cos θ − (5 + 2(1 − α) cos θ) 1 + 3 sin2 θ

If (2, 4) lies on the line P Q, then 4 − (7 + 2(1 − α) sin θ) 2 + 2k(1 − α) sin θ − (7 + 2(1 − α) sin θ) 1 = , where k =  2 − (5 + 2(1 − α) cos θ) 1 + 2k(1 − α) cos θ − (5 + 2(1 − α) cos θ) 1 + 3 sin2 θ

⇒α=1−

3 (8+6k) sin θ−(10+6k) cos θ

= f (θ), say.

Here f (θ) must lie in [0, 1] and hence the admissible domain of f (θ) is D f = [63◦ , 222.66◦ ]. The maximum value of f (θ) over D f occurs at 157.32◦ and L P1 P2 . Hence, the value is 0.8352 for the possibility of containment of (2,4) on  μ((2, 4)| L P1 P2 ) = μ((2, 4)| L P1 P2 ) = 0.8352. Note 3.2 In finding the membership grade at a point (x, y), say, there are several lines on  L 2P (0) containing (x, y). Among these lines, there exists one line with the highest membership grade that offers supremum1 of the set {α : (x, y) lies on the 1

This supremum exists in the set considered, since  L 2P (α) is a compact set. (Observation 3.2.2).

60

3 Fuzzy Line

1 and P 2 with membership value α}. We call this line line joining same points on P (segment) as the adjoining line of the point (x, y). In Example 3.3, the adjoining line of (2, 4) is y = 1.2808x + 1.4385. Note that a fuzzy line can be perceived as a collection of line segments and halflines with varied membership values. Here the slope of all of these half-lines depends on the slope of the core line. Thus, the slope of the fuzzy line may be considered as the union of the slope of its crisp line segments joining same points. Thus, the membership value of a particular slope, m say, may be taken as the supremum of membership values of all the line segments with slope m. The membership of any L 2P is given by line segment l in    L 2P = min μ((x, y)| L 2P ). μ l| (x,y)∈l

Thus, the slope of a fuzzy line can be mathematically defined as follows. Definition 3.2.2 (Slope of a fuzzy line in two-point form ( m )). Let  L 2P be a   L 2P (0). The slope fuzzy line and l be a line segment (in between P1 and P2 ) in  of the fuzzy line is defined by its membership function as: L 2P ) : slope of l is m}. μ(m| m ) = sup{μ(l|

Example 3.4 (Slope of a fuzzy line). Consider the fuzzy line in Example 3.2. For a 2(1−α) cos θ (1−α) sin θ 1 (0) particular α ∈ [0, 1], same points are P : (1 + √ ,2 + √ )∈ P 2 2 1+3 sin θ

1+3 sin θ

2 (0) with membership grade and Q : (5 + 2(1 − α) cos θ, 7 + 2(1 − α) sin θ) ∈ P 5+2(1−α) sin θ[1− √

α ∈ [0, 1], where θ ∈ [0, 2π]. The slope of the line P Q is

the supremum and infimum of this quantity over [0, 2π] are 5+(1−α) and 4 respectively. By definition, if the slope of this line is denoted by m , then

1

]

1+3 sin2 θ 1 ] 1+3 sin2 θ

4+2(1−α) cos θ[1− √

;

5 , 4+2(1−α)



  5 + (1 − α)

5 m =μ m = α. μ   4 4 + 2(1 − α) Hence, the slope of the fuzzy line is given by: ⎧ 5 5 ⎪ ⎨3 − 2m if 6 ≤ m ≤ μ(m| m ) = 6 − 4m if 54 ≤ m ≤ ⎪ ⎩ 0 elsewhere.

5 4 3 2

Theorem 3.2.2 and the next subsections show that the slope of a fuzzy line passing through two fuzzy points is a fuzzy number with a singleton core. Thus, it follows

3.2 Construction of Fuzzy Line

61

that the slope of a fuzzy line is a fuzzy number whose core contains only one element. The core of the slope of a fuzzy line cannot have more than one number because otherwise it lacks inner conformity with conventional geometry. Therefore, the core of a fuzzy line must be a straight line and it cannot be a band of lines. Otherwise, this may be a fuzzy region and cannot be a fuzzy line. The following observations can be made in this regard.  1 where (x1 , y1 ) ∈ Observation 3.2.1 For any α ∈ (0, 1), m (α) = m : m = xy22 −y −x1  2 (α) are same points . 1 (α) and (x2 , y2 ) ∈ P P  L P1 P2 (α) = l : l is the line segment joining Observation 3.2.2 For any α ∈ (0, 1),   1 (α) and P 2 (α) . two same points in P Observation 3.2.3 For any α ∈ (0, 1), there exist two continuous curves f (x, y; α) = 0 and g(x, y; α) = 0 such that  L 2P (α) is bounded by them. As α increases, the region  L 2P (α) gradually decreases and for α = 1 both f = 0 and g = 0 coincide with the straight line  L 2P (1). Observation 3.2.4 The slope of the fuzzy line segment  L P1 P2 and the slope of the fuzzy line  L 2P are identical. 1 and P 2 , is unique. Observation 3.2.5  L 2P , passing through two particular points P

Theorem 3.2.2 The slope of the fuzzy line  L 2P passing through two fuzzy 2 is a fuzzy number. 1 and P points P

Proof We prove that the slope of the fuzzy line segment  L P1 P2 is a fuzzy number. Then theorem will be followed from Observation 3.2.4. 1 be fuzzy point at (a, b) and P 2 be a fuzzy L P1 P2 . Let P Let m  be the slope of  . Obviously, μ(m| m ) = 1. points at (c, d). Let m = d−b c−a Let α ∈ [0, 1]; θ1 (α) and θ2 (α) be the infimum and supremum, respectively, of the L P1 P2 . As angle of elevation with respect to the positive x-axis for the line segments in  2 (α) are convex compact sets, the angles of elevation of the straight lines 1 (α) and P P ˜ = [θ1 (α), θ2 (α)]. Therefore, in  L 2P (0) must constitute a fuzzy set θ˜ (say), with θ(α) ˜ μ(.|θ) is upper semi-continuous. L P1 P2 (β) ⊆  L P1 P2 (α). We note that,  θ(β) From Observation 3.2.3, for any β > α,  ⊆ θ(α), that is, [θ1 (β), θ2 (β)] ⊆ [θ1 (α), θ2 (α)] for 0 < α < β ≤ 1. This implies that θ1 (α) ≤ θ1 (β) ≤ θ2 (β) ≤ θ2 (α). Therefore, θ1 is an increasing function, θ2 is θ is a fuzzy a decreasing function, and θ1 (1) = θ2 (1) = tan−1 m at α = 1. Thus,    [θ1 (α), θ2 (α)] . number with support θ(0) = closur e 0 α and (ii) μ(l| L P S ) < α.

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3 Fuzzy Line

(i) As (x1 , y1 ) ∈ l, then μ(l| L P S ) = min μ((x, y)| L P S ) ≤ α. So, μ(l| L P S ) > α. (x,y)∈l

(ii) In this case, on contrary, we let μ(l| L P S ) < α. Then there exists (x2 , y2 ) in  L PS) : L P S ∩ l whose membership value offers a minimum of the set {μ((x, y)| (x, y) ∈ l}. L P S ) < α. Suppose that μ((x2 , y2 )| L PS) As μ(l| L P S ) < α, therefore μ((x2 , y2 )| = β. Thus, β < α. Here we note that (x2 , y2 ) lies on the line y − y1 = m(x − x1 ) and μ((x1 , y1 )|  m ). As μ((x1 , y1 )| L P S ) = α, there must exist (x3 , y3 ) ∈ l ∩ L P S ) = α = μ(m|  = α. Therefore, by Definition 3.2.4, μ((x2 , y2 )|  L PS) = P(0) with μ((x3 , y3 )| P)  m ∈ m (0) with β = sup γ : (y − y2 ) = m  (x − x2 ), where (x, y) ∈ P(0),   = γ = μ(m  | μ((x, y)| P) m ) ≥ α, since (x3 , y3 ) lies on the line y − y2 = m(x − x2 ). A contradiction arises. Hence, μ(l| L P S ) < α. Therefore, μ(l| L P S ) = α. Note 3.4 In Theorem 3.2.4, the membership grade of (x1 , y1 ) is taken with respect  to  L P S (not with respect to P). Note that on  L P S (0) if a line perpendicular to  L P S (1) is considered, then along that line there exists a fuzzy number on  L P S (0). For instance, in Fig. 3.3, along C D L P S can be a fuzzy number of L R-type (H/G/F) L R exists. The entire fuzzy line  visualized as a three-dimensional shape whose cross section across  L P S (1) is a fuzzy number such as (H/G/F) L R .  and the fuzzy numAlthough  L P S is constructed with the fuzzy point P  ber m , the surface of z = μ((x, y)| P(A)) must always remain in the interior of  will touch z = μ((x, y)| L P S ) and a portion of the surface of z = μ((x, y)| P(A))   the surface of z = μ((x, y)| L P S ). Apparently, there exist some points in P(A)(0)   whose membership grade in P(A) is less than that in L P S . For example, in Fig. 3.3 the points on the periphery (except K and L) of the circular region centered at A  have a membership value of ‘zero’ in P(A), whereas those points have positive  membership values in L P S .  4) be a fuzzy point with Example 3.6 (Point–slope form of a fuzzy line). Let P(5, a membership function that is a right circular cone with base {(x, y) : (x − 5)2 + (y − 4)2 ≤ 1} and slope m  = (tan 60◦ / tan 70◦ / tan 80◦ ), that is, m  = (1.732/  4) and with slope m  has 2.747/5.731). The fuzzy line  L P S passing through P(5,  4) is the shape shown in Fig. 3.3. A point with membership value α on P(5, (5 + (1 − α) cos θ, 4 + (1 − α) sin θ), where θ ∈ [0, 2π]. The numbers with membership grade α on m  are 1.732 + 1.015α and 5.731 − 2.984α. Here  L P S (0), by Definition 3.2.4, contains all the lines through (5 + (1 − α) cos θ, 4 + (1 − α) sin θ) with slope 1.732 + 1.015α or 5.731 − 2.984α for all possible values of α in [0, 1]. That is,

3.2 Construction of Fuzzy Line

 L P S (0) =





67



(x, y) :

α∈[0,1] θ∈[0,2π]

y − (4 + (1 − α) sin θ) = (5.731 − 2.984α)(x − (5 + (1 − α) cos θ))  or y − (4 + (1 − α) sin θ) = (1.732 + 1.015α)(x − (5 + (1 − α) cos θ))   = (x, y) : y − 1.732x + 6.660 ≥ 0, y − 5.731x + 18.837 ≤ 0   (x, y) : y − 1.732x + 2.660 ≥ 0, y − 5.731x + 30.473 ≥ 0

and  L P S (1) is the line y = 2.747x − 9.735. In Fig. 3.3, A = (5, 4), RS : y = 1.732x − 2.660, W Z : y = 1.732x − 6.660, U V : y = 5.731x − 30.473, M N : y = 5.731x − 18.837, and P Q : y = 2.747x − 9.735.

3.2.2.1

 P S) Construction of the Membership Function µ(.| L

To find the membership value at a point, (x, y) say, on  L P S we have to consider all the lines on  L P S (0) passing through the point (x, y). Among these, the line with the highest membership grade always offers the value for μ((x, y)| L P S ), since Definition 3.2.4 and Theorem 3.2.4 imply that  L P S ) : l ≡ y − y1 = m(x − x1 ), μ((x, y)| L P S ) = sup μ(l|  L P S ) = μ(m| m) . where μ((x1 , y1 )| We call this line that offers the supremum as the adjoining line of the point (x, y). The following example illustrates the evaluation of the membership value of a point on  L PS. Example 3.7 (Finding membership values in a point–slope form). Consider the fuzzy  5) with membership grade α line  L P S in Example 3.6. Let α ∈ [0, 1]. A point on P(4, is P : (5 + (1 − α) cos θ, 4 + (1 − α) sin θ), θ ∈ [0, 2π]. The values of m on m  with membership value α are 1.732 + 1.015α and 5.731 − 2.984α. Suppose we want to evaluate the membership grade at (6, 7). The equation for the line passing through P and with slope m is y − (4 + (1 − α) sin θ) = m(x − (5 + (1 − α) cos θ)). If this line passes through (6, 7), then (i) 7 − (4 + (1 − α) sin θ) = (1.732 + 1.015α)(6 − (5 + (1 − α) cos θ)) and (ii) 7 − (4 + (1 − α) sin θ) = (5.731 − 2.984α)(6 − (5 + (1 − α) cos θ)), where θ ∈ [0, 2π].

68

3 Fuzzy Line

Now (i) implies that  α = 0.4926 sec θ − 1.0150 − 0.717 cos θ + sin θ − 2.747 

0.1365 + 0.8751 cos θ + cos2 θ − 0.2690 sin θ cos θ − 0.7281 sin θ + 0.1325 sin2 θ



= f 1 (θ), say.

(ii) implies that   α = 1.4603 + sec θ − 0.5 − 0.1676 tan θ + 0.4603 1.18 + cos2 θ + cos θ(−2.5727−

 0.7281 sin θ) + 0.7909 sin θ + 0.1325 sin2 θ = f 2 (θ), say.

The membership value of (6, 7) on  L P S is μ((6, 7)| L P S ) = sup { f 1 (θ), f 2 (θ)} = 0.9580. θ∈[0,2π]

 The adjoining line for (6, 7) is y = 2.8723x − 10.2339. Here, D f1 : [99.3462◦ , 207.6603◦ ] and the maximum value of f 1 is 0.8573, attained at θ = 159◦ , D f2 : [39.9728◦, 291.4151◦ ] \ {90◦ , 270◦ } and the maximum of f 2 is 0.9580, achieved at θ = 161◦ . Observation 3.2.6 For any α ∈ (0, 1),  L P S (α) = {l : l is a line with slope m ∈  m (α) and passing through (x, y) ∈ P(α)}. L PS Now we define the y-intercept of  L P S using the intersecting region between    and the y-axis, that is, L P S (y-axis). The x-intercept can be defined similarly. Definition 3.2.5 (y-intercept of a fuzzy line in the point–slope form ( c)). Let  L P S . The y-intercept of  L P S can be L P S be a fuzzy line and l be a line in  defined by its membership function as μ(c| c) = sup{μ(l| L P S ) : y-intercept of l is c}.  We note that  the y-intercept of a fuzzy line L P S may not be a fuzzy number,  since L P S (y-axis) may not be a convex set always.

Note 3.5 For the fuzzy line constructed in point–slope form above, we observe that if the slope m  has a positive spread, then as we move along the core line  L P S (1)

3.2 Construction of Fuzzy Line

69

the spread of the fuzzy line widens with increasing distance from the location of  This fuzzy line with a widening spread cannot be perceived as a the fuzzy point P. collection of fuzzy points because fuzzy points must have a bounded support. Thus, we have the following result.

Theorem 3.2.5 To perceive a fuzzy line in point–slope form as a collection of fuzzy points, the slope m  must be a real constant.

Proof The proof is similar to that of Proposition 3.2.1. The next subsection investigates the slope–intercept form of fuzzy lines and a similar result to Theorem 3.2.5 is obtained.

 SI ) 3.2.3 Slope–Intercept Form ( L Let m  and  c be two given fuzzy numbers for which a fuzzy line  L S I has to be constructed with slope m  and y-intercept  c. We may consider that  L S I is the fuzzy  at (0, c), c =   line passing through the continuous fuzzy point P c(1) such that P  is at a distance of  c from the origin. However, there are many such P. We consider  with slope all the fuzzy lines  L P S passing through all possible such continuous P    m . We define L S I as the fuzzy line L P S that passes through such P and that has the smallest possible support area, because this particular  L P S is unique. If  is the area  of P(0), then as the infimum value of  is ‘zero’, we get    and has slope m  L P S : where  L P S passes through a point P  . L S I = lim  →0

Theorem 3.2.6 The fuzzy point on the y-axis at a distance of  c from the origin and with the smallest possible support area is the fuzzy number  c along the y-axis, and more precisely is the fuzzy number at (0, c) along the y-axis.  be the required fuzzy point with the smallest  (area of support of the Proof Let P  with fuzzy point). The theorem claims that there does not exist any point in P(0) a non-zero x-coordinate. More specifically, the theorem claims that , the area of  P(0), is ‘zero’.  is contained in the y-axis. To prove the theorem we have to show that support of P   is On contrary, let there exists one point (x, y) in P(0) such that x = 0. Since P(0)  convex, the existence of one point (x, y) in P(0) with x = 0 implies the existence of many such points (x, y) with x = 0. Hence,  > 0. However, this is contradictory, since the fuzzy number  c on the y-axis is itself a fuzzy point and the area of the support of this fuzzy point is ‘zero’.

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3 Fuzzy Line

Fig. 3.4 The fuzzy set on L converges to  c

Therefore, the required fuzzy point is the fuzzy number  c along the y-axis and this is the fuzzy (number) point at (0, c).  is a continuous In the above limiting formula for  L S I , it should be noted that P fuzzy point. Therefore, all the  L P S considered can be constructed by joining the line  and with slope in m segments through a point of P(0) (0) with the same membership  has area equal to ‘zero’ (i.e., P = value. In the limiting case when P c), all the fuzzy  will converge to  sets along different directions that lie on the support of P c. This  are depicted in Fig. 3.4. phenomenon in Theorem 3.2.6 and the limiting case of P  is a fuzzy point at a distance of  In Fig. 3.4, at C = (0, c), P c from the origin  such that the O. Let L be a line through C. Now we try to reduce the area of P(0)  will remain   will again be a fuzzy distance between O and P c and the changed P  point at C. In the limiting case, that is, when  (area of P(0)) tends to 0, the fuzzy  will be squeezed to  c, which is a fuzzy set along the y-axis on set along L on P(0)  P(0). That is, the fuzzy number (D1 /C/B1 ) along L will tend to the fuzzy number (D2 /C/B2 ). Apparently, the points B1 and D1 will tend to the points B2 and D2 when  → 0. This will happen for all possible L passing through C. Therefore, in  L P S , the combinations of the lines through B1 with slope m 1 (say) will be the combinations of the lines through B2 with slope m 1 in the limiting case. Similarly, the combinations of the lines through B1 with slope m 2 (say) will be the combinations of the lines through B2 with slope m 2 in the limiting case. However, the membership values of  and μ(B2 | c) will still always be the same. The case is similar for each point μ(B1 | P) on (D1 /C/B1 ).

3.2 Construction of Fuzzy Line

71

Thus, the definition    and with slope m  L PS :  L P S passes through P  L S I = lim  →0

gives the following definition of  L SI . Definition 3.2.6 (Slope–intercept form of a fuzzy line). Let m  and  c be  and y-intercept  c may two fuzzy numbers. The fuzzy line  L S I with slope m be defined by its membership function as μ((x, y)| L S I ) = sup{α : where (x, y) lieson the line with slope m ∈ m (0) and y − intercept c ∈  c(0) with μ(m | m ) = α = μ(c| c)}. More precisely, μ((x, y)| L S I ) = sup{α : y = mx + c where m ∈ m (0), c ∈  c(0) with μ(m| m ) = α = μ(c| c)}.

In general, for each m in m (0) there exist two values of c in  c(0) with the same membership grades. In addition, for each c there exist two values of m with the same membership values as c. From the definition, we observe that  L S I is constructed by taking the union of the lines with slope m that have exactly the same membership grade as the y-intercept. Figure 3.5 shows the method for constructing  L S I . The fuzzy L S I . The slopes number  c = (K /A/L) L R on the y-axis is used as the y-intercept of  of the lines RS and W Z are equal, m 1 say. The slopes of the lines M N and U V m ) = μ(m 2 | m ) = 0 and μ(K | c) = μ(L| c) = 0. As are equal, m 2 say. Here μ(m 1 | described above, for the point K there exist two slopes with the same membership value, and hence there are two lines RS and M N passing through K in  L S I (0). Similarly, for the point L, the lines U V and W Z passing through L must belong to  L S I (1) is the line P Q that passes through the core of  c. By drawing all the L S I (0).  lines through every point in  c(0), the fuzzy line  L S I can be formed and is represented by the shape in Fig. 3.5. Let l be a line in  L S I . We define L S I ). μ(l| L S I ) = min μ((x, y)| (x,y)∈l

The following theorem helps in finding the membership value of a line in  L SI . Theorem 3.2.7 Let l be a line in  L S I and there exist (0, c) ∈  c(0) and m∈m (0) with μ(m| m ) = α = μ((0, c)| L S I ) such that l ≡ y = mx + c. Then μ(l| L S I ) = α.

72

3 Fuzzy Line

Fig. 3.5 Slope–intercept form of a fuzzy line

Proof The proof is similar to that of Theorem 3.2.4. Note 3.6 In Theorem 3.2.7, the membership value of the point (0, c) is evaluated with respect to  L S I . However, in the slope–intercept form of fuzzy lines, μ((0, c)| c) = μ((0, c)| L S I ). From a geometric standpoint, we observe that if a line perpendicular to  L S I (1) is considered, then there exists a fuzzy number on  L S I (0) along the line. For example, in Fig. 3.5 a fuzzy number of L R-type (H/G/F) L R exists along the line C D. The whole fuzzy line  L S I can be visualized as a three-dimensional shape whose crosssection across  L S I (1) is a fuzzy number such as (H/G/F) L R . Note that the curve c) = of z = μ((x, 0)| c) coincides with z = μ((x, y)| L S I ) for y = 0, that is, μ((0, c)| c(0). μ((0, c)| L S I ) for all c in    Example 3.8 (Slope–intercept form of a fuzzy line). Let m  = − 25 / − 53 / − 56 and    c = 3/3.5/5 be two fuzzy numbers. Figure 3.5 shows the fuzzy line  L S I with slope m  and y-intercept  c. The numbers with membership value α on m  are 5(α − 3)/6 and −5(1 + α)/6. The numbers with membership value α on  c are (6 + α)/2 and (10 − 3α)/2. By Definition 3.2.6,  L S I (0) is the union of all the lines with slope 5(α − 3)/6 or −5(1 + α)/6 and y-intercept (6 + α)/2 or (10 − 3α)/2. That is,

3.2 Construction of Fuzzy Line

 L S I (0) =

 

(x, y) : y =

α∈[0,1]

y=

73

5(α − 3) (6 + α) 5(α − 3) (10 − 3α) x+ or y = x+ or 6 2 6 2

(6 + α) −5(1 + α) (10 − 3α)  −5(1 + α) x+ or y = x+ 6 2 6 2

  5 5 = (x, y) : y + x − 5 ≤ 0, y + x − 3 ≥ 0 2 6   5 5 (x, y) : y + x − 3 ≥ 0, y + x − 5 ≤ 0 2 6

and  L S I (1) is the line y + 53 x − 3.5 = 0. In Fig. 3.5, L = (0, 5), K = (0, 3), A = (0, 3.5), 5 5 5 M N : y = − x + 3, U V : y = − x + 5, W Z : y = − x + 5, 6 6 2 5 5 RS : y = − x + 3 and P Q : y = − x + 3.5. 2 3

3.2.3.1

 SI ) Construction of the Membership Function µ(.| L

To find the membership value at a point (x, y), say, we have to consider all the lines on  L S I (0) passing through the point (x, y). Among these, the line with the highest membership grade will offer the value of μ((x, y)| L S I ), since Definition 3.2.6 and Theorem 3.2.6 imply that L S I ) : l ≡ y = mx + c where μ(c| c) = μ((m| m ))}. μ((x, y)| L S I ) = sup{μ(l| We call this line that offers the supremum as the adjoining line of the point (x, y). Example 3.9 (Finding membership values in the slope–intercept form). Consider the fuzzy line  L S I in Example 3.8. Suppose we want to evaluate the membership  and c ∈  c be the numbers with membership value value at (1, 4) on  L S I . Let m ∈ m α. Then m = 5(α − 3)/6 or −5(1 + α)/6 and c = (6 + α)/2 or (10 − 3α)/2. If y = mx + c passes through (1, 4), then m + c = 4. That is, ⎫ 5(α − 3)/6 + (6 + α)/2 = 4⎪ ⎪ ⎪ 5(α − 3)/6 + (10 − 3α)/2 = 4⎬ −5(1 + α)/6 + (6 + α)/2 = 4⎪ ⎪ ⎪ ⎭ −5(1 + α)/6 + (10 − 3α)/2 = 4

which implies

⎧ α = 21/8 ⎪ ⎪ ⎪ ⎨ α = −9/4 ⎪ α = −11/2 ⎪ ⎪ ⎩ α = 1/14.

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3 Fuzzy Line

Thus, the only admissible value of α is 1/14 and the corresponding values of m and c are −75/84 and 137/28, respectively. Therefore, μ((1, 4)| L S I ) = 1/14. Here, 75 x + 137 . the adjoining line of (1, 4) is y = − 84 28 Observation 3.2.7 Similar to the way of defining the y-intercept of  L P S or  L 2P , c in if we define y-intercepts of  L S I , then the y-intercept will be exactly equal to  Definition 3.2.6. (α) Observation 3.2.8 For any α ∈ (0, 1),  L S I (α) = {l : l is a line with slope m ∈ m and y-intercept c ∈  c(α)}. Note 3.7 For the fuzzy line constructed in slope–intercept form above, we observe that if the slope m  has a positive spread, then as we move along the core line  L P S (1) the spread of the fuzzy line widens with increasing distance from the location of  This fuzzy line with a widening spread cannot be perceived as a the fuzzy point P. collection of fuzzy points because fuzzy points must have a bounded support. Thus, we have the following result.

Theorem 3.2.8 To perceive a fuzzy line in slope–intercept form as a collection of fuzzy points, the slope m  must be a real constant.

Proof The proof is similar to that of Proposition 3.2.1. In the next subsection we investigate the intercept form of a fuzzy line.

I) 3.2.4 Intercept Form ( L Suppose that  a and  b are two given fuzzy numbers and a fuzzy line  L I has to be constructed, where  a and  b are the x-intercept and y-intercept, respectively, of the 1 fuzzy line. We can consider that  L I passes through two continuous fuzzy points P 2 (on the y-axis) such that P 1 is at a distance of  (on the x-axis) and P a from the origin 2 is at  1 and and P b from the origin of the R2 plane. However, there are many such P 1 2 . Consider all the fuzzy lines  L 2P passing through all possible such continuous P P 2 . We define  1 and and P L I as the fuzzy line  L 2P that passes through fuzzy points P 2 such that they have the smallest possible support. This particular  L 2P happens to P 1 (0) and 2 is the area of P 2 (0), then obviously be unique. If 1 is the area of P  L I = lim

1 →0 2 →0



1 and P 2  L 2P passes through P L 2P : 

since the smallest possible values of 1 and 2 are ‘zero’s.



3.2 Construction of Fuzzy Line

75

Theorem 3.2.9 The fuzzy point on the x-axis at a distance  a from the origin and with the smallest possible area of support is the fuzzy number  a along the x-axis, and more specifically the fuzzy number at (a, 0). Proof The proof is similar to that of Theorem 3.2.6 and is omitted. 1 and P 2 are continuous. In the above limiting formula of  L I , the fuzzy points P 1  Therefore, all the L 2P considered can be constructed by joining same points of P    and P2 . In the limiting case when P1 has area equal to ‘zero’ (i.e., P1 =  a ), all the 1 will converge to  a. fuzzy sets along different directions that lie on the support of P 1 and P 2 2 . This phenomenon is depicted in Fig. 3.6, where P The same occurs for P are fuzzy points, at A and B, respectively, at distances of  a and  b from the origin O. L 1 and L 2 , which make the same angle with the line joining A and B, are two parallel lines passing through A and B, respectively. The pairs of points A1 and B1 , 1 and P 2 . Now we try to reduce the and C1 and D1 are same points with respect to P   a and the changed area of P1 (0) such that the distance between O and P1 will remain 

Fig. 3.6 Fuzzy sets on L 1 and L 2 converge to  a and  b, respectively

76

3 Fuzzy Line

1 will again be a fuzzy point at A. In the limiting case, that is, when 1 → 0, the P 1 (0) will converge to  a , the fuzzy set fuzzy set along L 1 (for each possible L 1 ) on P 1 (0). Apparently, the points A1 and C1 will tend to the points along the x-axis on P 2 and hence for the line L 2 : the points A2 and C2 , respectively. The same occurs for P B1 and D1 will tend to the points B2 and D2 when 2 → 0. This will occur for all possible L 2 passing through B. Therefore, the combination of the same points A1 and B1 will be the combination of the points A2 and B2 in the limiting case. Similarly, the combination of the points C2 and D2 will be the limiting combination of same points C1 and D1 . Thus, the definition   1 and P 2  L 2P :  L 2P passes through P L I = lim  1 →0 2 →0

gives the following definition of  LI. Definition 3.2.7 (Intercept form of a fuzzy line). Let  a and  b be two fuzzy numbers that are the x-intercept and y-intercept, respectively, of a fuzzy line  L I . The fuzzy line 1 (a, 0) = ‘ 2 (0, b) =  L P1 P2 , where P a along the x-axis’, and P L I can be defined as   ‘b along the y-axis’. Thus, if a ∈  a (0) and b ∈  b(0) are same points with membership value α, then     L I = L 1∞ L I L 2∞ , where  L I ) = μ((x, y)| L P1 P2 ) = sup α : (x, y) lies on the line with x- and (i) μ((x, y)| y-intercepts a ∈ a (0) and b ∈  b(0); a, b are same points with membership value  α and i as defined in  (ii)  L i∞ is the semi-infinite fuzzy line segment adjoined to P L 2P (i = 1, 2). More precisely,  L I ) = sup α : μ((x, y)|

x a

= 1, a ∈  a (0), b ∈  b(0) are same points  with μ(a| a ) = α = μ(b| b) .

+

y b

Note 3.8 Note that although  L I is defined as the limiting case of the two-point form, L 1∞ it can finally be obtained through the adjoining semi-infinite fuzzy line segments  and  L 2∞ on both extremities of  L I . Therefore, to obtain  L I , first  L I is formulated   via same points of  a and  b and then we consider  L 1∞  L I L 2∞ . Thus, similar to the two-point form of a fuzzy line,  L I can be perceived as a bi-infinite extension of LI. the fuzzy line segment  L I is constructed by taking the union of the lines By the definition, it is clear that  joining same points with respect to  a and  b. Figure 3.7 shows the way to construct  a on the x-axis is represented by the L R-type fuzzy numL I . The fuzzy number  ber (U/A/W ) L R . Similarly, the L R-type fuzzy number (V /B/Z ) L R on the y-axis represents  b. The membership grade of A (B) in  a ( b) is ‘one’ and it gradually

3.2 Construction of Fuzzy Line

77

Fig. 3.7 Intercept form of a fuzzy line

decreases to ‘zero’ at U (V ) and W (Z ) on the left and right, respectively. The points U and V are same points, since both lie on the left side of A and B, respectively, and μ(U | a ) = μ(V | b). Similarly, the pairs A and B, and W and Z , etc., are pairs of same points. Therefore, in  L I , there must exist lines joining U and V , A and B, and W and Z , etc. Here  L I (1) is the line P Q. The membership value on the line P Q is highest and slowly decreases on both sides as we move away from P Q. Let l be a line segment in  L I . We define L I ) = min μ((x, y)| L I ). μ(l| (x,y)∈l

Theorem 3.2.10 Let l be a line in  L I and there exist same points a and b L I ) = α such that l ≡ ax + by = 1. Then μ(l| LI) with μ((a, 0)| L I ) = μ((0, b)| = α.

Proof The proof is similar to that of Theorem 3.2.1.

78

3 Fuzzy Line

Note 3.9 In Theorem 3.2.10, membership values of the points (a, 0) and (0, b) are evaluated with respect to  L I . However, in the intercept form of a fuzzy line, b) = μ((0, b)| L I ). μ((a, 0)| a ) = μ((a, 0)| L I ) and μ((0, b)| We observe from Fig. 3.7 that if a line perpendicular to  L I is considered, then along that line there exists a fuzzy number of L R-type. For example, along the line C D L I . From a geometric viewpoint, the entire the fuzzy number (H/G/F) L R exists on  fuzzy line  L I can be visualized as a three-dimensional shape whose cross-section across  L I is a fuzzy number such as (H/G/F) L R . Note that the curve for z = μ((x, y)| a ) coincides with z = μ((x, y)| L I ) for a (0). Similarly, the curve for y = 0, that is, μ((a, 0)| a ) = μ((a, 0)| L I ) for all a in  b) = z = μ((x, y)| b) coincides with z = μ((x, y)| L I ) for x = 0, that is, μ((0, b)| b(0). In general, if  a = (a − α/a/a + β) L 1 R1 and  b= μ((0, b)| L I ) for all b in  L I , then (b − γ/b/b + δ) L 2 R2 are the x- and y-intercepts, respectively, of   μ((x, y)| L I ) = sup w :

x

+

y

=1 b − γ L −1 2 (w)  x y or + =1 . −1 −1 a + β R1 (w) b + δ R2 (w)

a−

αL −1 1 (w)

Observation 3.2.9 For any α ∈ (0, 1),  L I (α) = {l : l is a line segment with xintercept a ∈  a (α) and y-intercept b ∈  b(α), and a and b are same points }. Example 3.10 (Intercept form of a fuzzy line). Let  a and  b be two fuzzy numbers defined by: ⎧ 2 ⎪ ⎨(x − 2) if 2 ≤ x ≤ 3 2 μ(x| a ) = (4 − x) if 3 ≤ x ≤ 4 ⎪ ⎩ 0 elsewhere and  b = (3/4/5). The fuzzy line  L I with x-intercept  a and y-intercept  b have a  shape as in√Fig. 3.7. The same points with respect to  a and b with membership grade √ L I (0) is the α are 2 + α and 3 + α, or 4 − α, and 5 − α. By Definition 3.2.7,   union of the semi-infinite fuzzy line segments L 1∞ ,  L 2∞ , and all lines with slope √ √ and intercept 2 + α and 3 + α, or 4 − α and 5 − α, respectively. That is,  L 1∞ L I (0) = 



 L 2∞

 

(x, y) :

α∈[0,1]

 x x y y = 1, =1 √ + √ + 2+ α 3+α 4− α 5−α

  x y 3 x y 5 = (x, y) : + ≥ , + ≤ , x ≤ 0 3 4 4 3 4 4   y 2 x y 4 x (x, y) : + ≥ , + ≤ , y ≤ 0 3 4 3 3 4 3   x y x y (x, y) : + ≥ 1, + ≤ 1, x, y ≥ 0 2 3 4 3   y x y x (x, y) : + ≤ 1, + ≥ 1, x, y ≥ 0 2 3 4 5

3.2 Construction of Fuzzy Line

and  L I (1) is the line In Fig. 3.7,

x 3

+

y 4

79

= 1.

U = (2, 0), W = (4, 0), A = (3, 0), V = (0, 3), B = (0, 4) and Z = (0, 5). y x U V : + = 1, 2 3 y x W Z : + = 1 and 4 5 y x P Q : + = 1. 3 4

3.2.4.1

I) Construction of the Membership Function µ(.| L

To find the membership value at a point (x, y) ∈  L(0), we have to first check    L 1∞ (0), then whether (x, y) belongs to L 1∞ (0) or L I (0) or L 2∞ (0), since if (x, y) ∈  L 1∞ ) = μ((x, y)| L I ), and similarly for  L I (0) and  L 2∞ (0). For (x, y) ∈ μ((x, y)|  L 1∞ (0), there will exist only one half-line passing through (x, y) and parallel to  a) = L I (1). Suppose that this half-line intersects the x-axis at A. Clearly, μ(A| L 1∞ ) = μ((x, y)| L I ). A similar case occurs for (x, y) ∈  L 2∞ . If (x, y) ∈ μ((x, y)|  L I (0), then all the line segments in  L I (0) passing through (x, y) have to be considered. Among these, the line segment with the highest membership grade will offer the value of μ((x, y)| L I ), since Definition 3.2.7 and Theorem 3.2.10 imply that   y x L I ) : l ≡ + = 1 where a, b are same points . μ((x, y)| L I ) = sup μ(l| a b The following example illustrates how to find the membership value of a point on  LI. Example 3.11 (Finding membership values in the intercept form). Consider the fuzzy line in Example 3.10. Suppose we have to determine the membership grade at (3, −5). We test whether (3, −5) ∈  L I (0). Here,  L I (0) =

  (x, y) : α∈[0,1]

 3 3 5 5 = 1 or =1 . √ − √ − 2+ α 3+α 4− α 5−α

However, there exists no real α satisfying Thus, (3, −5) ∈ / L (0). I

3√ 2+ α

−

5 3+α

= 1 or

3√ 4− α

−

5 5−α

= 1.

80

3 Fuzzy Line

The line parallel to  L I (1), x3 + 4y = 1 and passing through (3, −5) is x3 + 4y + 41 = 0. This line intersects the x-axis and y-axis at A = (−3/4, 0) and B = (0, −1), respectively. However, μ(−3/4| a ) = 0 = μ(−1| b). Hence, (3, −5) ∈ / L 1∞ (0) and  L 2∞ (0). / L I (0). Therefore, μ((3, −5)| L I ) = 0. This occurs because (3, −5) ∈ Similar to the two-point form, the slope of the fuzzy line  L I can be defined as follows. Definition 3.2.8 (Slope of a fuzzy line in the intercept form ( m )). Let  L I be  a fuzzy line and l be a line segment in L I . The slope of the fuzzy line  L I is defined by its membership function as L I ) : slope of l is m}. μ(m| m ) = sup{μ(l| This definition of the slope m  of the fuzzy line  L I must give a fuzzy number, because l is a line segment of a line ax + by = 1 joining (a, 0) and (0, b), where a and b are same points with respect to the intercepts  a and  b. We note that the slopes of    L I and L I are identical. A natural question that arises is whether − ab gives the same value as m  or not. The answer is affirmative for the following reason. To form  L , all I

line segments joining the points (a, 0) and (0, b), where a and b are same points of   a and  b, are combined. However, in evaluating − ab , all the numbers − ab with varied membership values are to be collected, where b and a are inverse points with respect to − b and  a ; thus, a and b are same points with respect to  a and  b. Hence, m  is  exactly equal to the fuzzy number − ab .   −b : where b and Note 3.10 From the above discussion, it is evident that m =  a a are same points with respect to the fuzzy numbers  b and  a . This shows that if  θ) = sup α : we define the angle of inclination of  L I to the positive x-axis as μ(θ| where θ is the anglebetween the positive x-axis and a line segment l in  L I with membership value α , then from the evaluation of the membership value of a line segment l in  L I we obtain that m  = tan  θ.

3.3 Discussion and Comparison From our study, we obtained the following properties for fuzzy lines. (i) Fuzzy lines are bi-infinite extensions of fuzzy line segments. (ii) Fuzzy lines can be perceived as a collection of crisp points with different membership values or as an infinite collection of fuzzy points.

3.3 Discussion and Comparison

81

(iii) If a line perpendicular to the core of a fuzzy line is considered, then along that line there always exists a fuzzy number. (iv) The α-cuts of fuzzy lines are closed, connected and arc-wise connected subsets of R2 but are not necessarily convex. (v) A fuzzy line is always normalized, and more precisely its core is always a crisp line. (vi) A membership function will always be upper semi-continuous, which follows from the fact that an α-cut is closed. (vii) The slope and intercepts of fuzzy lines in two-point and intercept forms are fuzzy numbers with a singleton core. Let us now investigate interrelations between our different forms for fuzzy lines and then compare our proposed formulations with those described by Buckley and Eslami (1997). In finding interrelations between fuzzy lines, it is shown that how under the condiL 2P ,  L S I is an  L P S , and vice versa. In general,  L 2P or  LI tions stated above,  L I is an    cannot be equivalent to L P S or L S I . However, in the particular case when the slope of all four forms of a fuzzy line is a real constant, each of the forms is equivalent to another form, which directly follows from Proposition 3.2.1, Theorems 3.2.5 and 3.2.8. Theorem 3.3.1  L I is an  L 2P . Proof Let  a and  b be two fuzzy numbers that are the x-intercept and y-intercept, respectively, of  L I . The fuzzy number  a ( b) can be treated as a fuzzy point along the  0) and P(0,  b). x-axis (y-axis) (Fig. 3.6). We denote these two fuzzy points by P(a, α α α α  Let α ∈ [0, 1]. If  a (α) = [a1 , a2 ], then P(a, 0)(α) = [a1 , a2 ] × {0}. Similarly, if  b(α) = {0} × [b1α , b2α ]. If a fuzzy line  L 2P passing through b(α) = [b1α , b2α ], then   0) and P(0,  b) is considered, then clearly  L I (α) for any α. This P(a, L 2P (α) =  yields the desired result. 1 and In converting  L 2P to  L I , we consider a fuzzy line  L 2P passing through P    P2 . We consider P1 a fuzzy point on the x-axis and P2 a fuzzy point on the y-axis. 1 ∩ (x-axis) and Q 2 = P 2 ∩ (y-axis). Naturally, Q 1 and Q 2 are two fuzzy 1 = P Let Q numbers along the x- and y-axis, respectively. It can be easily perceived that if  1 and y-intercept L Q1 Q2 =  L P1 P2 , then  L 2P is the fuzzy line  L I with x-intercept Q  Q2. Theorem 3.3.2  L S I is an  L PS. Proof Let the slope and y-intercept of  L S I be m and c, respectively. The fuzzy number  c can be perceived as a fuzzy point along the y-axis. We denote this fuzzy point by  c)(α) = {0} × [c1α , c2α ]. If we  c). Let α ∈ [0, 1]. If  P(0, c(α) = [c1α , c2α ], then P(0,   consider a fuzzy line L P S passing through P(0, c) and with slope m, then clearly  L S I (α). This completes the proof. L P S (α) = 

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 in defining  In converting  L P S to  L S I , if the fuzzy point P L P S is a fuzzy point on  the y-axis whereby the fuzzy number along the y-axis on P(0), say c, has a maximum  along different lines through P(1),  spread among the fuzzy numbers on P(0) then  L S I , where the slope of  L S I is exactly the same as  L P S and the y-intercept L P S is an  c. of  L S I is  Now let us compare our proposed methodology with existing formulations of fuzzy lines. It is mentioned in the Introduction that various researchers have proposed ideas on what a fuzzy line is, but only (Buckley and Eslami 1997) described constructions of fuzzy lines. Therefore, we compare our proposed constructions with those reported by Buckley and Eslami. As a general observation, our four proposed formulations for fuzzy lines either facilitate evaluation of the membership functions or have less imprecise spread compared to that of Buckley and Eslami (1997). This is clarified in a pointwise comparison of the two approaches. • Two-Point Form 1 and P 2 is the According to our method, the fuzzy line  L 2P passing through P  union of the fuzzy line segment L P1 P2 and two semi-infinite fuzzy line segments adjoined to  L P1 P2 at its two extremities. As a result,  L 2P can be viewed as a biinfinite extension of the fuzzy line segment  L P1 P2 . By contrast, according to Buckley 1 and P 2 either uniformly increases or and Eslami (1997), the spread of  L 2P between P 2 (an epsilon1 and P uniformly decreases, but suddenly widens in the range beyond P butterfly shape (Guibas et al. 1989)). Thus, their fuzzy line is not a collection of fuzzy points. Conversely, our proposed  L 2P has bounded support across its core line and hence  L 2P can be viewed as a collection of fuzzy points. Our method considers only combinations of same points, whereas Buckley and Eslami combine all the points of the support of the fuzzy point. Eventually, the combinations in our definitions are subsets of the combinations taken by Buckley and Eslami (1997). Thus, our fuzzy line constructions have lesser spread and evaluation of the membership function for  L 2P is easier when compared to the approach of Buckley and Eslami (1997). • Point–Slope Form The definition of a fuzzy line in point–slope form according to Buckley and Eslami (1997) generates fuzzy points of infinitely large supports. This may not be appreciated. Since, according to the definition of fuzzy points, it cannot have an unbounded support. The problem may arise because of the positive spread of the fuzzy slope in Buckley and Eslami (1997). In our analysis of the point–slope form, Theorem 3.2.5 states that to define  L P S properly, the slope has to be constant. Our proposed method L P S can be with a (crisp) constant slope gives a proper shape to  L P S , from which  viewed as a collection of fuzzy points. Moreover, Definition 3.2.4 and Theorem 3.2.4 facilitate evaluation of the membership value of a point and a line, respectively, in the support of  L PS.

3.3 Discussion and Comparison

83

• Slope–Intercept Form Our proposed method defines a fuzzy line in slope–intercept form as the limiting case of the point–slope form. We note that a fuzzy line in point–slope form according to Buckley and Eslami (1997) has an unbounded imprecise region on either side of the core line. Thus, their  L S I cannot be perceived as a collection of fuzzy points due to Proposition 3.2.1 and Theorem 3.2.8. However, we demonstrated that to perceive a fuzzy line in slope–intercept form as a collection of fuzzy points, the slope m  must be a real constant. For a constant slope,  L S I according to both methods gives the same fuzzy line. However, our method identifies the membership function of  L S I more easily than the Buckley–Eslami method, since we find the membership value of any point in  L S I by solving a maximization problem (Sect. 3.2.3.1) and the constraint set for the optimization problem is a subset of the constraint set used by Buckley and Eslami. Thus, the support of our  L S I must always be a subset of the fuzzy line  L S I in Buckley and Eslami (1997) and their core lines are identical. Therefore, our proposed  L S I has a less imprecise spread than that of Buckley and Eslami (1997). • Intercept Form The method we used to construct  L I defines a fuzzy line in the intercept form as the limiting case of the two-point form. Thus, a fuzzy line in the intercept form is L I bi-infinitely. We observe that the obtained by extending the fuzzy line segment  L I (1). Because of this proposed  L I has a bounded imprecise part across the core line  bounded support, the proposed  L I can be perceived as a collection of fuzzy points. By contrast, the  L I used by Buckley and Eslami (1997) cannot be considered as a collection of fuzzy points since it has unbounded support on either side of  L I (1). Note also that when formulating  L I , our method uses only the line segment whose intercepts are same points of a given  a and  b. However, Buckley and Eslami used all possible lines whose intercepts are any numbers on the supports of  a and  b.  Therefore, our formulation of L I has a narrower spread than that of Buckley and Eslami (1997). By use of same points, evaluation of the membership function for  LI becomes easier than the Buckley–Eslami approach. Since, both methods use similar optimization problems to find μ of  L I and the constraint set we use is a proper subset of that used by Buckley and Eslami (1997). A question that arises from our different formulations of fuzzy lines is which definition of a fuzzy line is favored in our study. To answer this question, we note that each of our formulations depends on what information is known about the fuzzy line. If information is available for two fuzzy points on the fuzzy line, then the two-point form must be favored. If information is available for the slope and a fuzzy point, then the point–slope form of the fuzzy line must be favored, and so on.

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3.4 General Form of a Fuzzy Line As described above, a fuzzy line is perceived as union of crisp lines or line segments with different membership values. However, there are other ways too. For instance, a fuzzy line may be imagined as a collection of crisp points of different membership values or as a group of fuzzy points. Basically whatever way we define fuzzy line, it must be visualized as a straight infinitely long hazy band having one crisp straight line on its core with a smooth transition of membership values between the neighbouring points of that core line. To express this visualization mathematically, for each fuzzy line there must exist three functions f (x, y), g(x, y) and h(x, y), where f (x, y) = 0 and h(x, y) = 0 represent boundary curves of the support of the fuzzy line and the straight line g(x, y) = 0 represents the core. Obviously, g is a linear function and f and h are not necessarily linear. This concept leads to define the general form of a fuzzy line and also gives an equation to the fuzzy line. Definition 3.4.1 (General form ( L G ) Ghosh and Chakraborty 2015). In general form of fuzzy line, it is described as collection of crisp points with different membership values. Those points must lie in between two (boundary) curves and there must be one crisp line between those curves. Let those boundary curves be f (x, y) = 0 and h(x, y) = 0 and the crisp line be g(x, y) ≡ y − mx − c = 0. Then the fuzzy line  L G can be expressed by an equational form ( f (x, y)/g(x, y)/ h(x, y)) L R = 0, where L : [0, ∞) → [0, 1] and R : [0, ∞) → [0, 1] are suitable reference functions. Here the notation ( f /g/ h) L R = 0 means that membership grade of the points in  L G (0) gradually increases (through the function L) from 0 to 1 as on moving from the curve f (x, y) = 0 to g(x, y) = 0 and gradually decreases (through the function R) from 1 to 0 as on moving from g(x, y) = 0 to h(x, y) = 0. Let f and h be the functions having the properties that at each point (x, y) ∈  L G (0) for which f (x, y) = 0 = h(x, y) (i) (ii)

∂ f ∂ f ∂h , , and ∂∂hy exist and are ∂x ∂ y ∂x ∂f ∂h and ∂ y are non-vanishing. ∂y

continuous, and

Then from well-known Implicit Function Theorem of calculus, there exist two functions p(x) and q(x) such that f (x, y) = 0 ⇔ y = p(x) and g(x, y) = 0 ⇔ y = q(x). Let us consider a straight line perpendicular to  L G (1) ≡ y = mx + c. As explained, along x + my = k, which is perpendicular to  L G (1), there must exist one fuzzy number on the support of the fuzzy line  L G . In Fig. 3.1, the line x + my = k intersects y = p(x), y = mx + c and y = q(x) at F, G and H , respectively. It is to note that the above said fuzzy number on  L G (0) and along x + my = k can be expressed as (F/G/H ) L R .

3.4 General Form of a Fuzzy Line

85

 . Let F = (u 1 , v1 ) and H = (u 2 , v2 ). The points , c+mk 1+m 2   u + mv = k u + mv = k and (u 1 , v1 ) and (u 2 , v2 ) can be obtained by solving v = p(u) v = q(u), respectively.  Now membership value of any point (u, v) ∈  L G (0) {(x, y) : x + my = k} can be expressed by: Apparently, G =

 k−mc 1+m 2

⎧  ⎪ L d((u,v),G) if u 1 ≤ u ≤ k−mc , ⎪ d(F,G) 1+m 2 ⎪ ⎪ ⎪ c+mk ⎪ ⎪ ≤ v ≤ v1 , u + mv = k 1+m 2

  ⎨  d(G,(u,v)) k−mc μ (u, v; k)| LG = R if 1+m 2 ≤ u ≤ u 2 , d(G,H ) ⎪ ⎪ ⎪ ⎪ , u + mv = k v2 ≤ v ≤ c+mk ⎪ ⎪ 1+m 2 ⎪ ⎩ 0 elsewhere. Varying k ∈ R, membership value of any point in  L G can be obtained. 1 (1, 2) and P 2 (5, 7) where Example 3.12 Consider the fuzzy line passing through P  their membership functions are right circular cones with P1 (1, 2)(0) = {(x, y) : (x − 1 (5, 7)(0) = {(x, y) : (x − 5)2 + (y − 7)2 ≤ ( 1 )2 }. 1)2 + (y − 2)2 ≤ ( 21 )2 } and P 2 This fuzzy line joining A = (1, 2) and B = (5, 7) is depicted in Fig. 3.8. In the figure, the equations of the lines are given as follow: AB : 5x − 4y + 3 = 0, I J : 5x − 4y + 6.202 = 0 and C D : 5x − 4y − 0.202 = 0. The lines I J and C D respectively determine the boundary functions f (x, y) and h(x, y) of support of the fuzzy line. So, f (x, y) ≡ 5x − 4y + 6.202 = 0, g(x, y) ≡ 5x − 4y + 3 = 0, and h(x, y) ≡ 5x − 4y − 0.202 = 0. Here  L G (0) = {(x, y) : f (x, y) ≥ 0, h(x, y) ≤ 0} and for each point (x, y) ∈  L G (0) satisfying which 5x − 4y + 6.202 = 0 or 5x − 4y − 0.202 = 0, ∂ f ∂ f ∂h , , and ∂∂hy are constant ∂x ∂ y ∂x and ∂∂hy are non-vanishing.

(i) all of (ii)

∂f ∂y

and hence continuous, and

So, there exist p(x) and q(x) such that f (x, y) ≡ y − p(x) = 0 and h(x, y) ≡ y − q(x) = 0. It is easily observed that p(x) = 14 (5x + 6.202) and q(x) = 14 (5x −   0.202).  The equation of L G is 5x − 4y + 6.202 / 5x − 4y + 3 / 5x − 4y − 0.202 L R = 0 with L(t) = R(t) = max{0, 1 − |t|}. Its membership function can be constructed as follows. The line RS ≡ 4x + 5y − k = 0 is a general line  perpendicular to P Q or AB, L G can be (k ∈ R). Along RS, the fuzzy number H/G/F L R in the support of  found, where H , F and G are the points of intersection RS with I J , C D and P Q,

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3 Fuzzy Line

Fig. 3.8 Fuzzy line in the Example 3.12

respectively. This fuzzy number is same with the following two fuzzy numbers up to a rigid translation,2 1 (1, 2)(0) along M N , and (i) the fuzzy number lies on P 2 (5, 7)(0) along U V . (ii) the fuzzy number lies on P The points F, G and H are (−0.756 + 0.098k, 0.605 + 0.122k), (−0.366 + 0.098k, 0.293 + 0.122k) and (0.025 + 0.098k, −0.020 + 0.122k), respectively. Therefore, by varying k ∈ R, membership function of  L G can be obtained as:    1 − 2d(G, (u, v)) if (u, v) ∈  L G (0) ∩ F H μ (u, v; k)| LG = 0 otherwise. Theorem 3.4.1 Fuzzy line (by the Definition 3.4.1) passing through n fuzzy points 1 , P 2 , . . . , P n is unique. P

2A

translation is said to be rigid if it preserves relative distances—that is to say: if P1 and Q 1 are transformed to P2 and Q 2 , then the distance from P1 to Q 1 is equal to the distance from P2 to Q 2 .

3.4 General Form of a Fuzzy Line

87

1 , P 2 , . . . , Proof Suppose there exist two fuzzy lines  L G and  L G passing through P n . P L G are distinct, there must exist one point (x0 , y0 ) ∈  L G such that As  L G and   (x0 , y0 ) ∈ / L G or μ((x0 , y0 )| L G ) = μ((x0 , y0 )| L G ). / L G , then it implies that there do not exist same points in two consecIf (x0 , y0 ) ∈ 2 , . . . , P n such that (x0 , y0 ) lies on the line segment 1 , P utive fuzzy points amongst P L G implies that joining those two same points. This is impossible, since (x0 , y0 ) ∈  i and (x2 , y2 ) ∈ P i+1 , for some i, such there must exist two same points (x1 , y1 ) ∈ P that (x0 , y0 ) lies on the line segment joining (x1 , y1 ) and (x2 , y2 ). Therefore, the / L G cannot occur. possibility that (x0 , y0 ) ∈ L G ) = μ((x0 , y0 )| L G ) is also impossible, since membership Again μ((x0 , y0 )| L G and  L G are evaluated by the same Definition value of (x0 , y0 ) on both fuzzy lines  3.4.1. L G , then that point must belong to  L G Hence, if a point (x0 , y0 ) belongs to    satisfying μ((x0 , y0 )| L G ) = μ((x0 , y0 )| L G ). Changing the role of  L G and  L G , yields  L G . Hence the theorem is proved. LG =  Theorem 3.4.2 Let  L be a fuzzy line and l be a line perpendicular to  L(1). Along l, there always exists a fuzzy number on  L. Proof Let  L = {(x, μ(x| L)) : x ∈ R2 } be the fuzzy line. The line l can also be presented by {(x, μ(x|l)) : x ∈ R2 } where μ(x|l)  = 1 for x ∈ l, and 2‘zero’ otherwise. {(x, μ(x|l)) : x ∈ R } is a fuzzy set, Obviously, the set {(x, μ(x|  L)): x ∈ R2 }  We whose μ is evaluated by the t-norm as ‘min’. Let us denote this fuzzy set as A.  is a fuzzy number along l. have to prove that A  Let 0 < α ≤ 1. The set A(α) is a line segment or union of line segments which are subsets of l. It will be proved that it is exactly a line segment and cannot be union of line segments. Let Aα and Bα be the points of intersection of l with f (x, y; α) = 0 and h(x, y; α) = 0, respectively, where f (x, y; α) = 0 and h(x, y; α) = 0 are  boundary curves of the α-cut of  L. It will be shown that A(α) = Aα Bα . Let ∃  / A(α). So, μ(K | L) < α. Therefore, K ∈ / L(α). But this is not K ∈ Aα Bα but K ∈ possible, because all α-level sets of  L are closed convex sets and for 0 < α < β ≤ 1,   L(β) ⊆  L(α). Therefore, A(α) = Aα Bα which is a compact and convex set and for    is upper 0 < α < β ≤ 1 we get A(β) ⊆ A(α). Thus, membership function of A  is a compact set. semi-continuous and A(0)  = min{μ(G|l), If G be the point of intersection of l and  L(1), then μ(G| A)  is a fuzzy number along the line l. μ(G| L)} = 1. Hence, the fuzzy set A Theorem 3.4.2 helps to think a fuzzy line  L as a collection of fuzzy numbers at each point of  L(1) or collection of fuzzy points at each point of  L(1). From the Theorem 3.4.2 it is obtained that along a line l, which is perpendicular to  L, a fuzzy number exists in  L(0). However, the spread of all of those fuzzy numbers may not be equal in general. If spreads on both the sides of the core are equal, the fuzzy lines may be called as symmetric and otherwise non-symmetric.

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3 Fuzzy Line

Definition 3.4.2 (Symmetric and non-symmetric fuzzy lines). If all the fuzzy numbers, situated on a fuzzy line  L and along the perpendicular lines of  L(1), have equal spreads, then  L is said to be a symmetric fuzzy line and otherwise non-symmetric. Note 3.11 Symmetric fuzzy line may be visualized as a figure obtained by pulling a fuzzy number (point) continuously along the line  L(1) with the restriction that core of the fuzzy number (point) is always retained on  L(1). Thus, evaluating membership value of the points on a symmetric fuzzy line is quite easier since once a fuzzy number along a perpendicular line on the support of the fuzzy line becomes known, the entire fuzzy line can be captured. It may be noted that if a fuzzy line  L is perceived as a group of fuzzy points, then  which belong to  boundary of supports of the fuzzy points P L touch the boundary 1 and P 2 is considered,   of L(0). For example, if a fuzzy line L P1 P2 passing through P   = λP 1 + (1 − λ) P 2 (0 ≤ λ ≤ 1) whose supports the fuzzy points lie on L P1 P2 are P touch the boundary curves of  L(0). However, there exist several fuzzy points whose core lie on  L(1) and supports of them are subsets of  L(0). It is to be noted that all of these fuzzy points are partially contained in the fuzzy line  L. Thus, in a nutshell—it is observed that fuzzy lines are perceived as one of the following: (i) collection of crisp points, (ii) collection of fuzzy points, or (iii) collection of crisp line segments or half-lines or lines. However any of the three considerations depends on the other two, since fuzzy points or crisp lines are collection of crisp points and intersection of two curves is a point or set of points. Unique characterization of fuzzy lines or the identification of a fuzzy set to be called as a fuzzy line is given in the following theorem. Theorem 3.4.3 (Characterization theorem). A fuzzy set is a fuzzy line if and only if its core is a crisp straight line and along any line perpendicular to the core line there must exist a fuzzy number on the support of the fuzzy line. Proof Using previous theorem it can be proved that the first part of the theorem is true. It may be observed that under the assumption of the theorem, the union of all the fuzzy numbers along the perpendicular lines to the core line, gives a fuzzy line in general form. So the converse part is true. Hence the result follows. It is now necessary to define slope and intercepts of a fuzzy line. Definition 3.4.3 (Slope of a fuzzy line). Let  L be a fuzzy line and l¯ be a line segment  in L(0). Let us consider that membership value of l¯ on  L as inf{μ(x| L) : x ∈ l}. Slope of the fuzzy line may defined by its membership function as: ¯ μ(m| m ) = sup{μ(l| L) : slope of l¯ is m}.

3.4 General Form of a Fuzzy Line

89

Example 3.13 (Slope of a fuzzy line). Let  L be the fuzzy line passing through 2 (7, 9). Supports of P 1 (2, 3) and P 2 (7, 9) are 1 (2, 3) and P the fuzzy points P {(x, y) : (x − 2)2 + (y − 3)2 ≤ 41 } and {(x, y) : (x − 7)2 + (y − 9)2 ≤ 1}, respec1 (2, 3) and P 2 (7, 9) have the shapes of right circular tively. Membership functions of P cones with vertices at (2, 3) and (7, 9), respectively. L 1∞ Fuzzy line  L is determined by 

  L 12 L 2∞ .

L 1∞ and  L 2∞ has slope 1.2. Each of the half-lines in the infinite fuzzy half-lines  2 with membership value α ∈ [0, 1] are S 1 = (2 + 1 and P Same points of P θα 1−α 2 cos θ, 3 + 2 sin θ) and Sθα = (7 + (1 − α) cos θ, 9 + (1 − α) sin θ), respectively, for each θ ∈ [0, 2π]. 1−α 2

1 2 Sθα is Slope of the line segment Sθα

6+ 1−α 2 sin θ . 5+ 1−α 2 cos θ

If m  be the slope of the fuzzy line  L, then according to the Definition 3.4.3, mem6 + 1−α sin θ 2 on m  is α. bership value of sup 1−α θ∈[0,2π] 5 + 2 cos θ  1−α 6− 2 sin b(α) , Therefore, core of m  is 1.2 and for each α ∈ [0, 1], m (α) = 5+ 1−α 2 cos b(α) √  1−α 244−(1−α)2 +12 6+ 2 sin b(α) where b(α) = 2 tan−1 . (1−α)−10 5+ 1−α cos b(α) 2

Next, y-intercept of a fuzzy line is defined. Similarly, the x-intercept can be defined. Definition 3.4.4 (y-intercept of a fuzzy line). y-intercept of a fuzzy line  L,  c say, can be defined by its membership function as:     μ (c, 0)| L if (c, 0) ∈  L(0) (y-axis) μ(c| c) = 0 otherwise, i.e.,  c= L



(y-axis).

Example 3.14 (y-intercept of a fuzzy line). Let us evaluate the y-intercept of the 1 (2, 3) and P 2 (7, 9) in the Example 3.13. According to fuzzy line  L passing through P √     the Definition 3.4.4, y-intercept of L is the triangular fuzzy number 35 − 1061 35 35 + √  61 . 10 Let us now study fuzzy distance between a fuzzy point and a fuzzy line.

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3 Fuzzy Line

Definition 3.4.5 (Distance and vertical distance between a fuzzy point and a fuzzy  be a fuzzy point and   and  line). Let P L be a fuzzy line. Distance between P L can be defined as:    

(dl , α) : dl = inf Aα (du , α) : du = sup Aα , α∈[0,1]

 where Aα = {d((x1 , y1 ), (x2 , y2 )) : (x1 , y1 ) ∈  L(α) and (x2 , y2 ) ∈ P(α)}. The tuples (dl , α) or (du , α) mean that ‘the distance dl or du is adjoined with membership value α’. Here the distance metric d((x1 , y1 ), (x2 , y2 )) is the Euclidean distance metric. Instead of the Euclidean distance, if the function d  ((x1 , y1 ), (x2 , y2 )) = |y1 − y2 | is considered to measure the distance between (x1 , y1 ) and (x2 , y2 ), then the fuzzy set = D

 

(dl , α) : dl = inf Aα



(du , α) : du = sup Aα



α∈[0,1]

 and the fuzzy line  may be called as the vertical distance between the fuzzy point P L.  the vertical distance between a fuzzy point and a fuzzy line is a Theorem 3.4.4 D, fuzzy number. Proof Similar to Theorem 2.3.  6) be a Example 3.15 Let  L be the fuzzy line (y − 1/y − 2/y − 3) = 0. Let P(1,  2 fuzzy point whose graph of μ is a right elliptical cone with base (x, y) : (x−1) + (1/2)2  (y−6)2 ≤ 1 , i.e., (1/3)2   2 (y−6)2 (y−6)2 (x−1)2 1 − if (x−1) 2 + (1/3)2 2 + (1/3)2 ≤ 1 (1/2) (1/2)  μ((x, y)| P(1, 6)) = 0 elsewhere.  6)(α) = {(x, y) : Therefore,  L(α) = {(x, y) : 1 + α ≤ y ≤ 3 − α} and P(1, + (y−6) ≤ (1 − α)2 }. (1/3)2 2

(x−1)2 (1/2)2





  16−4α 8+4α  16−4α  D = μ D = α. Precisely, the So, Aα = [ 8+4α , ] and hence μ 3 3 3 3    vertical distance D between P and L given by:  = μ(x| D)

⎧ 3x−8 ⎪ ⎨ 4

16−3x ⎪ 4

⎩

0

if 83 ≤ x ≤ 4 if 4 ≤ x ≤ 16 3 otherwise.

3.5 Conclusion

91

3.5 Conclusion This chapter proposed some construction procedures for fuzzy lines using the concepts of same and inverse points. The reference frame of R2 has been used to measure imprecision of a fuzzy line. Four different forms of a fuzzy line and their basic definitions, construction of their membership functions, α-cuts, slopes, and intercepts are analyzed. The interrelations and conversions from one form to another are also demonstrated. For all the forms, the proposed definitions facilitate evaluation of membership functions and have a lesser spread than previous line definitions (Buckley and Eslami 1997). Mathematical description of general form of fuzzy line is also suggested. In the next chapters, this geometry has been extended to fuzzy triangle and nonlinear fuzzy curves.

References Buckley, J.J., Qu, Y.: Solving systems of linear fuzzy equations. Fuzzy Sets Syst. 43(1), 33–43 (1991) Buckley, J.J., Eslami, E.: Fuzzy plane geometry I: points and lines. Fuzzy Sets Syst. 86, 179–187 (1997) Chakraborty, D., Ghosh, D.: Analytical fuzzy plane geometry II. Fuzzy Sets Syst. 243(1), 84–109 (2014) Ghosh, D., Chakraborty, D.: On general form of fuzzy lines and its application in fuzzy line fitting. J Intell Fuzzy Syst. 29(2), 659–671 (2015) Guibas, L., Salesin, D., Stolfi, J.: Epsilon geometry: building robust algorithms from imprecise computations. In: Proceedings of the 5th Annual Symposium on Computational Geometry, Saarbrüchen, Germany, ACM Press, pp. 208–217 (1989)

Chapter 4

Fuzzy Triangle and Fuzzy Trigonometry

4.1 Introduction The counterpart of a crisp triangle, C, in Euclidian geometry, is a fuzzy triangle. It is helpful to visualize that a fuzzy transform of C as the result of execution of the instruction—Draw C by hand with an unprecisiated spray pen. Here the fuzzytransformation is an one-to-many function. In this chapter, concepts about fuzzy triangle, fuzzy triangular properties and some basics of fuzzy trigonometry are propose (Ghosh and Chakraborty 2018). After defining a fuzzy triangle, its side lengths, vertex angles, area and perimeter have been studied. All the proposed concepts introduced here depend on the newly defined concepts of same and inverse points (Ghosh and Chakraborty 2012).

4.2 Fuzzy Triangle 1 , P 2 and P 3 are given and a Let us suppose that three distinct fuzzy points P  1 P2 P3 ) has to form. A construction procedure may be designed fuzzy triangle (ΔP 2 1 , P as follows. Considering three same points u, v and w in the support of P 3 respectively, let us construct a triangle  having vertices as u, v and w. and P 1 ) = α, then obviously μ(v| P 2 ) = α, μ(w| P 3 ) = α. We may put memberIf μ(u| P  1 P2 P3 can be considered as union  1 P2 P3 is also α. Now ΔP ship value of  in ΔP of all of these ’s—crisp triangles with different membership grades. Thus a formal definition of a fuzzy triangle may be given by its membership function as:  1 P2 P3 ) = sup {α : x ∈ , where  is constructed by the same points u ∈ μ(x|ΔP 2 (0) and w ∈ P 3 (0) as vertices; μ(u| P 1 ) = μ(v| P 2 ) = μ(w| P 1 ) = α}. 1 (0), v ∈ P P L P1 P2 ∪ Remark 4.1 Fuzzy triangle defined in the above definition is exactly equal to   L P2 P3 ∪  L P3 P1 . © Springer Nature Switzerland AG 2019 D. Ghosh and D. Chakraborty, An Introduction to Analytical Fuzzy Plane Geometry, Studies in Fuzziness and Soft Computing 381, https://doi.org/10.1007/978-3-030-15722-7_4

93

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 1 P2 P3 , whose vertices are three Example 4.1 Let us consider the fuzzy triangle, ΔP 2 (5, 7) and P 3 (6, 1). Let the membership functions are right 1 (1, 2), P fuzzy points P elliptical/circular cone with supports 1 (1, 2)(0) = {(x, y) : (x−1)2 + (y − 2)2 ≤ 1}, P 4 2 (5, 7)(0) = {(x, y) : (x − 5)2 + (y − 7)2 ≤ 4} and P 3 (6, 1)(0) = {(x, y) : (x − 6)2 + (y − 1)2 ≤ 1}. P  1 P2 P3 . Let us now evaluate membership value of (2, 4) in the fuzzy triangle ΔP 2 (5, 7) and P 3 (6, 1) 1 (1, 2), P The same points with membership value α ∈ [0, 1] on P are   cos θ sin θ) , , 2 + √2(1−α) Aα,θ : 1 + √2(1−α) 2 2 2 2 4 sin θ+cos θ

4 sin θ+cos θ

Bα,θ : (5 + 2(1 − α) cos θ, 7 + 2(1 − α) sin θ) and Cα,θ : (6 + (1 − α) cos θ, 1 + (1 − α) sin θ) respectively, where θ ∈ [0, 2π]. Apparently, there is a possibility that (2, 4) may lie on the line segment  L¯ P1 P2 , but   (2, 4) cannot lie on the line segments L¯ P2 P3 and L¯ P1 P3 . Now the condition that (2, 4) lies  L¯ P1 P2 or on the line segment Aα,θ Bα,θ (for some θ ∈ [0, 2π] and α ∈ [0, 1]) is: 4−(7+2(1−α) sin θ) 2−(5+2(1−α) cos θ)

⇒α=1−

=

2+2k(1−α) sin θ−(7+2(1−α) sin θ) , where 1+2k(1−α) cos θ−(5+2(1−α) cos θ)

3 (8+6k) sin θ−(10+6k) cos θ

k=√

1

4 sin2 θ+cos2 θ

= f (θ), say.

Here f (θ) must lie in [0, 1], and hence admissible domain of f (θ) is D f = [63◦ , 222.66◦ ]. Maximum value of f (θ) over D f occurred at 157.32◦ and the value is 0.8352, the possibility of containment of (2,4) on  L¯ P1 P2 . Thus, the point (2, 4) lies on the triangle Aα,θ Bα,θ Cα,θ for α = 0.8352 and θ = 157.32◦ , i.e, A ≡ (0.7726, 2.1193), B = (4.7081, 7.1531) and C ≡ (5.8541, 1.0766).  1 P2 P3 ) = 0.8352. Hence, μ((2, 4)|ΔP In the Fig. 4.1, construction of a fuzzy triangle has been displayed. Different 2 and P 3 . Deeper grey shading 1 , P grey level sets represent different α-cuts of P 1 , represents higher value of α. Totally black points P1 , P2 and P3 are core of P 3 respectively. For ten different values of α (α = 0.1, 0.2, . . . , 0.9, 1.0), 2 and P P 2 (α) and P 3 (α) are shown. The lines A1 B1 , A2 B2 and A3 B3 are parallel and 1 (α), P P 1 , P 2 and passing through P1 , P2 and P3 respectively. Let us consider any α-cut of P 3 , say for example α = 0.4. We note that intersection points of the boundaries of P 2 (0.4) and P 3 (0.4) with the lines A1 B1 , A2 B2 and A3 B3 respectively are 1 (0.4), P P S1 , S2 , S3 and T1 , T2 , T3 . Due to the definition of same points, the points S1 , S2 , S3 1 , P 2 and P 3 with membership value and T1 , T2 , T3 are same points with respect to P

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Fig. 4.1 Construction of a fuzzy triangle

 1 P2 P3 is union of all α = 0.4. According to the definition of fuzzy triangle, ΔP triangles like S1 S2 S3 , T1 T2 T3 , etc. with membership value α = 0.4. Note 4.1 It is to observe that if support of any two vertices of a fuzzy triangle have non-empty intersection, there may be found several same points which are coincident. Corresponding to those same points, the crisp triangle in the support of the fuzzy triangle reduces to a crisp line segment. Another case may happen that though the  1 P2 P3 are different but two or more of supports of vertices of a fuzzy triangle ΔP their core points are identical. In this case since the core of the fuzzy triangle is a crisp line segment, it can form a fuzzy triangle. These are all degenerate cases of fuzzy triangle. So to get a fuzzy triangle, we need to have three fuzzy points having distinct core points. In the following theorem, the α-cut of a fuzzy triangle has been found.  1 P2 P3 be a fuzzy triangle. Its α-cut is the set {x : x ∈  Theorem 4.2.1 Let ΔP 1 (α), where  is a crisp triangle whose vertices are three same points u ∈ P   v ∈ P2 (α) and w ∈ P3 (α)}.

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  1 P2 P3 = α∈[0,1] {α : Proof The theorem is followed form the observation that ΔP 1 , P 2 and P 3 with membership α is a triangle with vertices as same points of P value α}. Note 4.2 The result of the Theorem 4.2.1 directly shows that fuzzy triangle joining three fuzzy points having three distinct core points is unique, since once vertices of fuzzy triangle are changed, several crisp triangles in the support of the fuzzy triangle which eventually construct the fuzzy triangles are going to change and hence fuzzy triangle will have different membership function.

Definition 4.2.1 (Side lengths and vertex angles of a fuzzy triangle) Length of  1 P2 P3 is defined by fuzzy distance (Definition the side of the fuzzy triangle ΔP 2 ), D(  P 2 , P 3 ) and D(  P 3 , P 1 ) respec P 1 , P 2.1.3) between the vertices, i.e., D( p1 and  p2 respectively. The vertex angles of tively; let us denote them as  p3 ,   1 P2 P3 may be defined as the fuzzy angles (Definition 4.5 in Ghosh and ΔP L P1 P2 ,  L P2 P3 ),  ∠( L P2 P3 ,  L P3 P1 ) and  ∠ ( L P3 P1 ,  L P1 P2 ); Chakraborty (2012))  ∠(    and the notations ∠P2 , ∠P3 and ∠P1 respectively may be used to represent them. It is to note that vertex angle  ∠Pi is situated opposite to the side with length  pi , i = 1, 2, 3. Now let us try to find side lengths of a fuzzy triangle. Length of the sides of  1 P2 P3 may be defined by fuzzy distances (Definition 2.1.3) the fuzzy triangle ΔP 2 ), D(  P 2 , P 3 ) and D(  P 3 , P 1 ). Let us denote them  P 1 , P between the vertices, i.e., D(  1 P2 P3 may be defined as p1 and  p2 respectively. The vertex angles of ΔP as  p3 ,   L P1 P2 ,  L P2 P3 ),  ∠( L P2 P3 ,  L P3 P1 ) and  ∠( L P3 P1 ,  L P1 P2 ); and the notations  ∠P2 ,  ∠P3 ∠(  and ∠P1 respectively may be used to represent. It is to note that vertex angle  ∠Pi is situated opposite to the side with length  pi , i = 1, 2, 3. 1 (1, 0), P 2 (2, 0) and  1 P2 P3 be a fuzzy triangle whose vertices P Example 4.2 Let ΔP  P3 (1.5, 4) are as follows. 1 (0) = {(x, y) : (x − 1)2 + 1 is a right circular cone with base P The shape of P 1 2 y ≤ 4 } and vertex (1, 0). 2 is a right circular cone with base P 2 (0) = {(x, y) : (x − 2)2 + y 2 ≤ The shape of P 1 } and vertex (2, 0). 4 3 is a right elliptical cone with base P 3 (0) = {(x, y) : (x − 1.5)2 + The shape of P 2 (y − 2) ≤ 1} and vertex (1.5, 2). 2 (2, 0) and 1 (1, 0), P The same points with membership value α ∈ [0, 1] on P 3 (1.5, 2) are P   cos θ, (1−α) sin θ , Bα,θ : (2 + (1−α) cos θ, (1−α) sin θ) and Aα,θ : 1 + (1−α) 2 2 2 2 Cα,θ : (1.5 + (1 − α) cos θ, 2 + (1 − α) sin θ) respectively, where θ ∈ [0, 2π]. To calculate length of the side  p3 , first let us obtain the pair of inverse L¯ P1 P2 , i.e.,   1 and P 2 are Aα,θ  points in P1 and P2 . The inverse points with membership value on P

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97

and Bα,π+θ . Here, min d(Aα,θ , Bα,π+θ ) = α and max d(Aα,θ , Bα,π+θ ) = 2 − α 0≤θ≤2π

0≤θ≤2π

p3 ∀α ∈ [0, 1]. Thus,  p3 (α) = [α, 2 − α] ∀α ∈ [0, 1]. Hence, membership value of  will be obtained as ⎧ ⎪ if 0 ≤ d ≤ 1 ⎨d μ(d| p3 ) = 2 − d if 1 ≤ d ≤ 2 ⎪ ⎩ 0 elsewhere. Similarly, length of the sides  p1 and  p2 respectively will be L¯ P2 P3 and  L¯ P3 P1 , i.e.,  obtained as ⎧ 2 ⎪ ⎨ 3 d − 0.3744 if 0.5615 ≤ d ≤ 2.0616 p2 ) = 2.3744 − 23 d if 2.0616 ≤ d ≤ 3.5616 μ(d| p1 ) = μ(d| ⎪ ⎩ 0 elsewhere.  1 P2 P3 , let us first To evaluate the vertex angle  ∠P2 of the fuzzy triangle ΔP calculate the angle between the line segments Aα,θ Bα,θ and Bα,θ Cα,θ joining same points of the vertices. Here ∠(Aα,θ Bα,θ , Bα,θ Cα,θ ) = tan−1

2+ (1−α) sin θ 2 . −0.5+ (1−α) cos θ 2

∠P2 (0) = We obtain that the core of the angle  ∠P2 is 75.9627◦ and support is 

−1 2+ (1−α) sin θ

 ◦ ◦ 2

tan

= [51.7839 , 90 ]. 0≤α≤1,0≤θ≤2π −0.5+ (1−α) cos θ 2 Similarly, core and support of the angle  ∠P1 are 28.0749◦ and [22.0228◦ , ◦ 157.6549 ] respectively.  1 P2 P3 =  L P1 P2 ∪  L P2 P3 ∪  L P3 P1 , since the fuzzy line segments We note that ΔP are also defined by collection of crisp line segments adjoining same points of the  1 P2 P3 are defined as length of the fuzzy extreme fuzzy points. Side lengths of ΔP    line segments L P1 P2 , L P2 P3 and L P3 P1 . Obviously, side lengths of a fuzzy triangle are fuzzy numbers, because distance between two fuzzy points, measured by inverse  1 P2 P3 points, is a fuzzy number (Ghosh and Chakraborty 2012). Vertex angles of ΔP    are the angles between fuzzy line segments L P1 P2 , L P2 P3 and L P3 P1 . It is worthy to mention that vertex angles of a fuzzy triangle having vertices as three continuous fuzzy points are fuzzy numbers, since angle between two fuzzy line segments joining two fuzzy points can be easily shown as fuzzy number. But we observe that support of the fuzzy number obtained by adding the vertex angles of a fuzzy triangle may contain angle more than 180◦ . The following example gives one example supporting this observation.  1 P2 P3 , whose vertices are three Example 4.3 Let us consider the fuzzy triangle, ΔP    fuzzy points P1 (3, 2), P2 (1, 1) and P3 (1, 0). Membership functions of them are right circular/elliptical cone with supports as:

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2 1 (3, 2)(0) = {(x, y) : (x−3) + (y−2) ≤ 1}, P 32 0.52 2  P2 (1, 1)(0) = {(x, y) : (x − 1) + (y − 1)2 ≤ 41 } and 3 (1, 0)(0) = {(x, y) : (x − 1)2 + y 2 ≤ 1 }. P 4 2

 1 P2 P3 ∠( L P1 P2 ,  L P2 P3 ) of the fuzzy triangle ΔP Let us suppose vertex angle  ∠P2 =  has to evaluate. 2 (1, 0) and 1 (3, 2), P The same points with membership value α ∈ [0, 1] on P  P3 (1, 1) are sin θ), Aα,θ : (3 + 3(1 − α) cos θ, 2 + 1−α 2 1−α Bα,θ : (1 + 1−α cos θ, 1 + sin θ) and 2 2 Cα,θ : (1 + 1−α cos θ, 1−α sin θ) respectively, where θ ∈ [0, 2π]. 2 2  ◦ Thus,  ∠P2 (0) = θ∈[0,2π],α∈[0,1] |∠(Aα,θ Bα,θ , Bα,θ C α,θ )| = [102.5306 , ◦ 194.0362 ]. A geometric visualization of the scenario is given in Fig. 4.2. Remark 4.2 As vertex angles of the fuzzy triangle are defined as angles between the side line segments of the fuzzy triangle, there is a possibility that vertex angle may contain more than 180◦ angle on its support. It is shown in the Fig. 4.2 that ∠(A0,π B0,π , B0,π C0,π ) = 194.0362◦ . However, we note that if the vertex angle  ∠P2 would have defined by the collection of the angle ∠Bα,θ of the triangles ∠P2 cannot be more Aα,θ Bα,θ Cα,θ for all possible θ and α, the measurement of  than 180◦ on its support.

Fig. 4.2 Addition of the vertex angles may contain more that 180◦ on its support

4.2 Fuzzy Triangle

99

In another way, we may define vertex angle of a fuzzy triangle as follows. Let us 1 , P 2 and P 3 . The vertex angle  1 P2 P3 having vertices as P consider a fuzzy triangle ΔP   1 , ∠P2 = {∠ABC : where ABC is a triangle, A ∈ P ∠P2 may be defined as:  2 , C ∈ P 3 are three same points}. Similarly,  1 ,  3 can be defined. By this B∈P ∠P ∠P definition of fuzzy vertex angle, for the triangle considered in the Example 4.3 we obtain that  ∠P2 (0) = [102.5306◦ , 180◦ ]. However, by this definition also addition ◦ of three vertex angles may  contain more than 180 angle. Thus, in either definition  ∠P2 +  ∠P3 = {∠A + ∠B + ∠C : where ABC is a triangle having ∠P1 +  2 and P 3 }, since right hand side is the crisp number 1 , P vertices as same points of P 180◦ and left hand side is a fuzzy number. Now, let us try to investigate perimeter and area of a fuzzy triangle. Definition 4.2.2 (Perimeter of a fuzzy triangle) Let us consider a fuzzy trian 1 P2 P3 . Fuzzy perimeter of the considered fuzzy triangle may be defined gle ΔP by the following ways. (Method 1) Let us denote the fuzzy perimeter as  δ1 . It may be defined by the membership function: μ(δ| δ1 ) = sup {α : δ is the perimeter of the triangle 2 (0) and w ∈ P 3 (0) as its ver1 (0), v ∈ P formed by three same points u ∈ P 2 ) = μ(w| P 3 ) = α}. 1 ) = μ(v| P tices with μ(u| P (Method 2) In this method, let us denote the fuzzy perimeter as  δ2 . It may be p1 +  p2 +  p3 . defined by:  δ2 = 

Remark 4.3 Here addition  a + b of two fuzzy numbers  a and  b will be performed by applying the concept of same point as defined in Ghosh and Chakraborty (2012). The definition in Ghosh and Chakraborty (2012) for addition of two fuzzy numbers  says that  a + b = {x + y : x, y are same points in  a,  b }. In fact, this addition and extended addition give the same result as shown in Ghosh and Chakraborty (2012). Observation 4.2.1 Fuzzy perimeter obtained by above two methods are not equal,  1 P2 P3 and the evaluation i.e.,  δ1  =  δ2 . It is easily followed from the formation of ΔP  of distance between two fuzzy vertices (points). ΔP1 P2 P3 is formed by taking union of  1 P2 P3 , all crisp triangles whose vertices are same points of the fuzzy vertices of ΔP whereas distance between two fuzzy vertices is evaluated by combining distances p2 +  p3 cannot between inverse points. Thus, addition of the side lengths, i.e.,  p1 +  be equals to the union of all the perimeter of the crisp triangles on the support of  1 P2 P3 . ΔP The following example explores the fact that  δ1  =  δ2 . 1 (1, 0), P 2 (2, 0)  1 P2 P3 be a fuzzy triangle whose vertices P Example 4.4 Let ΔP 3 (1.5, 2) are as follows. All of them has membership function as right circular and P 2 (0) = {(x, y) : 1 (0) = {(x, y) : (x − 1)2 + y 2 ≤ 1 }, P cone with support sets are: P 4 1 3 (0) = {(x, y) : (x − 1.5)2 + (y − 2)2 ≤ 1}. (x − 2)2 + y 2 ≤ 4 }, P

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1 (1, 0), P 2 (2, 0) and The same points with membership value α ∈ [0, 1] on P 3 (1.5, 2) are: P   cos θ, (1−α) sin θ , Aα,θ : 1 + (1−α) 2 2 cos θ, (1−α) sin θ) and Bα,θ : (2 + (1−α) 2 2 where θ ∈ [0, 2π]. Cα,θ : (1.5 + (1 − α) cos θ, 2 + (1 − α) sin θ) respectively,  δ2 , we get  δ1 (0) = α∈[0,1],θ∈[0,2π] |d(Aα,θ , Bα,θ ) + Thus from definition of  δ1 and  δ2 (0) =  p1 (0) +  p2 (0) + d(Bα,θ , Cα,θ ) + d(Cα,θ , Aα,θ )| = [4.1623, 6.0990] and  δ1 (0).  p3 (0) = [0.5616, 3.5616] + [0.5616, 3.5616] + [0, 2] = [1.1232, 9.1232] =  The results of the following two theorems give information to get α-cuts, and hence membership functions  δ1 and  δ2 Theorem 4.2.2  δ1 is a fuzzy number and  δ1 (α) = {δ : δ is the perimeter of the 2 (α) and w ∈ P 3 (α) as its 1 (α), v ∈ P triangle constructed by the same points u ∈ P vertices}. Proof Similar to earlier stated Theorem 2.3. Theorem 4.2.3  δ2 is a fuzzy number and  δ2 (α) =  p1 (α) +  p2 (α) +  p3 (α) ∀ α ∈ [0, 1]. Proof Theorem directly follows from the addition of fuzzy numbers using same points. Now let us define define area of a fuzzy triangle.  of a fuzzy triangle Definition 4.2.3 (Area of a fuzzy triangle) Fuzzy area (Δ)  1 P2 P3 may be defined by its membership function as: μ(|Δ)  = sup {α :  ΔP 2 (0) 1 (0), v ∈ P is the area of the triangle constructed by the same points u ∈ P 1 ) = μ(v| P 2 ) = μ(w| P 3 ) = α}. 3 (0) as its vertices with μ(u| P and w ∈ P

 of the fuzzy triangle in the Example 4.4. Example 4.5 Let us calculate the area (Δ) Area of the triangle Aα,θ Bα,θ Cα,θ for a particular value of θ and α is 21 |2 + 21 (1 −  is α) sin θ|. Thus, support of Δ  = Δ(0)  is 1. and core of Δ

 θ∈[0,2π]

 α∈[0,1]

1 3 5 1 |2 + (1 − α) sin θ| = [ , ] 2 2 4 4

4.2 Fuzzy Triangle

101

The results of the following theorem gives information to get α-cuts, and hence  membership function, of Δ.  is a fuzzy number and Δ(α)  = { :  is the area of the triangle Theorem 4.2.4 Δ 1 (α), v ∈ P 2 (α) and w ∈ P 3 (α) as its vertices}. constructed by the same points u ∈ P Proof Similar to Theorem 5.1 of Ghosh and Chakraborty (2012). In the next, we will introduce similarity of fuzzy triangles (Ghosh and Chakraborty 2013) and some basic fuzzy trigonometric functions using the proposed fuzzy triangle. It has been shown that several well-known trigonometric identities do not hold with proper equality for fuzzy angle.

4.3 Similarity of Fuzzy Triangles In classical trigonometry, two triangles are said as similar if their shapes are alike but sizes are different. That is, a triangle and its enlarged (magnified) versions are similar. Let us now try to generalize this idea in finding similar fuzzy triangles. To do so, we observe that if we enlarge a fuzzy triangle, side lengths of the fuzzy triangle before and after its enlargement can be found easily. But unlike to classical triangles, after the enlargement of a fuzzy triangle, imprecision of its sides also gets enlarged. This happens due to presence of imprecision in the sides. So, in finding similarity of fuzzy triangles, imprecision of the sides are also to be accounted. To measure imprecision  1 P2 P3 say, let us consider one of its side say of the sides of a fuzzy triangle ΔP  L P1 P2 . We consider a line l(x, y) perpendicular to  L P1 P2 (1) at (x, y) ∈  L P1 P2 (1). As  L P1 P2 is a fuzzy line segment, along the line l(x, y) there must exists one fuzzy number (Ghosh and Chakraborty 2012) given by l(x, y) ∩  L P1 P2 . We denote this l3 (x, y) fuzzy number by  l3 (x, y). Thus, corresponding to each (x, y), the function  always gives one fuzzy number. We will say the function defined by (x, y) →  l3 (x, y) L P1 P2 . Similarly, there will be two more as the imprecision function of the side  imprecision functions  l1 (x, y) and  l2 (x, y) corresponding to the sides  L P2 P3 and  L P1 P3 respectively. It is noticeable that when a fuzzy triangle is enlarged, then all of its imprecision functions are magnified by some constant multiplication. Thus, two fuzzy triangles may be said as similar fuzzy triangles if all of their corresponding side lengths and corresponding imprecision functions are constant multiplication of the other.

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4 Fuzzy Triangle and Fuzzy Trigonometry

 1 P2 P3 and ΔQ  1 Q2 Q3 Definition 4.3.1 (Similarity of fuzzy triangles) Let ΔP are two fuzzy triangles.  p1 ,  p2 ,  p3 and  q1 ,  q2 ,  q3 being their side lengths and  l pi ,  lqi for i = 1, 2, 3 are their imprecision functions of the corresponding sides.  1 Q 2 Q 3 are said to be similar if there exists  1 P2 P3 and ΔQ Fuzzy triangles ΔP k ∈ R such that (i)  pi = k qi , ∀ i and lqi (xqi t , yqi t ) ∀ t ∈ [0, 1] for each i = 1, 2, 3, where t (ii)  l pi (x pi t , y pi t ) = k L P2 P3 and (x p1 t , y p1 t ) ∈ is taken as follows. For i = 1, let us consider   L P2 P3 (1). Then the point (x p1 t , y p1 t ) corresponds to that t for which (x p1 t , y p1 t ) = t P2 + (1 − t)P3 , t ∈ [0, 1]. Similar relation will be applied L P1 P2 ,  L P3 P1 ,  L Q1 Q2 ,  L Q 2 Q 3 and  L Q3 Q1 . for the sides   1 P2 P3 and If above two conditions hold true, then one of the fuzzy triangles ΔP  ΔQ 1 Q 2 Q 3 is enlarged or contracted version of another. For enlargement |k| ≥ 1 and for lessening |k| < 1. Remark 4.4 Here question may arise whether the constant k would be fuzzy. Answer is that k will be crisp always, since magnification of one fuzzy triangle can be done by some crisp constant multiplication of all the points on the support of the fuzzy triangle and conversely. i for i = 1, 2, 3 are six fuzzy points. If two i and Q Theorem 4.3.1 Let us suppose P  1 Q 2 Q 3 are similar, then the crisp triangles joining  1 P2 P3 and ΔQ fuzzy triangles ΔP the same points, with membership value α, of the corresponding vertices of the fuzzy triangles are also similar triangles for each α ∈ [0, 1].  1 Q 2 Q 3 are fuzzily similar triangles. Therefore, impre 1 P2 P3 and ΔQ Proof Here ΔP  1 P2 P3 are some constant multiplication cision function and length of each side of ΔP  1 Q2 Q3. of the imprecision function and side length of the corresponding side of ΔQ So, one of the fuzzy triangles is enlarged version of another. Once a fuzzy triangle is enlarged, its all the vertices also magnified by the same measure. Thus there exists i . i = k Q some constant k ∈ R such that P A fuzzy triangle can be viewed as a collection of crisp triangles whose vertices are same points of fuzzy vertices. Thus enlargement of fuzzy triangle means enlargement of corresponding crisp triangles on its support. Let (x1 , y1 ), (x2 , y2 ) and (x3 , y3 ) are 2 and P 3 and 1 , P same points with membership value α ∈ [0, 1] of the fuzzy points P  1 P2 P3 joining these three same points.  pα is the crisp triangle in the support of ΔP Then, it is easy to observe that (kx1 , ky1 ), (kx2 , ky2 ) and (kx3 , ky3 ) are same points 2 and Q 3 ; and the crisp triangle 1 , Q with membership value α of the fuzzy points Q qα with vertices as (kx1 , ky1 ), (kx2 , ky2 ) and (kx3 , ky3 ) will have membership  1 Q 2 Q 3 . Hence the result. value α in the fuzzy triangle ΔQ It is worthy to mention that in the definition of similarity of fuzzy triangles (Definition 4.3.1), side lengths of the fuzzy triangles are in a constant ratio and corresponding

4.3 Similarity of Fuzzy Triangles

103

imprecision functions of the sides are also in a constant ratio. Thus, the definition essentially reflects S-S-S (Side-Side-Side) rule to find similar crisp triangles, since for the core of two similar fuzzy triangles the Definition 4.3.1 reduces to S-S-S rule. However, there are other rules also to investigate similarity of crisp triangles. Let us now try to investigate similarity of fuzzy triangles by these rules, like S-A-S (SideAngle-Side), A-A-A (Angle-Angle-Angle) and A-A-S (Angle-Angle-Side) rules. To examine so, we need to find a construction procedure of fuzzy triangle when its two sides and one angle is given. A construction procedure may be given as follows. Procedure is explained with an example. Suppose we have to construct a fuzzy triangle whose two sides are  3 = (1/3/5) √ √ √ √  and 2 2 = ( 2 /2 2 /3 2) and the angle between those two sides is ‘around  say whose membership 45◦ ’; the fuzzy point of intersection of those two sides is P function is a right circular cone with support {(x, y) : x 2 + y 2 ≤ 1} and vertex at (0, 0). First let us construct the core of the fuzzy triangle. Taking x-axis as one of its side, we obtain that core is the triangle with vertices O : (0, 0), Q : (3, 0), R : (2, 2). Now let us try to obtain a fuzzy point having core at (3, 0) and  3 distance apart from  0). We note that there are infinitely many such fuzzy points. For example some of P(0, 1 (3, 0) with support {(x, y) : (x − 3)2 + y 2 ≤ 1} and membership functhem are R 0.5 (3, 0) with support {(x, y) : (x − 3)2 + y 2 /0.52 ≤ tion is a right circular cone, R  (3, 0) with support 1} and membership function is a right circular cone, and R 2 2 2 {(x, y) : (x − 3) + y / ≤ 1} and membership function is a right circular cone  say, along x-axis is itself a fuzzy where  ∈ (0, 1]. The fuzzy number (2/3/4), R   (3, 0) is  point at (3, 0) which is 3 distance apart from P(0, 0). The fuzzy point R shown in the Fig. 4.3. √ Similarly, there are several fuzzy points having core at (2, 2) and  2 2 dis 0). For instance some of them are Q 1 (2, 2) with support tance apart from P(0, 2 2 2 {(x, y) : (x + y − 4) /2 + (x − y) ≤ 1} and membership function is a right cir0.5 (2, 2) with support {(x, y) : (x + y − 4)2 /22 + (x − y)2 /0.52 ≤ 1} cular cone, Q  (2, 2) with support and membership function is a right circular cone, and Q 2 2 2 2 {(x, y) : (x + y − 4) /2 + (x − y) / ≤ 1} and membership function is a right circular cone where  ∈ (0, 1]. The triangular fuzzy number ((1, 1)/(2, 2)/(3, 3)), √  say, along the line y = x is itself a fuzzy point at (2, 2) which is  Q 2 2 distance  0). Fuzzy point Q  (2, 2) is shown in the Fig. 4.3. apart from P(0,  Q  R with 0 <  < 1, lengths of the sides  L P Q Here for the fuzzy triangles ΔP √   and L are  3 and 2 2 respectively and angle between those sides is ‘around 45◦ ’. P R

This shows that there are infinitely many fuzzy triangles whose two side lengths are √  3 and  2 2 and fuzzy angle between them is ‘around 45◦ ’. Let us measure the angle  ∠Q  P R . We know its value is ‘around 45◦ ’, but let  Q  us try to evaluate its appropriate value. The same points of the fuzzy points P, 1  √ and R with membership value 0 are (cos θ, sin θ), (2 + 2 2 [(2 − ) cos θ + (2 + + ) cos θ + (2 − ) sin θ]) and (3 + cos θ,  sin θ).    , θmax ] where θmin = Thus support of the angle  ∠Q  P R will be given by [θmin

) sin θ], 2 +

1 √ [(2 2 2

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4 Fuzzy Triangle and Fuzzy Trigonometry

√  and R  having distance   0) Fig. 4.3 Two fuzzy points Q 3 and  2 2 respectively from P(0,

 min f (θ), θmax = max f (θ)

0≤θ≤2π

and f (θ) = tan

−1

0≤θ≤2π 2+ 2√1 2 [(2+) cos θ+(2−) sin θ]−sin θ

2+ 2√1 2 [(2−) cos θ+(2+) sin θ]−cos θ

− tan−1

(−1) sin θ . 3

  , θmax ] is an  dependent interval. Therefore, as  varies, support Obviously, [θmin  of ∠Q  P R is also varies. Hence if support of angle ‘around 45◦ ’ and its membership  Q  R for function is known previously, there may not exists any fuzzy triangle ΔP ◦ ◦  which ∠Q  P R is exactly equals to the given 45 (around 45 ). From this example, we note that for given two fuzzy numbers a and  b, a fuzzy angle   θ and a fuzzy point P, a fuzzy triangle may not be found whose two side lengths  and  are  a and  b and corresponding vertex and vertex angle are P θ respectively.  For example, in the above example if  a and b would have been (1/2/3) and  θ= (35◦ /45◦ /55◦ ), then no fuzzy triangle can be found.

Thus, in general, for a given vertex angle and two given side lengths a fuzzy triangle cannot always be determined. So, S-A-S rule cannot always be generalized in fuzzy environment. Main difficulty comes in implementing the S-A-S rule is that only two sides and one angle cannot determine vertices of the fuzzy triangle uniquely.

4.3 Similarity of Fuzzy Triangles

105

Similarly, A-A-A rule cannot be generalized. The main reason for A-A-A rule cannot be generalized is that addition of three vertex angles may not always fixed. Yes, of course this value is ‘around 180◦ , but the spreads differ for different fuzzy triangles. However in A-A-S rule we have the following theorem. Theorem 4.3.2 If for two fuzzy triangles two vertex angles are equal, and length of the sides opposite to the other vertex angle of them are equal or they are constant multiplication of another (i.e. the membership functions of these sides are constant multiplier to another) then the fuzzy triangles are similar.

Proof Let us recall that fuzzy line segments are collection of crisp line segments with varied membership values. According to the assumption of the theorem, for the given sides of the fuzzy triangles, we note that corresponding to each and every crisp line segment lying on the given side of a fuzzy triangle there must exists one crisp line segment, with same membership value, on the corresponding given sides of the  1 P2 P3 another fuzzy triangle. Now let us suppose the given fuzzy triangles are ΔP  1 Q 2 Q 3 ; for them the side lengths and imprecision functions are given for and ΔQ 1 and L P1 P2 and  L Q 1 Q 2 respectively; the given vertex angles are Θ the given side      Θ2 ; the angle Θi is the vertex angle ∠Pi and ∠Q i for i = 1, 2. Let l pα and lqα are L P1 P2 and  L Q1 Q2 two line segments with membership value α on the support of  respectively. Now let us consider two triangles  pα and qα whose two sides are 1 l pα and lqα and two vertex angle on the two extremities of l pα and lqα are θ1 ∈ Θ  and θ2 ∈ Θ2 with membership values α ∈ [0, 1]. We observe that  pα and qα are  1 Q 2 Q 3 = ∨α∈[0,1] qα , the  1 P2 P3 = ∨α∈[0,1]  pα and ΔQ similar triangle. As ΔP   fuzzy triangles ΔP1 P2 P3 , ΔQ 1 Q 2 Q 3 are similar. In the foregoing paragraphs, study has been made for different rules to determine similarity of fuzzy triangles. Let us try to investigate in respect of their area. In particular, how the area of fuzzy triangles are changing when one fuzzy triangle is enlarged. It is to note that as fuzzy triangle is collection of crisp triangles with different membership value and this vertices of this crisp triangles are same points of the vertices of the fuzzy triangle, enlargement of the fuzzy triangles effectively means enlargement of the vertices of the fuzzy triangles. i are fuzzy points such that Q i = k P i , where k is some i and Q Theorem 4.3.3 If P real constant, then following results hold.  1 Q 2 Q 3 (1) and   1 P2 P3 (1), then (kx, ky) ∈ ΔQ l pi (x, y) = k lqi (i) If (x, y) ∈ ΔP   (kx, ky) where l pi and lqi for some i ∈ {1, 2, 3} are imprecision function as defined in the Definition 4.3.1.  1 Q 2 Q 3 are similar.  1 P2 P3 and ΔQ (ii) Fuzzy triangles ΔP   1 P2 P3 and ΔQ  1 Q 2 Q 3 respec (iii) If Δ p and Δq are area of the fuzzy triangles ΔP 2  tively, then Δq = k Δ p .

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4 Fuzzy Triangle and Fuzzy Trigonometry

Proof  1 P2 P3 (1), then (kx, ky) ∈ ΔQ  1 Q 2 Q 3 (1) is triv(i) First part that if (x, y) ∈ ΔP   ially true. For the second part that l pi (x, y) = k lqi (kx, ky) for i ∈ {1, 2, 3}, we lqi (kx, ky) for i = 1 and similar will be the case will prove that  l pi (x, y) = k lq1 (kx, ky), we let (x1 , y1 ), (x2 , y2 ) belong for i = 2, 3. To prove  l p1 (x, y) = k 3 2 with membership value α ∈ [0, 1] and (x3 , y3 ), (x4 , y4 ) belong to P to P with membership value α ∈ [0, 1]. Let us also suppose that (x1 , y1 ), (x3 , y3 ) are same points and (x2 , y2 ), (x4 , y4 ) are same points. Let l13 and l24 are line segments joining (x1 , y1 ), (x3 , y3 ) and (x2 , y2 ), (x4 , y4 ) respectively. We also k k and l24 are line segments joining k(x1 , y1 ), k(x3 , y3 ) and k(x2 , y2 ), suppose l13 k(x4 , y4 ) respectively. Without loss of generality let the line segments l13 and l24 lie on the different sides of the line joining P1 , P2 . For all the assumptions k please refer to the Fig. 4.4. We note that distance between the line segments l13

i = k P i , for k = 3  1 P2 P3 and ΔQ  1 Q 2 Q 3 where Q Fig. 4.4 Fuzzy triangles ΔP

4.3 Similarity of Fuzzy Triangles

107

k and l24 is k times of the distance between l13 and l24 . Thus support of the imprecision functional value l p1 (x, y) will be k times of the support of the imprecision  L¯ P1 P2 . Hence the result. functional value lq1 (kx, ky) for each (x, y) ∈  (ii) This part is clear from the Theorem 4.2.1.  1 P2 P3 , it is the union of all of the crisp triangles  (iii) From the construction of ΔP having vertices as three same points (x1 , y1 ), (x2 , y2 ) and (x3 , y3 ) (say) of the vertices of the fuzzy triangle. We note that if Δ is the area of the crisp triangle , then k 2 Δ will be the area of the triangle having vertices as k(x1 , y1 ), k(x2 , y2 ) and k(x3 , y3 ). Hence the result follows.

Remark 4.5 In this theorem we note that if three vertices of a fuzzy triangle are some constant multiplication (i.e., enlarged or contracted version) of the vertices of another fuzzy triangle, then both the fuzzy triangles will be similar. Thus we may implement this result to determine similarity of fuzzy triangles. This rule may be refereed as V-V-V (Vertex-Vertex-Vertex) rule for fuzzy similarity of fuzzy triangles. It is noteworthy that V-V-V rule also applicable for crisp triangles, since two triangles having vertices (x1 , y1 ), (x2 , y2 ), (x3 , y3 ) and k(x1 , y1 ), k(x2 , y2 ), k(x3 , y3 ) (for some non-zero constant k) are similar. Remark 4.6 From the proof of the above theorem, we obtain that two fuzzy triangles are similar if and only if corresponding to each crisp triangle in the support of a fuzzy triangle there exists one crisp triangle on the support of the another fuzzy triangle with same membership value as the prior triangle and vice versa. Remark 4.7 The result of the above theorem also gives that ‘areas of similar fuzzy triangles are (real) constant multiplication of other’. Note 4.3 It is observed that taking the crisp triangle P1 P2 P3 as prototype, l2 (x, y) fuzzy triangle can be obtained by f -transformation (x, y) →  l1 (x, y), or  l3 (x, y). Thus the defined concept of fuzzy triangle is similar to Zadeh (2009). In the study of similarity of fuzzy it is observed that S-S-S and A-A-S rules can be generalized in the fuzzy environment, but S-A-S and A-A-A rules cannot be generalized.

4.4 Fuzzy Trigonometry  1 P2 P3 To define fuzzy trigonometric functions, let us suppose a fuzzy triangle ΔP    is given and we have to find sin ∠P2 , cos ∠P2 , tan ∠P2 , etc. Here a definition of sin  ∠P2 is studied and other functions can be derived in a similar way. 2 (0), P 3 (0) respectively 1 (0), P Let a, b, c be three same points taken from P 2  (where, a, b, c ∈ R ). Now let us consider the triangle abc in ΔP1 P2 P3 . Let θ be the angle between ab, bc; and n be the foot of perpendicular from a to the

108

4 Fuzzy Triangle and Fuzzy Trigonometry

line bc. Obviously, sin θ = d(a,n) , where d(, ) is the usual Euclidean distance metric. d(a,b)   3 ) = α. If μ(a| P1 ) = α, then μ(b| P2 ) = α and μ(c| P  1 P2 P3 are α, α Since membership values of a, b and n in the fuzzy triangle ΔP and greater or equals to α respectively, membership value of sin θ in sin  ∠P2 may  1 P2 P3 . Thus, be assigned as minimum of membership value of a, b and n in ΔP μ(sin θ| sin  ∠P2 ) = α. ∠P2 is Now, sin ∠P2 can be defined as union of the above sin θs. Therefore, sin defined as follows.  1 P2 P3 , Definition 4.4.1 (Fuzzy sine function) Let for a fuzzy triangle ΔP    ∠P2 = Θ. Then sin Θ may be defined by the membership function: 1 (0), b ∈ P 2 (0), c ∈  = sup{α : s = sin θ = d(a,n) where a ∈ P μ(s| sin Θ) d(a,b) 3 (0) are same points with membership value α and n is the foot of perP pendicular from a to the line joining b and c}.  is a fuzzy number. In the proof, In the next, it is proved that above defined sin Θ 1 (α), :a∈P for 0 < α ≤ 1, the notation A(α) is used to represent the set { d(a,n) d(a,b) 2 (α), c ∈ P 3 (α) are same points and n is the foot of perpendicular from a to b∈P the line joining b and c}. Similar to Theorem 5.1 of Ghosh and Chakraborty (2012)  we can prove that A(α) = sin Θ(α). Before proving the theorem we will observe  1 P2 P3 , sin  ∠P2 may have singleton one surprising fact that for a fuzzy triangle ΔP support.  3),  1 P2 P3 with vertices P(2, Example 4.6 Let us consider a fuzzy triangle ΔP  5) and P(6,  7). All of these three fuzzy points have right circular cone as memP(4, bership functions having bases (x − 2)2 + (y − 3)2 ≤ 41 , (x − 4)2 + (y − 5)2 ≤ 41 and (x − 6)2 + (y − 7)2 ≤ 41 ; and vertices at (2, 3), (4, 5) and (6, 7) respectively. The same points with respect to the above fuzzy points can be represented by a cos θ, 3 + (1−α) sin θ), b = (4 + (1−α) cos θ, 5 + (1−α) sin θ) and c = = (2 + (1−α) 2 2 2 2 (1−α) (1−α) (6 + 2 cos θ, 7 + 2 sin θ) respectively with 0 ≤ θ ≤ 2π, 0 ≤ α ≤ 1. ∠P2 = π4 and sin  ∠P2 is the crisp For any a, b and c: ∠(ab, bc) = π4 . Apparently,  1 1  number √2 . Hence, support of sin ∠P2 is the singleton set { √2 }. Note 4.4 It can be easily perceived that if membership functions and supports of 1 , P 2 , P 3 are identical up to a translation then all of sin  P ∠P1 , sin  ∠P2 and sin  ∠P3 must have singleton support, since in this situation the angles  ∠P1 ,  ∠P2 and  ∠P3 are crisp angles.  evaluated by the Definition 4.4.1 is a fuzzy number. Theorem 4.4.1 sin Θ 2 and P 3 and consider a fuzzy 1 , P Proof Let us take three different fuzzy points P  L P1 P2 and  L P2 P3 . triangle using them as vertices. Let Θ be the fuzzy angle between 

4.4 Fuzzy Trigonometry

109

1 , P 2 , P 3 are fuzzy points, their α-cuts P 1 (α), P 2 (α), P 3 (α) are non-empty As P compact subset of R2 . Hence supremum and infimum of A(α) are attainable at A(α). That is, if those elements are u(α), l(α) respectively then l(α) ∈ A(α) and u(α) ∈ A(α). Therefore, A(α) ⊆ [l(α), u(α)]. We will prove that A(α) = [l(α), u(α)] for 0 < α ≤ 1. To prove this, it is sufficient to prove that A(α) is convex, closed and bounded set. 1 (0), Boundedness of A(α) is trivially true, because it is assumed that the sets P 3 (0) have empty intersection. 2 (0) and P P Now as l(α) ∈ A(α) and u(α) ∈ A(α), obviously convexity of A(α) will imply its closedness. We will prove that A(α) is convex.  is the singleton {s0 } where s0 = sin ∠(P1 P2 , It is easy to notice that core of sin Θ P2 P3 ). We argue that A(α) contains all the points of [l(α), s0 ] and also of [s0 , u(α)]. If s0 = l(α) = u(α) then result is trivially true. If not, then let λ ∈ (0, 1) and t1 , t2 ∈ A(α) with t1 < t2 < s0 . Obviously, l(α) < t1 < λt1 + (1 − λ)t2 < t2 < s0 . Let θ1 = sin−1 (t1 ), θ2 = sin−1 (t2 ), θα = sin−1 (l(α)) and θλ = sin−1 (λt1 + (1 − λ)t2 ), where sin−1 not necessarily represents principle value. We took θ1 such a manner that 0 ≤ θ1 < θ0 ≤ π. The similar restriction also followed for θ2 , θλ and θα . It can be easily  is observed that 0 ≤ θα < θ1 < θλ < θ2 < θ0 ≤ π. As membership function of Θ  Hence sin(θλ ) ∈ A(α), i.e. λt1 + (1 − λ)t2 ∈ continuous, it follows that θλ ∈ Θ(α). A(α). So [l(α), s0 ] ⊆ A(α). Similarly we can prove that [s0 , u(α)] ⊆ A(α). Hence A(α) = [l(α), u(α)], a closed bounded interval. Therefore, membership  is upper semi-continuous. function of sin Θ i (α) for i = 1, 2, 3. Therefore, A(β) i (β) ⊆ P Let 0 < α ≤ β ≤ 1. Apparently, P ⊆ A(α), i.e., [l(β), u(β)] ⊆ [l(α), u(α)]. This implies, l is an increasing function and u is a decreasing function. On the other hand, apparently, A(0) = [l(0), u(0)]  is and A(1) = {θ0 }. Obviously, membership value of sin θ0 in the fuzzy set sin(Θ) one. Hence the result. Here a natural question may arise, whether there exists any relation between the  evaluated by extension principle and by the Definition 4.4.1? Result value of sin Θ of the following theorem finds this relation. Theorem 4.4.2 Let us consider a fuzzy triangle constructed by three different 1 , P 2 , P 3 . Let Θ  be the fuzzy angle between  fuzzy points P L P1 P2 and  L P2 P3 and   is evaluated by the extension principle. Then S(α) = sin(Θ)(α), where sin(Θ) S(α) is identically equal to A(α) for 0 ≤ α ≤ 1. = Proof The theorem is followed from the fact that sin Θ  {sα : sα = sin Θα }.

 α∈[0,1]

A(α) =

 α∈[0,1]

Therefore, the trigonometric sine functions evaluated by extension principle and by the Definition 4.4.1 are identical.  the other fuzzy trigonometric functions, like In a similar way of defining sin Θ, cosine, tangent, etc. can also be defined for fuzzy angles.

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4 Fuzzy Triangle and Fuzzy Trigonometry

 may have discontinuous membership funcHere it is surprising to note that sin Θ  is continuous. Following example is an counter tion even if membership function of Θ example of this fact.  = (0/ π /π). Then sin Θ  has following membership function Example 4.7 Let Θ 4 which is discontinuous.  = μ(s| sin Θ)

⎧ 4 sin−1 (s) ⎪ ⎨ π

−1

⎪ ⎩

if 0 ≤ s
α. Proof It can be argued that (i) μ(E|C 2 ) < α. Thus, there exists (x4 , y4 ) in C 2 (0) such that (i) On contrary, let μ(E|C   (x4 , y4 ) ∈ E and μ((x4 , y4 )|C2 ) < α. Let μ((x4 , y4 )|C2 ) = β. As (x4 , y4 ) ∈ E and E is a circle joining three same points with membership value α, 2 ) = sup{δ : where (x, y) lies on the circle joining three same μ((x4 , y4 )|C

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5 Fuzzy Circle

Y P3 ( A3 )

S3

R3 L3

P1 ( A1 )

A1 L1

R1

S1

Q3

A3

P2 ( A2 )  2

Q

 1

Q

L2

R2

S 2

A2 X

O E C2 (1)

F

1 (A1 ), P 2 (A2 ) and P 3 (A3 ) Fig. 5.5 Fuzzy circle through P

points with membership value δ} ≥ α. But β < α. A contradiction arises. Thus, 2 ) < α. μ(E|C  = min{α : where (x, y) lies on E and (ii) This part is obvious, since μ(E|C) 2 ) = α} and all the points (x1 , y1 ), (x2 , y2 ) and (x3 , y3 ) lie on E. μ((x, y)|C 2 ) = α. Hence, μ(E|C Figure 5.6 displays the constructed fuzzy circle through the three fuzzy points considered in the Fig. 5.5. Membership values are shown by optical density of the grey levels. Totally black circle passing through A1 , A2 and A3 represents the core 2 shows that on C 2 if a line perpendicular to C 2 (1) is considered, 2 (1). The figure of C C  then along that line there exists a fuzzy number on the support set C2 (0). For example,  ∩ L S is if the line L S is considered in the Fig. 5.6, then intersecting line segment C an L R-type fuzzy number (Q/E/R) L R along L S. Thus, the whole fuzzy circle can be visualized as a three-dimensional figure (basically a subset of R2 × [0, 1]) whose 2 (1) is a fuzzy number such as (Q/E/R) L R . cross section across C 2 ), for any point (x0 , y0 ) It may be noted that due to definition of μ((x, y)|C  i ), i = 1, 2, 3. Therefore,  in Pi (0), μ((x0 , y0 )|C2 ) is greater or equal to μ((x, y)| P   3 , the surface of z =  though C2 is constructed with the three fuzzy points P1 , P2 and P 2 ),  μ((x, y)| P(Ai )) must always remain in the interior of the surface z = μ((x, y)|C  and a portion of the surface of z = μ((x, y)| P(Ai )) will touch the surface of z =

5.3 Construction of Fuzzy Circle

133

2 in the Example 5.6 Fig. 5.6 Fuzzy circle C

2 ). Apparently, there exist some points in P(A  i )(0) whose membership μ((x, y)|C   grade in P(Ai ) are less than that in C2 . For example, in the Fig. 5.6 the point V = 1 , whereas the figure directly (−0.5000, 0.5670) has membership value zero on P  shows that μ(V |C2 ) is a positive value. 2 can be observed as a union of fuzzy points also. To see The fuzzy circle C 2 (0). Take a convex region on the so, consider a fuzzy number (Q/E/R) L R on C  support set C2 (0) in such a way that all the points on the line segment Q R, except Q and R, are interior points of this convex region. Now it can be defined that a fuzzy  say on such a convex region by the membership function μ((x, y)| P)  = point P   μ((x, y)| (Q/E/R) L R ) when (x, y) ∈ Q R, μ((x, y)| P) ≤ μ((x, y)|C2 ) otherwise, θ ) = 1 only at E. Several such fuzzy points at (Q/E/R) L R can be and μ((x, y)| P 2 (1), many fuzzy obtained through changing the convex region. Varying E on C 2 .  points on C2 (0) will be obtained. Collection of all these fuzzy points is the entire C  Thus, C2 is a collection of fuzzy points. 1 (−1, 1), P 2 (3, 1) and P 3 (1, 4) be three Example 5.6 (Fuzzy circle (Method 2)) Let P 1 , fuzzy points whose membership functions are right circular cones. The bases of P 2 2 (y−1) (x−3) 2 2 3 are {(x, y) : (x + 1) + 2 and P ≤ 1}, {(x, y) : 1/8 + (y − 1) ≤ 1} and P 1/9 2 2 {(x, y) : (x − 1) + (y − 4) ≤ 1}, respectively. Vertices of membership functions

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5 Fuzzy Circle

1 , P 2 and P 3 are (−1, 1), (3, 1) and (1, 4), respectively. For a particular α in of P 2 and P 3 are 1 , P [0, 1], same points with membership value α on P α α , y1θ )= Q α1θ : (x1θ

Q α2θ Q α3θ



− 1 + (1 − α) 

cos θ

, 1 + (1 − α) 

sin θ



, 1 + 8 sin2 θ 1 + 8 sin2 θ

sin θ cos θ α α , 1 + (1 − α) √ and : (x2θ , y2θ ) = 3 + (1 − α) √ 1 + 7 cos2 θ 1 + 7 cos2 θ α α : (x3θ , y3θ ) = (1 + (1 − α) cos θ, 4 + (1 − α) sin θ) where θ ∈ [0, 2π].

The circle, E θα say, passing through Q α1θ , Q α2θ and Q α3θ can be determined by the equation x 2 + y 2 + 2gθα x + 2 f θα y + cθα = 0,   α 2  α  α 2 − x + y 1  1θ 1θ   α 2  2y1θ 1  α 2 α α where gθ = k  −x2θ + y2θ  2y2θ 1 , f θα =  − x α 2 + y α 2 2y α 1  3θ 3θ 3θ  α    2x 2y α − x α 2 + y α 2  1θ 1θ 1θ 1θ     α α α 2 α 2  2y2θ −x2θ + y2θ cθα = k1  2x2θ   2x α 2y α − x α 2 + y α 2  3θ 3θ 3θ 3θ

 α  α 2   α 2   2x − x + y 1θ 1θ 1θ   1  α 2 1  α α 2 2x2θ −x2θ + y2θ  1 , k   2x α − x α 2 + y α 2 1  3θ 3θ 3θ

 α   2x 2y α 1  1θ   1θ α α 2y2θ 1 . and k =  2x2θ  2x α 2y α 1  3θ 3θ

1 , P 2 and P 3 is the union of all possible circles The fuzzy circle passing through P E θα , i.e., 2 = C





{(x, y) : x 2 + y 2 + 2gθα x + 2 f θα y + cθα = 0}.

α∈[0,1] θ∈[0,2π]

2 is the circle 2 is depicted in the Fig. 5.6. Core of C The figure of the fuzzy circle C {(x, y) : (x − 1.000)2 + (y − 1.8334)2 = 2.16672 }.

2 ) 5.3.2 Construction of the Membership Function µ(.| C 2 ) may not always be simple to be evaluate because The closed form of μ((x, y)|C the membership value at a particular point is the supremum of a set of real numbers obtained by solving nonlinear equations. It may be noted that Definition 5.3.2 and 2 ) = sup{α : where (x, y) lies in a circle passing Theorem 5.3.2 imply μ((x, y)|C 2 and P 3 with membership value α}. Thus, to  through three same points in P1 , P  obtain μ((x, y)|C2 ), three generic same points with membership value α ∈ [0, 1] must first be taken. Then all possible α values for which (x, y) lies on the circle

5.3 Construction of Fuzzy Circle

135

joining three same points with membership values are identified. Evaluation of α may require solving nonlinear equation. The supremum of all of these α values is the 2 ). The circle corresponding to which the supremum membership value of μ((x, y)|C is attained is called as the adjoining circle of the point (x, y). The following example illustrates how to find membership value of a point in a fuzzy circle passing through three fuzzy points. 2 in Example 5.6. Suppose it is required to Example 5.7 Consider the fuzzy circle C 2 . Let Q α , Q α and Q α be same points evaluate the membership value at (3, 3) on C 1θ 2θ 3θ with membership value α ∈ [0, 1]. Here Q α1θ , Q α2θ and Q α3θ have same expressions as that in Example 5.6. The circle, E θα , passing through Q α1θ , Q α2θ and Q α3θ can be determined by the equation x 2 + y 2 + 2gθα x + 2 f θα y + cθα = 0, where gθα , f θα and cθα are as displayed in Example 5.6. If the point (3, 3) lies on the circle E θα , then it must be that 18 + 6gθα + 6 f θα + cθα = 0, which implies that A(θ)(1 − α)3 + B(θ)(1 − α) + C(θ) = 0 for some functions A(θ), B(θ) and C(θ). Being a third degree equation in α, this equation will give three parametric values of α. Let α = f 1 (θ), f 2 (θ) and f 3 (θ). Satisfying the constraint that α is real valued and must lie in [0, 1], some domain of f 1 (θ), f 2 (θ) and f 3 (θ) will be obtained. Let this domain be D. Hence according to the Definition 5.3.2, 2 ) = maxθ∈D⊂[0,2π] { f 1 (θ), f 2 (θ), f 3 (θ)} = 0.7800 (this maximum value μ((3, 3)|C is attained for θ = 62.6415◦ . The adjoining circle for the point (3, 3) is x 2 + y 2 − 2.0747x − 3.9838y + 1 , 0.1841 = 0. This circle passes through the same points (−0.99645, 1.0687) ∈ P 3 which are corresponding to α = 2 and (3.0616, 1.1191) ∈ P (1.1011, 4.1954) ∈ P 0.7800 and θ = 62.6415◦ . 2 (α) = {(x, y) : where (x, y) lies on a circle passing through three Theorem 5.3.3 C 2 (α) and P 3 (α)}. 1 (α), P same points on P Proof Define a set A(α) = {(x, y) : where (x, y) lies on a circle passing through  2 (α) and P 3 (α)}. It will be proved that C 2 (α) = A(α) three same point on P(α), P  for all α ∈ (0, 1]. It will then imply that C2 (α) = A(α) for all α ∈ [0, 1]. 2 (α) ⊆ A(α) and A(α) ⊆ C 2 (α) for all α ∈ (0, 1]. It will be shown that C  Take any α in (0, 1]. To prove C2 (α) ⊆ A(α), we take a point, (x, y) say, on 2 ) ≥ α. Let μ((x, y)|C 2 ) = β. Then either β > α or β = α. 2 (α). Then μ((x, y)|C C 2 ) (Definition 5.3.2) a γ must In the case β > α, due to the definition of μ((x, y)|C be obtained with α < γ ≤ β such that (x, y) ∈ A(γ).

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5 Fuzzy Circle

i (γ) ⊆ From the construction of A(α), it can be seen that A(γ) ⊆ A(α) since P 2 (α) ⊆ i (α) for each i = 1, 2, 3. Therefore, (x, y) lies in A(α). Thus, in this case, C P A(α). 2 ) = In the case when β = α, take a point (x, y) in A(α). We recall that μ((x, y)|C 1 (0), sup{t : (x, y) lies on the circle passing through three same points (x1 , y1 ) ∈ P 2 (0) and (x3 , y3 ) ∈ P 3 (0) with membership value t } = β = α. There(x2 , y2 ) ∈ P 1 (0), (x2n , y2n ) ∈ P 2 (0) fore, there must exist sequence of same points (x1n , y1n ) ∈ P 3 (0) such that and (x3n , y3n ) ∈ P (i) (x, y) lies on the circles through (x1n , y1n ), (x2n , y2n ) and (x3n , y3n ) for each n, 1 ) = μ((x2n , y2n )| P 2 ) = μ((x3n , y3n )| P 3 )) = tn for each n and (ii) μ((x1n , y1n )| P (iii) {tn } be a non-decreasing sequence that converges to β. Therefore, for any > 0, there exists k ∈ N such that β − < tn for all n ≥ K . Thus for any n ≥ K it must be that (x, y) in A(tn ) ⊆ A(β − ). But is arbitrary. So, (x, y) lies in A(β). Hence, in this case, (x, y) belongs to 2 (α) ⊆ A(α). A(α). So, C 2 (α). Let (x, y) be in A(α). Then from the It can now be argued that A(α) ⊆ C  ≥ α. Thus,  definition of A(α) and μ((x, y)|C2 ) it can be noticed that μ((x, y)|C)   (x, y) belongs to C2 (α) and hence A(α) is a subset of C2 (α). 2 (α) = A(α) for all α in (0, 1] and hence for all α in [0, 1]. Therefore, C 2 (α) is closed set, since so Note 5.3 It may be noted from the Theorem 5.3.3 that C 2 (α) is an arc-wise connected set. i (α), i = 1, 2, 3. Also C are P 2 . Now it will be tried to define and analyze center for the fuzzy circle C 2 ) Let C 2 be a fuzzy circle passing Definition 5.3.3 (Center of the fuzzy circle C 2 and P 3 . Center of C 2 , C  say, can be defined 1 , P through three fuzzy points P by its membership function as  = sup{α : where c is center of a circle passing through three μ(c|C) 2 and P 3 with membership value α}. 1 , P same points on P

2 ) Consider the fuzzy circle C 2 in Example Example 5.8 (Center of the fuzzy circle C α α  2 and 5.6. The circle, E θ , passing through three same points Q 1θ ∈ P1 , Q α2θ ∈ P α  Q 3θ ∈ P3 is determined by the equation x 2 + y 2 + 2gθα x + 2 f θα y + cθα = 0, where gθα , f θα and cθα are as given in Example 5.6.

5.3 Construction of Fuzzy Circle

137

 say, of C 2 is given by According to the Definition 5.3.3, the center, C = C



{(−gθα , − f θα )}.

θ∈[0,2π] α∈[0,1]

 is depicted in Fig. 5.7. Core of C  is (1, 1.8334). Different grey level Fuzzy center C  represents sets represent various α-cuts. A little darker curve in the support of C   boundary of C(0.3). The set C(0.3) has appeared to be a non-convex curve.   As C(0.3) in the Example 5.8 is non-convex (in fact each C(α) is non-convex for α in [0, 1)), fuzzy center of a fuzzy circle may not always be a fuzzy point. Though  looses its convexity, the following theorem shows that α-cut of the fuzzy center C  the α-cuts C(α) are compact and they form a family of nested compact sets.  be the center of a fuzzy circle C 2 passing through three fuzzy Theorem 5.3.4 Let C 2 and P 3 . Then for each α in [0, 1], 1 , P points P  (i) C(α) = {c : where c is center of a circle passing through three same points in 2 (α) and P 3 (α)}, 1 (α), P P 2 and P 3 are collinear then C(α)  1 , P is compact (ii) if no pair of same points in P and connected, and  2 ) ⊆ C(α  1 ). (iii) for 0 ≤ α1 ≤ α2 ≤ 1, C(α Proof (i) The proof is similar to that of Theorem 5.3.3. α α 1 (α), (x α , y α ) ∈ P 2 (α) and (x α , y α ) ∈ P 3 (α) be same (ii) Let (x1θ , y1θ )∈ P 2θ 2θ 3θ 3θ α α α , y1θ ), points with membership value α. The circle, E θ say, passing through (x1θ α α α α (x2θ , y2θ ) and (x3θ , y3θ ) can be determined by the equation x 2 + y 2 + 2gθα x + 2 f θα y + cθα = 0

where gθα =

    − x α 2 + y α 2 2y α  1θ 1θ   1θ    − x α 2 + y α 2 2y α  2θ 2θ   2θ    − x α 2 + y α 2 2y α 3θ 3θ  3θ  2x α 2y α 1    1θ 1θ      2x α 2y α 1    2θ 2θ      2x α 2y α 1  3θ 3θ



1    1    1

, f θα =

    2x α − x α 2 + y α 2  1θ 1θ 1θ      2x α − x α 2 + y α 2  2θ 2θ 2θ      2x α − x α 2 + y α 2 3θ  3θ 3θ   2x α 2y α 1   1θ 1θ      2x α 2y α 1   2θ 2θ      2x α 2y α 1  3θ 3θ



1   

1   

1

.

From part (i) and Definition 5.3.3 we have  C(α) =

 

β

β

− gθ , − f θ

 : β ∈ [0, α], θ ∈ [0, 2π] .

 β β Define a function F : [0, α] × [0, 2π] → R2 by F(β, θ) = − gθ , − f θ . i (α) is a compact convex set, its boundary must be a continuous curve, i = As P α α 1, 2, 3. Therefore, the functions xiθ and yiθ are continuous for each i = 1, 2, 3.

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1 , P 2 and P 3 in the Example 5.8 Fig. 5.7 Center of the fuzzy circle through P

1 , P 2 and P 3 are collinear. Thus, Due to the hypothesis, no pair of same points in P β β expressions of gθ and f θ directly show that the function F(β, θ) is continuous in [0, α] × [0, 2π]. As [0, α] × [0, 2π] is a compact and connected set, being a continuous image, the set F([0, α] × [0, 2π]) must be a compact and connected set. Therefore  C(α) = F([0, α] × [0, 2π]) is compact and also connected. i (α1 ) for each i = 1, 2, 3. i (α2 ) ⊆ P (iii) This part is followed from part (i) and P 2 will be defined and investigated. Next, radius of the fuzzy circle C 2 ) Let C 2 be a fuzzy circle passing Definition 5.3.4 (Radius of the fuzzy circle C 2 and P 3 . Radius of C 2 ,  1 , P r say, can be defined through three fuzzy points P by its membership function as μ(r |˜r) = sup{α : where r is radius of a circle passing through three 2 and P 3 with membership value α}. 1 , P same points on P

2 ) Consider the fuzzy circle C 2 in Example Example 5.9 (Radius of the fuzzy circle C 2 is the union of all the circles E α for θ ∈ [0, 2π] and α ∈ [0, 1]. Let r˜ be the 5.6. C θ 2 . According to the Definition 5.3.4, radius of C

5.3 Construction of Fuzzy Circle

r˜ =

139

   rαθ : where rαθ is radius of the circle E αθ θ∈[0,2π] α∈[0,1]

=

   2 2 gαθ + f αθ − cαθ .

θ∈[0,2π] α∈[0,1]

Support of r˜ is [1.9721, 2.4120] and the core is {2.1667}. Suppose membership value of a point, r = 2.36 say, in r˜ (0) = [1.9721, 2.4120] has to be identified. 2 2 Corresponding to r = 2.36 it must be that gαθ + f αθ − cαθ = 2.362 . This equation may be satisfied by various values of α and θ. By definition of μ(.|˜r ), the value 2 2 μ(2.36|˜r ) is the supremum of the set of all possible α which satisfy gαθ + f αθ − cαθ = 2.362 . Here one can verify that μ(2.36|˜r ) = 0.1937. One can easily get different α-cuts of r˜ as r˜ (0.1) = [1.9894, 2.3856], r˜ (0.2) = [2.0067, 2.3596], r˜ (0.3) = [2.0246, 2.3339], r˜ (0.4) = [2.0429, 2.3087], r˜ (0.5) = [2.0621, 2.2839], r˜ (0.6) = [2.0820, 2.2595], r˜ (0.7) = [2.1024, 2.2356], r˜ (0.8) = [2.1233, 2.2121] and r˜ (0.9) = [2.1447, 2.1891]. 1 , P 2 and P 3 are collinear, then radius Theorem 5.3.5 If no pair of same points in P     r˜ of the fuzzy circle C2 passing through P1 , P2 and P3 is a fuzzy number. Proof In this proof the notations in the proof of Theorem 5.3.4 will be followed. Theorem 5.3.4 and the definition of r˜ show that r˜ (α) = {r : where r is the radius of E θα for θ ∈ [0, 2π]}. It may be noted that r˜ (α) =

 

  2 2 R(β, θ) where R(β, θ) = gβθ + f βθ − cβθ .

θ∈[0,2π] β∈[0,α]

1 , P 2 and P 3 are collinear, Due to the assumption that no pair of same points in P β β expressions of gθ and f θ imply that the function R(β, θ) is continuous in [0, α] × [0, 2π]. As [0, α] × [0, 2π] is a compact and connected set, being a continuous image, the set R([0, α] × [0, 2π]) must be a compact and connected subset of the real line. Therefore, r˜ (α) is a closed and bounded interval for each α ∈ [0, 1]. Let r˜ (α) = [a(α), c(α)] and r˜ (0) = [a, c]. Thus, μ(r |˜r) = 0 for all r not in [a, c]. It is obvious from the description of r˜ (α) that for 0 ≤ α1 ≤ α2 ≤ 1, r˜ (α2 ) ⊆ r˜ (α1 ) i (α1 ) for i = 1, 2, 3. Therefore, as α increases, a(α) increases and i (α2 ) ⊆ P since P c(α) decreases. Now, for all t ∈ R, the set {r : μ(r | r ) ≥ t} is closed and bounded. Therefore, the  is upper semi-continuous. membership function of D

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1 (1) = (a1 , b1 ), P 2 (1) = (a2 , b2 ) and P 3 (1) = (a3 , b3 ). Now, if b be the Let P r (1) = {b}. radius of the circle passing through (a1 , b1 ), (a2 , b2 ) and (a3 , b3 ) then  Hence, r˜ is a fuzzy number.

5.4 Discussion and Comparison In the proposed study, following properties of fuzzy circles are obtained: (i) A fuzzy circle is always normalized and its core is a crisp circle. (ii) If a line perpendicular to the core of fuzzy circle is considered, then a fuzzy number can be found along that line. (iii) Fuzzy circle can be viewed as a collection of crisp points with varied membership values or as a collection of fuzzy points. (iv) For a fuzzy circle, its α-cuts must be closed, connected and arc-wise connected. However, α-cuts may not always be convex. (v) Membership function of a fuzzy circle is upper semi-continuous. Since, all its α-cuts are closed. (vi) Center of a fuzzy circle passing through three fuzzy points may not be a fuzzy point. However, the fuzzy center satisfies all the properties of a fuzzy point except convexity for α-cuts. (vi) Radius of a fuzzy circle passing through three fuzzy points must be a fuzzy number. In the given construction procedures for fuzzy circles, two formulations were proposed. It is now proposed investigate interrelations between the two forms. It may be noted that the formulated fuzzy circle by Method 1 cannot be equivalent to the fuzzy circle by Method 2 and vice versa, in general. Since, by Method 2 center of a fuzzy circle passing through three fuzzy points may not turn out to be a fuzzy point, whereas in Method 1 it is a prior assumed that center of a fuzzy circle is a fuzzy point. Similarly, in conversion of Method 1 to Method 2, it can be seen that for 1 (α) cannot be perceived as collection of crisp circles each α in [0, 1], the α-cut C 1 (α) with different membership values. Because, in the Fig. 5.4, it is observed that C β β is union of all possible curves which are locus of the points Uθ or Vθ of membership value β for all β in [0, α]. Though there is no general equivalence relation between the fuzzy circles by Method 1 and Method 2, following theorem shows a special case 1 and conversely C 1 is equivalent to C 2 . 2 is equivalent to C when C 1 (a1 , b1 ), P 2 (a2 , b2 ) and P 3 (a3 , b3 ) be three fuzzy points whose Theorem 5.4.1 Let P 2 2 2 bases are {(x, y) : (x − a1 ) + (y − b1 ) ≤ r }, {(x, y) : (x − a2 )2 + (y − b2 )2 ≤ r 2 } and {(x, y) : (x − a3 )2 + (y − b3 )2 ≤ r 2 }, respectively. Let membership func1 , P 2 and P 3 be right circular cones and no pair of same points in P 1 , P 2 tions of P  and P3 are collinear. Then 2 is a fuzzy point and its radius is a fuzzy number, (i) center of the fuzzy circle C 1 and 2 is a C (ii) C 1 is a C 2 . (iii) C

5.4 Discussion and Comparison

141

1 , P 2 and P 3 are Proof Same points with membership value α for the fuzzy points P α , y α ) = (a + r (1 − α) cos θ, b + r (1 − α) sin θ), (x1θ 1 1 1θ

α , y α ) = (a + r (1 − α) cos θ, b + r (1 − α) sin θ) and (x2θ 2 2 2θ

α , y α ) = (a + r (1 − α) cos θ, b + r (1 − α) sin θ), respectively, for θ ∈ [0, 2π]. (x3θ 3 3 3θ

α α α α The center, (−gθα , − f θα ) say, of the circle passing through (x1θ , y1θ ), (x2θ , y2θ ) α α θ θ and (x3θ , y3θ ) is given by −gα = −g1 + r (1 − α) cos θ and − f α = − f 1 + r (1 − α) sin θ, where      a 2 + b2 2b 1   2a a 2 + b2 1   1  1  1 1 1 1   2   2 2 2  a + b 2b 1   2a a + b 1  2  2 2  2  2 2   2    a + b2 2b3 1   2a3 a 2 + b2 1  3 3 3 3  g1 =  and f 1 =  .    2a1 2b1 1   2a1 2b1 1       2a 2b 1   2a 2b 1  2 2  2 2         2a3 2b3 1   2a3 2b3 1   of the fuzzy circle C 2 is given by its According to Theorem 5.3.4, center C constituent α-cuts      − g1 + r (1 − β) cos θ, − f 1 + r (1 − β) sin θ : θ ∈ [0, 2π] . C(α) = β∈[0,α]



 = β for More precisely, μ (−g1 + r (1 − β) cos θ, − f 1 + r (1 − β) sin θ)  C each θ ∈ [0, 2π]. 2 is given by According to Theorem 5.3.4, in this result, the fuzzy radius r˜ of C r˜ (α) =

 

 2 2 gαθ + f αθ − cαθ .

θ∈[0,2π] β∈[0,α]

r is trivially a symmetric fuzzy number with respect to its core R =  Thus,  g1 2 + f 1 2 − c1 , where g1 , f 1 are as mentioned above and c1 is c10 . Let λ and β be left and right spread of r˜ . Then, r˜ can be presented by (r − λ/r/r + β) L R for  2 2 some L and R. Here λ and β are both equal to g00 + f 00 − c00 − g1 2 + f 1 2 − c1 .  and radius r˜ . Therefore, from Theorem 5.3.1, 1 be the circle with center C Now let C 1 (α) = C

 γ∈[0,α], θ∈[0,2π]

 (− f 1 + (R − λL −1 (γ)) cos θ, −g1 + (R − λL −1 (γ)) sin θ),  (− f 1 + (R + β R −1 (γ)) cos θ, −g1 + (R + β R −1 (γ)) sin θ) .

Similarly, Theorem 5.3.3 gives,   2 (α) = (x, y) : (R − λL −1 (α))2 ≤ (x + f 1 )2 + (y + g1 )2 ≤ (R + β R −1 (α))2 . C

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5 Fuzzy Circle

1 (α) and C 2 (α) are the same. Therefore, C 1 = C 2 . It may be noted that C 1 , P 2 and Note 5.4 In the Theorem 5.4.1, if radius of the bases of the fuzzy points P 2 may not be a fuzzy point. For instance, 3 are different, then the fuzzy center of C P 2 (1, 4) and P 3 (3, 1) are taken whose circular bases 1 (−1, 1), P if three fuzzy points P  is non-convex have radii 1, 2 and 4 respectively, then one can easily see that C(0)  is not a fuzzy point. and hence C Now let us compare the proposed fuzzy circles with the existing formulations. It is reported in introduction that several ideas on fuzzy circles or fuzzy conics are proposed but construction procedure of fuzzy circles is given only in the work by Buckley and Eslami (1997b). Thus, the comparison will be made only with Buckley and Eslami. The paragraphs that follow include a pointwise comparison. 1 by Method 1 • Fuzzy circle C In order to define fuzzy circle, Method 1 generalizes basic definition of classical circle. For a given fuzzy center C˜ and a radius r˜ , Definition 5.3.1 defines a fuzzy circle as collection of fuzzy numbers along different lines which have exactly r˜ distance from the center. On the other hand, according to the idea of Buckley and Easlami, when a fuzzy center C˜ and a radius r˜ are given, a fuzzy circle can be defined as  ˜ α∈[0,1] {E : where E is a circle with center at C(α) and radius at r˜ (α)}. Buckley and Easlami’s definition does not follow that distance between the fuzzy points / fuzzy numbers on the support of the constructed fuzzy circle is exactly r˜ distance apart from the fuzzy center. In contrast, the proposed definition follows. For example, ˜ 0) say take a fuzzy number ( 21 /2/4) say for fuzzy radius r˜ and a fuzzy point C(0, ˜ as center of the fuzzy circle, where C(0, 0) has right circular cone as membership function and base as {(x, y) : x 2 + y 2 ≤ 1}. Support of the fuzzy circles by the proposed definition and by Buckely-Eslami’s definition are {(x, y) : 21 ≤ x 2 + y 2 ≤ 4} and {(x, y) : x 2 + y 2 ≤ 5} respectively. Along the line siny θ = cosx θ , the fuzzy num  ˜ ber on C(0, 0) is A˜ θ = (− cos θ, − sin θ) (0, 0) (cos θ, sin θ) . On the support of the fuzzy circle by Definition 5.3.1, the fuzzy number along siny θ = cosx θ   is B˜ θ = ( 21 cos θ, 21 sin θ) (2 cos θ, 2 sin θ) (3 cos θ, 3 sin θ) . According to Buckley-Eslami’s fuzzy circle, the fuzzy number along siny θ = cosx θ on the support   of the fuzzy circle is B˜ θB E = (0, 0) (2 cos θ, 2 sin θ) (5 cos θ, 5 sin θ) . It may ˜ A˜ θ , B˜ B E ) = (0/2/6) = r˜ . ˜ A˜ θ , B˜ θ ) = ( 1 /2/4) = r˜ . But D( be noted that D( θ 2 2 by Method 2 • Fuzzy circle C    The fuzzy circle passing through three  fuzzy points P1 , P2 and P3 , according to the  Definition 5.3.2, is given by C2 = {E : where E is a fuzzy circle passing through 2 (0) and P 3 (0)}. Thus, evaluation of membership value 1 (0), P three same points in P  μ((x0 , y0 )|C2 ) of a particular point (x0 , y0 ) is obtained by taking supremum of the membership values of same points that lie on the circles on which (x0 , y0 ) lies. 1 , In contrast,Buckley-Eslami’s approach defines a fuzzy circle passing through P    P2 and P3 as {F : where F is a fuzzy circle passing through three points in P1 (0), 3 (0)}. Therefore, evaluation of membership value μ((x0 , y0 )|C 2 ) of a 2 (0) and P P

5.4 Discussion and Comparison

143

particular point (x0 , y0 ) is obtained by taking supremum over the minimum of the membership values of the three points that lie on the circles on which (x0 , y0 ) lies. It may be noted that the proposed definition takes the union of the circles joining only same points to form a fuzzy circle. On the other hand, the Buckley-Eslami’s fuzzy circle considers the union of the circles that pass through three points in the supports of the fuzzy points. Therefore, the fuzzy circle according to the proposed method has lesser spread than that of Buckley and Eslami. 2 ), both the methods Moreover, it may be observed that to evaluate μ((x0 , y0 )|C essentially solves a constrained optimization problem (the supremum). Then constraint set in the proposed method is subset of that of Buckley-Eslami. Therefore, fuzzy circle in the proposed method is a subset of Buckley-Eslami’s fuzzy circle. A question may arise from the proposed definitions of fuzzy circles that out of the two proposed fuzzy circles which one is favored in this study. To answer, it may be noted that each of the two proposed formulations depends on the information about the fuzzy circle. The fuzzy circle in Method 1 is favored when information about the fuzzy center and the fuzzy radius is known. If information about three fuzzy points on the fuzzy circle is known, then the Method 2 is favored.

5.5 Concluding Remarks In this chapter, at the outset it was focused on finding a fuzzy number which has a fixed fuzzy distance from a given fuzzy number. This study is used to define fuzzy circles thereafter. Two different formulations of fuzzy circles are proposed. The Method 1 basically extended the conventional definition of classical circle. Method 2 defined a fuzzy circle through three given fuzzy points. For each of the methodologies, αcuts and construction of membership functions are given in detail. Properties of the formulated fuzzy circles have been discussed under the proposed methodologies. In a particular situation, Theorem 5.4.1 finds equivalence between the fuzzy circles formulated by the two methods. Future research may find the equivalence for more generalized situations.

References Buckley, J.J., Eslami, E.: Fuzzy plane geometry I: points and lines. Fuzzy Sets Syst. 86, 179–187 (1997a) Buckley, J.J., Eslami, E.: Fuzzy plane geometry II: circles and polygons. Fuzzy Sets Syst. 87, 79–85 (1997b) Ghosh, D., Chakraborty, D.: Analytical fuzzy plane geometry III, Fuzzy Sets Syst. 283, 83–107 (2016) Swokowski, E.W., Cole, J.A.: Algebra and Trigonometry With Analytic Geometry, 4th edn. Cengage Learning (2007) Wang, X., Ruan, D., Kerre, E.E.: Mathematics of Fuzziness–Basic Issues, vol. 245. Studies in Fuzziness and Soft Computing, Springer, Berlin (2009)

Chapter 6

Fuzzy Parabola

6.1 Introduction This chapter attempts to construct fuzzy parabola in detail. Depending on the available information about the fuzzy parabola, two different formulations are discussed. Construction of the membership functions of fuzzy parabolas are detailed with numerical illustrations. Different results and interrelations on the attributes of fuzzy parabolas are presented. It is needed to give attention for three fuzzy curves which are obtained by cutting a fuzzy right circular cone by a crisp plane. Thus the derived intersecting curves are called fuzzy conic sections, or briefly, fuzzy conics. The derived three fuzzy conics are called fuzzy parabola, fuzzy ellipse, and fuzzy hyperbola. Here we attempt to show the construction procedure of fuzzy parabola only.

6.2 Construction of Fuzzy Parabola In conventional Euclidean geometry, according to the information (attributes) available for a parabola, we can describe a parabola in two different ways. If the focus, (u, v) say, and the directrix, ax + by + c = 0 say, are known, the parabola is obtained as a collection of points that are equidistant from the focus and the directrix. That is, through the algebraic equation (ax + by + c)2 = (x − u)2 + (y − v)2 . a 2 + b2 If only five (degrees of freedom of a parabola) points on the parabola are known, we obtain the parabola by determining five parameters, i.e., a, b, g, f and c in its generic algebraic equation © Springer Nature Switzerland AG 2019 D. Ghosh and D. Chakraborty, An Introduction to Analytical Fuzzy Plane Geometry, Studies in Fuzziness and Soft Computing 381, https://doi.org/10.1007/978-3-030-15722-7_6

145

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6 Fuzzy Parabola

ax 2 + 2hx y + by 2 + 2gx + 2 f y + c = 0, with h 2 = ab. We give two different procedures for the construction of fuzzy parabolas. (1) The first approach uses the basic definition of a parabola: a parabola is the set of points whose distances from a fixed point (focus) and from a given line (directrix) are equal. (2) The second approach considers five fuzzy points on the plane and formulates a fuzzy parabola based on given five fuzzy points.

6.2.1 Method 1  b) be a fuzzy point and  Let F(a, L a fuzzy line. We can attempt to construct a fuzzy  b) and directrix  parabola with focus F(a, L, in the following three possible ways. F P 2 and  F P 3. We denote the resulting three fuzzy parabolas by  F P 1,  Definition 6.1 (Method 1.1)  and directrix  by the The fuzzy parabola  F P 1 with focus F  L maybe defined   such that D  P,  F  =D  P,   collection of the fuzzy points P’s L .

Although this method generalizes the conventional idea to define a parabola, it   is very restrictive and not well-defined  for arbitrary  F and L. Because, there may  satisfying D  P,  F  =D  P,    and  not exist fuzzy point P L for any choice of F L. It is worthy to mention that the relation stated in the definition can not be crisp, it must be fuzzy or imprecise in nature. Because of this fact, it will lead to the case  with the identical core will be there. Resulting where existence of infinitely many Ps’ thereby a non-unique fuzzy parabola. For instance, we follow two examples given below.   Example 6.1 Consider the fuzzy line  L ≡ x + 23 /x + 1/x + 21 = 0, and the fuzzy  0) whose membership function is right circular cone with base point F(1, 

(x, y) : (x − 1)2 + y 2 ≤

1 4



and vertex (1, 0).      such that D  P,  F  =D  P,   If possible let there exists a fuzzy point P L .  across the core is ζ. Suppose that the maximum spread of F      P,  F  and D  P,   Then, spread of the fuzzy numbers D L are 14 + ζ and 1 + ζ, respectively.      P,  F  and D  P,   Since the spreads are unequal, D L cannot be identical.      satisfying D  P,  F  =D  P,   Hence, there does not exist P L .  and  Hence, it is not possible to define the fuzzy parabola  F P 1 with the given F L.

6.2 Construction of Fuzzy Parabola

147

Fig. 6.1 Construction of fuzzy parabola  F P1

Example 6.2 Consider the fuzzy line  L ≡ (x + 2/x + 1/x) = 0, and the fuzzy point  0) whose membership function is right circular cone with base F(1, 

 (x, y) : (x − 1)2 + y 2 ≤ 1

and vertex (1, 0).  and  For these F L, the core parabola  F P 1 (1) is y 2 = 4x. 0 (4, 4),  Notice that at the point (4, 4) ∈ F P 1 (1), (Fig. 6.1), we have a fuzzy point P with membership function    1 − (x − 4)2 + (y − 4)2 if (x − 4)2 + (y − 4)2 ≤ 1  μ (x, y)| P0 (4, 4) = 0 elsewhere,      P 0 , F  =D  P 0 ,  where, D L = (3/4/7) L R , for some reference functions L and R.

148

6 Fuzzy Parabola

ξ at (4, 4), We indicate that, in fact, there exist infinitely many fuzzy points, P ξ > 0, with membership function

 y−4 2   1 − ( x−4 )2 + ( 1+ξ ) if (x − 4)2 + (y − 4)2 ≤ (1 + ξ)2 1+ξ ξ (4, 4) = μ (x, y)| P 0 elsewhere,      P ξ , F  =D  P ξ ,  all of which satisfy D L = (3 − ξ/5/7 + ξ) L ξ Rξ . ξ (4, 4) are obtained by applying the expansion map The fuzzy points P (x, y) −→ (4 + k(x − 4), 4 + k(y − 4)) , where k = 1 + ξ, ξ > 0 0 (4, 4). on the fuzzy point P    P ξ ,  L = (3 − ξ/5/7 + ξ) L ξ Rξ is trivial and clearly The calculation to obtain D evident from Fig. 6.1.    = (3 − ξ/5/7 + ξ) L ξ Rξ is as follows.  P ξ , F The details to obtain D  0) and P ξ (4, 4) is The line joining the cores of F(1, L≡

x−1 3

= 4y .

 0) and P(4,  4) are the circles The boundary of the supports of F(1, C ≡ (x − 1)2 + y 2 = 1 and C2 ≡ (x − 4)2 + (y − 4)2 = (1 + ξ)2 , respectively. The intersection points of the line L and the circle C1 are ( 25 , − 45 ) and ( 85 , 45 ). The intersection points of the line L and the circle C2 are Rξ :

    4 + 35 (1 + ξ), 4 + 45 (1 + ξ) and Sξ : 4 − 35 (1 + ξ), 4 − 45 (1 + ξ) .

 0) and P ξ (4, 4) We notice from Fig. 6.1 the nearest and farthest inverse points on F(1, are, respectively,   ( 25 , − 45 ) and Rξ : 4 + 35 (1 + ξ), 4 + 45 (1 + ξ) ,   and ( 58 , 45 ) and Sξ : 4 − 35 (1 + ξ), 4 − 45 (1 + ξ) . The distance between nearest two inverse points is 

4−

11+3ξ 4

2

+ 4−

8+4ξ 5

2

= 3 − ξ.

The distance between farthest two inverse points is 

4+

1+3ξ 4

2

+ 4+

8+4ξ 5

2

= 7 + ξ.

6.2 Construction of Fuzzy Parabola

149

   P ξ , F  = (3 − ξ/5/7 + ξ) L ξ Rξ for some reference functions L ξ and Rξ . Thus, D ξ at (4, 4) Arbitrarinessof ξ > 0 shows that there are infinitely many fuzzy points P  P ξ , F  = (3 − ξ/5/7 + ξ) L ξ Rξ . A similar observation is applied at any satisfying D point on the core parabola  F P 1 (1): y 2 = 4x.  and directrix Thus, the a unique fuzzy parabola  F P 1 , by Definition 6.1, with focus F  L cannot be defined.

 be a fuzzy point and  Theorem 6.1 Let F    L, a fuzzy line. If there is a fuzzy point  b) that satisfy D  P,  F  =D  P,   P(a, L then there exist infinitely many fuzzy     k s’ say, such that D  P k , F  =D  P k ,  points, P L .

Proof We recall that        and Q   P,  F  = d Fθα , Pθα : Fθα , Pθα are inverse-points on P D α∈[0,1] θ∈[0,2π]

with membership value α

 (6.1)

and        P,   d Fθα , L α : Fθα , L α are inverse point-and-curve on D L = α∈[0,1] θ∈[0,2π]

  and  P L with membership value α .

(6.2)

Suppose that Pθα = x αpθ , y αpθ and Fθα = x αf θ , y αf θ are two inverse-points with  and F,  respectively; the points P α and F α , respectively, membership value α on P θ θ lie on the lines y−b x −c y−d x −a = and = , respectively, cos θ sin θ cos θ sin θ  is (c, d). where the core point of F We apply the expansion map (x, y) −→ (a + k(x − a), b + k(y − b)) , k > 1  b). Let P k (a, b) be the resulting fuzzy point. Applying on the fuzzy point P(a, α  and expansion map on the point Pθ inverse-points with membership value α on F  Pk (a, b) are

150

6 Fuzzy Parabola



 and the point P α = a + k(x α − a), b + k(y α − b) ∈ P k . Fθα = x αf θ , y αf θ ∈ F kθ pθ pθ

Again we notice that if     d Pθα , Fθα = d Pθα , L α , then

  α   α , Fθα = d Pkθ , Lα . d Pkθ

Hence, by Eqs. (6.1) and (6.2), we obtain that      =D  P k ,   P k , F L , D and thus the theorem is proved through the arbitrariness of k > 1.  and fuzzy directrix Remark 6.1 Theorem 6.1 indicates that for a given fuzzy focus F   L, two cases may arise to define a fuzzy parabola F P 1 .  satisfying D(  F,  P)  = D(  F,   (Case 1) There may not exist fuzzy point P L). Thus, a fuzzy parabola  F P 1 is not possible to define in this case. (Case 2) At each point on the core parabola  F P 1 there exists infinitely many fuzzy  satisfying D(  F,  P)  = D(  F,   points P’s L). In this case, it is not possible to identify a unique fuzzy parabola  F P 1. Thus, Definition 6.1 is not any appropriate way to define fuzzy parabola.

Definition 6.2 (Method 1.2)  b) and directrix  L may bedefined The fuzzy parabola  F P 2 with focus F(a,   P θ , F)  =D  P θ ,  θ ’s such that D( L , by the collection of the fuzzy points P θ ’s are constructed, Fig. 6.2, satisfying the following three conditions: where P θ contains a fuzzy number Q θ (xθ , yθ ) along the line x−a = y−b , where (i) P cos θ sin θ the point (xθ , yθ ) lie on the core parabola  F P 2 (1),  π (xθ , yθ ) along the perpendicular line to θ contains a fuzzy number P (ii) P θ, 2  the core line L(1), and θ is the smallest convex set containing (iii) support set of the fuzzy point P   Q θ (xθ , yθ ) and Pθ, π (xθ , yθ ). 2

Figure 6.2 describes the construction of the fuzzy parabola  F P 2 with focus y−b x−a  = is the line that passes through F(a, b) and directrix  L. In the figure, L θ ≡ cos θ sin θ  and makes an angle θ with the positive x-axis. Consider the point, the core of F Pθ : (xθ , yθ ), of intersection of the line L θ and the core parabola  F P 2 (1). Then, construct a line L θ, π that is perpendicular to the line  L(1) and passes through 2

6.2 Construction of Fuzzy Parabola

151

Fig. 6.2 Construction of fuzzy parabola  F P2

the point Pθ . Let  L π be the fuzzy number along the line L θ, π which is ob2 2 θ = (Aθ /Pθ /Cθ ) L θ Rθ and P  π = tained by the intersection of L θ, π and  L. Let Q θ, 2 2 (Dθ /Pθ /Bθ ) L θ Rθ be two fuzzy numbers along the lines L θ and L θ, π , respectively, 2  P  π,  F,  Q θ ). The fuzzy point, P θ say, at Pθ with smallest such that D( L π ) = D( θ,

2

2

θ , has support set the quadrangle possible compact support that contains  L θ and P θ , D(  P θ ,   F,  P θ ). L) = D( Aθ Bθ Cθ Dθ . It is easy to observe that for the fuzzy point P θ ’s constitute the fuzzy parabola  F P 2. The collection of all such fuzzy points P The following theorem shows that for a given Pθ , there are infinitely such fuzzy θ ’s at the point Pθ . point P

152

6 Fuzzy Parabola

1 and A 2 be two continuous fuzzy numbers along an idenTheorem 6.2 Let A 2 , the tical line with cores (a1 , b1 ) and (a2 , b2 ), respectively. Suppose for A left and right spreads are l2 and r2 , respectively, and the left and right refer 2 =  A 1 , A ence functions are L and R, respectively. Furthermore, assume D 2 be the fuzzy number obtained by applying the ex(d − l/d/d + r ) L R . Let A pansion transformation (x, y) −→ (a2 + k(x − a2 ), b2 + k(y − b2 )) , for some k > 1, 1 and A 2 is given by 2 . Then, the distance between A on the fuzzy number A (d − l − ξl2 /d/d + r + ξr2 ) L R , where k − ξ = 1. 1 and Proof Without loss of generality, we prove the result for two fuzzy numbers A    A2 on the real line with cores a1 and a2 , respectively, and A2 is the fuzzy number given by the transformation x −→ a2 + k(x − a2 ). Let x1α and x2α be two inverse points, with membership value α, on the fuzzy 2 . We assume x2α ≥ x1α , without loss of generality. 1 and A numbers A Then, x2α − x1α = d − l L −1 (α) or d + r R −1 (α). We prove for the case x2α − x1α = d − l L −1 (α). The other case will be similarly followed. According to the considered expansion transformation and the definition of inverse points, evidently x1α and a2 + k(x2α − a2 ) are inverse points, with membership value 2 . 1 and A α, on A The distance between these two inverse points is (a2 + k(x2α − a2 )) − x1α = (x2α − x1α ) + (x2α − a2 )(k − 1) = (d − l L −1 (α)) − ξl2 L −1 (α), where ξ = k − 1. As x1α and a2 + k(x2α − a2 ) are inverse points, the membership value of the differ A 1 , A 2 ) is α. ence (a2 + k(x2α − a2 )) − x1α on the fuzzy number D( Therefore,    A 1 , A 2 ) = α μ (a2 + k(x2α − a2 )) − x1α | D(    A 1 , A 2 ) = α =⇒ μ (d − l L −1 (α)) − ξl2 L −1 (α)| D(     2 ) = L d − x , for d − (l + ξl2 ) ≤ x ≤ d.  A 1 , A =⇒ μ x| D( l + ξl2

6.2 Construction of Fuzzy Parabola

153

Similarly,  2 ) = L  A 1 , A μ x| D( 



x −d r + ξr2

 , for d ≤ x ≤ d + (r + ξr2 ).

Thus,

⎧ d−x ⎪ L l+ξl ⎪ 2 ⎨

   A 1 , A 2 ) = R x−d μ x| D( r +ξr2 ⎪ ⎪ ⎩ 0

if d − (l + ξl2 ) ≤ x ≤ d if d ≤ x ≤ d + (r + ξr2 ) elsewhere.

Hence, the result follows. 1 = (1/2/3) and A 2 = (6/7/8). Example 6.3 Consider the fuzzy numbers A 2 are l2 = 1 and r2 , respectively. The left and right spreads of the fuzzy number A 2 are L(x) = R(x) = max{0, 1 − |x|}, 0 ≤ x ≤ 1. The reference functions of A   2 is D  A 1 , A 2 = (3/5/7). 1 and A The distance between A Thus, according to the notations in Theorem 6.2, l = 2, d = 5, r = 2. 2 = By the transformation x → 7 + 2(x − 7), i.e., ξ = 2, the fuzzy number A 2 = (5/7/9). (6/7/8) is transformed to A 2 is (2/5/8). 1 and A The distance between A We note that (2/5/8) is identical to (d + (l − ξl2 ) / d / d + (r + ξr2 )). Remark 6.2 We refer to the description, the paragraph immediately after Definition 6.2, of the construction procedure of the fuzzy parabola  F P 2 . Fuzzy parabola θ ’s whose constituents are  F P 2 is observed as the collection of the fuzzy points P  π = (Dθ /Pθ /Bθ ) L  R  . The basic requirement on the θ = (Aθ /Pθ /Cθ ) L θ Rθ and P Q θ, θ θ 2  π and Q θ is that construction of P θ,

2

 F,  Q θ ).  P  π, L π ) = D( D( θ, 2

2

 With the help of Theorem 6.1, one can find infinitely many P

θ,

identical core Pθ satisfying

π 2

 with the and Q θ

 F,  Q θ ).  P  π ,  L π ) = D( D( θ, 2

2

θ is not a unique fuzzy point at Pθ . In fact, there Thus, the required fuzzy point P  b) and are infinitely many such fuzzy points at Pθ . Hence, for a given focus F(a,   directrix L, a unique fuzzy parabola F P 2 cannot be determined.

154

6 Fuzzy Parabola

Definition 6.3 (Method 1.3)  b) and directrix  For a given fuzzy focus F(a, L, a fuzzy parabola  F P 3 may be defined by the collection of parabolas C P’s with varied membership values. Mathematically,  F P3 =     C Pα  α∈[0,1]

 C Pα is a parabola with focus (xα , yα ) ∈ F(0), and directrix a Line      = α = μ Lα |  L α on the support of  L so that μ (xα , yα ) | F L .

 and directrix  The construction of the fuzzy parabola  F P 3 , for a given focus F L, is shown in Fig. 6.3. In the procedure, for an α ∈ [0, 1], we consider a point (xθα , yθα )  with membership value α and a line L α in  L with membership value α. The in F two crisp parabolas with foci (xθα , yθα ), and directrices the two L α ’s are E θα and Fθα , respectively. The parabolas E θα and Fθα are C Pα ’s in Definition 6.3 which are drawn by the dotted curves. Collection of all E θα and Fθα ’s for the complete range of α ∈ [0, 1] constitute the fuzzy parabola  F P 3.  0) be the fuzzy point that has membership value right circular Example 6.4 Let F(1, cone with base {(x, y) ∈ R2 : (x − 1)2 + y 2 ≤ 1} and vertex (1, 0).  0) is the collection of the following points with membership The fuzzy point F(1, values α α , y1θ ) = (1 + (1 − α) cos θ, (1 − α) sin θ), α ∈ [0, 1], θ ∈ [0, 2π]. (x1θ

Let  L be the fuzzy line (x + 2/x + 1/x) = 0, which is the collection of the following crisp lines L α with membership values L α : x + (2 − α) = 0 or x + α = 0, α ∈ [0, 1].  and directrix  L. Suppose that  F P 3 be the fuzzy parabola with focus F According to Definition 6.3, the fuzzy parabola  F P 3 is the collection of crisp α α , y1θ ) and directrix is the line parabolas, C Pα ’s say, whose vertex is the point (x1θ L α . Such a C Pα is given by y=

1 2((1−α) sin θ−kα )

  (x − 1 − (1 − α) cos θ)2 − kα2 + (1 − α)2 sin2 θ ,

where kα = α − 2 or −α. The fuzzy parabola  F P 3 is depicted in Fig. 6.3. Remark 6.3 It is evident from Fig. 6.3 that, although, the fuzzy parabola  F P 3 , by  and  Definition 6.3, is uniquely defined for any given F L,  F P 3 cannot be observed as

6.2 Construction of Fuzzy Parabola

155

Fig. 6.3 Construction of fuzzy parabola  F P3

a collection of fuzzy points. Because, the spread of the fuzzy parabola gets wider and F P 3 (1). Therefore, wider and hence  F P 3 has unbounded spread across the core of  Definition 6.3 is not an appropriate way to define a fuzzy parabola, since it lacks the fundamental property of a parabola that it is a collection of fuzzy points. The entire discussion and analysis of this Sect. 6.2.1 shows that none of the three definitions is an appropriate idea to define a fuzzy parabola. The next subsection attempts to construct a fuzzy parabola that passes through five given fuzzy points. It seems the following Method 2 is an appropriately defined fuzzy parabola, because it does not have any of the above deficiencies.

156

6 Fuzzy Parabola

6.2.2 Method 2 i (ai , bi ), i = 1, 2, . . . , 5 be given five fuzzy points whose cores lie on a crisp Let P parabola, C P say. In order to construct a fuzzy parabola that passes through these five fuzzy points, we first need to construct a fuzzy parabolic segment,  F P 1···5 say, 2 , . . ., P 5 . Then, we can extend  1 , P F P 1···5 to infinity on both sides to obtain joining P the fuzzy parabola  F P. To extend  F P bi-infinitely, we consider two hypothetical fuzzy points with a core 1 (a1 , b1 ) and the other is on C P, one is an infinite distance away on the side of P 5 (a5 , b5 ). Let these two fuzzy points also an infinite distance away on the side of P 5∞ , respectively. Let  1∞ and P F P 1∞ and  F P 5∞ be the semi-infinite fuzzy be P 1 , and P 5 and P 5∞ , respectively. Then, the 1∞ and P parabolic segments joining P fuzzy parabola  F P can be defined by  FP =  F P 1∞



 F P 1···5



 F P 5∞ ,

F P 1∞ and  F P 5∞ are defined as where the middle part  F P 1···5 and the infinite tails  follows. • Construction of  F P 1···5  F P 1···5 =

  α∈[0,1]

F Pα : where F Pα is a crisp parabola that passes through five same i (ai , bi ), i = 1, 2, . . . , 5 with membership value α . points on P

Mathematically, through membership function, the part  F P 1···5 can be defined by    μ (x, y)|  F P 1···5 = sup α : where (x, y) lies on F Pα that passes through five same i , i = 1, 2, . . . , 5 with membership value α . points on P

As per the definition, the fuzzy parabolic-segment  F P 1···5 is observed as a collection of crisp points with various membership values. However, the definition of   membership function μ (x, y)|  F P 1···5 shows that a fuzzy parabola is the union of all i , i = 1, . . . , 5. crisp parabolas that pass through five same-points on the supports of P F P 5∞ • Construction of  F P 1∞ and  The construction of the infinite tails essentially depend on the consideration of 1∞ and P n∞ 1∞ and  F P 5∞ . How can we obtain the fuzzy points P the fuzzy points P and what does the phrase ‘infinite distance away’ actually mean? Answers to these

6.2 Construction of Fuzzy Parabola

157

questions are presented in the following Proposition 6.1. The proposition states that 1∞ and P 5∞ may not be of for construction of  F P 1∞ , the shape and position of P 5∞ are two hypothetical fuzzy points. 1∞ and P due importance. The fuzzy points P To construct  F P 1∞ and  F P 5∞ , the only required information is that the supports 5∞ must be compact sets and their cores must lie on the core parabola 1∞ and P of P C P. F P 5∞ , we need to take the union of all the parabolic segTo obtain  F P 1∞ and  1∞ , and P 5 and P 5∞ , respectively. However, 1 and P ments joining same-points of P according to Proposition 6.1, these parabolic segments are always parallel to the core F P 5∞ parabola  F P(1). Thus, the semi-infinite fuzzy parabolic segments  F P 1∞ and  on  F P are two bunches of semi-infinite crisp parabolic segments with varied membership values and the half-parabolic segments must be parallel to the crisp parabola C P. Proposition 6.1 To consider a fuzzy parabola  F P as a collection of fuzzy F P 5∞ must points, all the semi-infinite crisp parabolic segments in F P 1∞ and be parallel to the core parabola  F P(1). Proof Assume on the contrary that there exists a semi-infinite crisp parabolic segment F P 1∞ which is not parallel to the core parabola  F P(1). According to the formulation of  F P 1∞ (0), it is union of the semi-infinite parabolic 1∞ . Therefore, correspond1 and P segments joining same-points of the fuzzy points P 1 (0) and (x1∞ , y1∞ ) ∈ ing to F P 1∞ , there must exist two same points (x1 , y1 ) ∈ P 1∞ (0) that are two extremities of F P 1∞ . P F P(1), either F P 1∞ intersects  F P(1) or the distance Since F P 1∞ is not parallel to  between the point (x1∞ , y1∞ ) and the semi-infinite parabolic segment F P 1∞ must be infinitely large. 1 (0) and (x1∞ , y1∞ ) ∈ P 1∞ (0) The former case evidently implies that (x1 , y1 ) ∈ P are not same points. Because, in this case, (x1 , y1 ) and (x1∞ , y1∞ ) lie on two different sides of  F P(1), and hence lie on two different sides of the line joining core points 1∞ . 1 and P of P 1∞ cannot 1∞ is unbounded and hence P The latter case implies that the support of P be a fuzzy point. Thus, both cases are impossible. A contradiction arises. F P(1). Therefore, all the semi-infinite parabolic Hence, F P 1∞ must be parallel to  F P(1). In a similar manner, we can show segments in  F P 1∞ (0) must be parallel to  F P(1). that all the semi-infinite parabolic segments in  F P 1∞ (0) must be parallel to  Hence, the result follows. To construct  F P, we first consider the middle fuzzy parabolic segment  F P 1···5 1 , P 2 , . . ., P 5 , and that is union of the parabolic segments joining five same points in P then the semi-infinite fuzzy parabolic segments  F P 1∞ and  F P 5∞ are adjoined on F P 1∞ and  F P 5∞ can be evaluated using the following either side of  F P 1···5 . Here  formulations:

158

6 Fuzzy Parabola

Fig. 6.4 Construction of fuzzy parabola in Method 2



1 (0) and F P1∞ (x, y) is a semi-infinite F P1∞ (x, y): where (x, y) ∈ P  1 parabolic segment parallel to  F P(1) with membership value (x, y) on P

 F P 1∞ =

and   5 (0) and F P5∞ (x, y) is a semi-infinite F P 5∞ = F P5∞ (x, y): where (x, y) ∈ P  5 . parabolic segment parallel to  F P(1) with membership value (x, y) on P 1 , In Fig. 6.4 shows the method for constructing  F P through five fuzzy points: P     P2 , P3 , P4 and P5 . The regions under the ellipse centered at A1 , ellipse centered at A2 , circle centered at A3 , square centered at A4 and ellipse centered at A5 are the 2 , P 3 , P 4 and P 5 , respectively. The grey-shaded regions 1 , P supports of the points P inside the supports of the fuzzy points represent different α-cuts. The variation of the membership grades for the fuzzy points is indicated by the intensity of the grey levels. The darker grey-level indicates a higher membership function. The membership i is one and it decreases gradually to zero on the periphery of the grade of Ai in P  support of Pi for each i = 1, 2, 3, 4, 5.

6.2 Construction of Fuzzy Parabola

159

In Fig. 6.4, the third darkest inner ellipses or square centered at Ai ’s are the α-cuts i (α). We consider five lines: L iθ which passes through Ai , for each i = 1, 2, 3, 4, 5. P i (Ai )(α), being These five lines have an angle θ with the positive x-axis. Because P i (Ai )(α), the line α-cut of a fuzzy point, is convex and Ai is an interior point of P i (Ai )(α) at exactly two points. Let these L iθ must intersect with the boundary of P α α and Riθ . Thus, Q α1θ , Q α2θ , Q α3θ , Q α4θ and Q α5θ constitwo intersecting points be Q iθ tute a set of five same-points with membership value α. Similarly, the collection of α α α α α , R2θ , R3θ , R4θ and R5θ is also a set of five same-points with membership value α. R1θ α α α be the parabola that passes through five Q iθ ’s, and F PθL be the parabola Let F PθU α α α and Riθ that passes through five Riθ ’s. Since membership value of all the points Q iθ α α is α, we put a membership grade of α to the parabolas F PθL and F PθU on the fuzzy parabola  F P. α and Through varying θ in [0, 2π] and α in [0, 1], several parabolas such as F PθL α F PθU will be obtained. According to the definition, the fuzzy parabolic-segment α α  and F PθU with membership value F P 1···5 is the collection of all the parabolas F PθU α, i.e.,    α α  , F PθU F PθL . F P 1···5 = θ∈[0,2π] α∈[0,1]

Let F P be any parabola in the support of the fuzzy parabola  F P. We define the membership value of the parabola F P in  F P by μ(F P|  F P) =

min

(x,y)∈F P

μ((x, y)|  F P).

The following theorem shows how to obtain the membership value of a parabola i ’s, i = 1, 2, . . . , 5. F P in  F P using the same-points in P Theorem 6.3 Suppose that F P is a parabola in  F P and five same-points i (0), with μ((xi , yi )| F P) = α for all i = 1, 2, . . . , 5 exist such that (xi , yi ) ∈ P F P is the parabola that passes through the five (xi , yi )’s. Then, μ(F P| F P) = α.

Proof We argue that (i) μ(F P|  F P) < α and (ii) μ(F P|  F P) > α. (i) By contrast, let μ(F P|  F P) < α. Thus, by the definition of μ(F P|  F P), there exist (x0 , y0 ) in  F P such that (x0 , y0 ) ∈ F P and μ((x0 , y0 )|  F P) < α. Let F P) = β. As (x0 , y0 ) ∈ F P and F P is a parabola that joins the five μ((x0 , y0 )|  F P) = sup{δ : where (x, y) same-points with membership value α, μ((x0 , y0 )|  lies on the parabola that joins the five same-points with membership value δ} ≥ α. However, β < α; therefore, a contradiction arises. Hence, μ(F P|  F P) < α.

160

6 Fuzzy Parabola

Fig. 6.5 A fuzzy parabola in Method 2 (towards completing the fuzzy parabola in Fig. 6.4)

(ii) This part is obvious, since μ(F P|  F P) = min{α : where (x, y) lies on F P and μ((x, y)|  F P) = α}, and all the points (xi , yi ), i = 1, 2, . . . , 5 lie on F P. Therefore, μ(F P|  F P) = α. The complete fuzzy parabola  F P is depicted in Fig. 6.5. The region between the curves f 0L and f 0U is the support of the fuzzy parabola  F P. The core parabola is the i lies. curve C P on which the five core points Ai of the fuzzy points P i ), for each i ’s, the surfaces of z = μ((x, y)| P Although  F P is constructed by P  i = 1, 2, . . . , 5, always lies in the interior of z = μ((x, y)| F P) and a portion of i ) touches the surface of z = μ((x, y)|  F P). Hence, there the surface z = μ((x, y)| P i is less than the membership  exist some points in Pi (0) whose membership grade at P grade at  F P. For instance, in Fig. 6.5, the points on the periphery of the elliptic region 1 , whereas all those points 1 have a membership value of zero for the fuzzy point P P U L  F P. except where P1 (0) touches f 0 and f 0 have positive a membership value for  i (0), μ((x0 , y0 )|  F P) is greater than In fact, it is to note that for any point (x0 , y0 ) in P i ), i = 1, 2, . . . , 5. or equal to μ((x, y)| P

6.2 Construction of Fuzzy Parabola

161

Note that on  F P, if a line perpendicular to the core parabola C P ≡  F P(1) is considered, then along that line there exists a fuzzy number on  F P(0). For example, if the line C D is considered in Fig. 6.5, then along C D, a fuzzy number of L R type, (F/G/H ) L R say, can be found. Thus, the whole fuzzy parabola can be visualized as a three-dimensional figure (basically a subset of R2 × [0, 1]) whose cross-section across  F P is a fuzzy number such as (F/G/H ) L R . The fuzzy parabola  F P can also be observed as a union of fuzzy points. Toward F P. Let us take a this visualization, let us consider a fuzzy number (F/G/H ) L R on  convex region on the support set  F P(0) in such a manner that all of the points on the line segment F H , except F and H , are interior points of this convex region. We de on such a convex region by the membership function μ((x, y)| P)  fine a fuzzy point P  ≤ μ((x, y)|  F P) other= μ((x, y)| (F/G/H ) L R ), when (x, y) ∈ F H , μ((x, y)| P)  = 1 only at G. Several such fuzzy points at (F/G/H ) L R can wise, and μ((x, y)| P) be obtained by changing the convex region. By varying G on  F P(1) ≡ C P, we will obtain many fuzzy points on  F P(0). The collection of all these fuzzy points is the 2 is a collection of fuzzy points. 2 . Thus, C entire C √ √ 1 (5, −4), P 2 (3, −2 2), P 3 (1, 0), P 4 (2, 2) and P 5 (4, 2 3) be Example 6.5 Let P five fuzzy points. The membership functions of these five fuzzy points are right circular/elliptical√cones with bases {(x, y) : (x − 5)2 + (y + 4)2 ≤ 1}, {(x, y) : (x − 3)2 + 4(y + 2 2)2 ≤ 1}, {(x, y) : (x − 1)√2 + y 2 ≤ 1}, {(x, y) : 4(x − 2)2 + (y − of 2)2 ≤ 1} and {(x, y) : (x − 4)2 + (y − 2 3)2√≤ 1}, respectively. The vertices √ the membership functions are (5, −4), (3, −2 2), (1, 0), (2, 2) and (4, 2 3), respectively. 2 , P 3 , 1 , P For a particular α ∈ [0, 1], the same points with membership value α on P 5 are 4 and P P α α , y1θ ) = (5 + (1 − α) cos θ, −4 + (1 − α) sin θ) , Q α1θ : (x1θ  √ α α α Q 2θ : (x2θ , y2θ ) = 3 + (1 − α) √ cos θ 2 , −2 2 + (1 − α) √ 1+3 sin θ

sin θ



1+3 sin2 θ

,

α α Q α3θ : (x3θ , y3θ ) = (1 + (1 − α) cos θ, (1 − α) sin θ) ,

α α cos θ √ sin θ Q α4θ : (x4θ , and , y4θ ) = 2 + (1 − α) √3 cos , 2 + (1 − α) 2 θ+1 3 cos2 θ+1

√ α α Q α5θ : (x5θ , y5θ ) = 4 + (1 − α) cos θ, 2 3 + (1 − α) sin θ .

The parabola, E θα say, that passes through Q α1θ , Q α2θ , Q α3θ , Q α4θ , and Q α5θ can be determined by the equation aθα x 2 + 2h αθ x y + bθα y 2 + 2gθα x + 2 f θα y + cθα = 0, with h αθ 2 = aθα bθα , where

(6.3)

162

6 Fuzzy Parabola

 α yα  −x1θ 1θ     −x α y α  2θ 2θ  2h αθ  α α yα aθ = α  −x3θ 3θ kθ   α  −x y α  4θ 4θ    −x α y α 5θ 5θ  α  x1θ    α x  2θ  α  α h gθα = αθ  x3θ kθ    xα  4θ    xα

α y1θ

2

α x1θ

α y2θ

2

α x2θ

α y3θ

2

α x3θ

α y4θ

2

α x4θ

α y5θ

2

α x5θ

α 1  y1θ    α y2θ 1    2h αθ α 1  , bα = y3θ θ  kθα   α y4θ 1    α y 1 5θ

α 1  y1θ    α 1 y2θ    α 1, y3θ    α 1 y4θ    α y 1

2

α y1θ

2

α yα − x1θ 1θ

2

α y1θ

2

α yα − x2θ 2θ

2

α y1θ

2

α yα − x3θ 3θ

2

α y1θ

2

α yα − x4θ 4θ

2

α y1θ

2

α yα − x5θ 5θ

2

α y1θ

2

α x1θ

2

α y1θ

2

α α x2θ y2θ

2

α y1θ

2

α yα x3θ 3θ

2

α y1θ

2

α α x4θ y4θ

2

α y1θ

2

α x5θ

5θ

 α  x1θ    α x  2θ  α  α 2h cθα = αθ  x3θ kθ    xα  4θ    xα 5θ

f θα

5θ

α y1θ

α y5θ

 α  x1θ    α x  2θ   α x  3θ    xα  4θ    xα 5θ

 α  x1θ    α x  2θ  α hθ  α = α  x3θ kθ    xα  4θ    xα 5θ

2

α yα α − x1θ 1θ x 1θ

2

α yα α − x2θ 2θ x 2θ

2

α yα α − x3θ 3θ x 3θ

2

α yα α − x4θ 4θ x 4θ

2

α yα α − x5θ 5θ x 5θ

2

α y1θ

2

α x1θ

2

α y1θ

2

α x2θ

2

α y1θ

2

α x3θ

2

α y1θ

2

α x4θ

2

α y1θ

2

α x5θ

 α α y α   x1θ − x1θ 1θ       α α α x − x2θ y2θ   2θ     α y α  and k α =  x α − x3θ θ 3θ   3θ     α α   xα − x4θ y4θ   4θ      xα α yα  − x5θ 5θ 5θ

α 1  y1θ    α y2θ 1    α 1, y3θ    α y4θ 1    α y 1 5θ

α y α 1  − x1θ 1θ    α yα 1  − x2θ 2θ    α yα 1  , − x3θ 3θ    α yα 1  − x4θ  4θ   α α −x y 1 5θ 5θ

2

α y1θ

2

α x1θ

2

α y1θ

2

α x2θ

2

α y1θ

2

α x3θ

2

α y1θ

2

α x4θ

2

α y1θ

2

α x5θ

α 1  y1θ    α y2θ 1    α 1. y3θ    α y4θ 1    yα 1  5θ

i ’s, i = 1, 2, . . . , 5, is the The fuzzy parabolic segment  F P 1···5 that passes through P union of all possible parabolic-segments E θα ’s that lies in between Q α1θ and Q α5θ ’s. That is,  F P 1···5 =





α∈[0,1] θ∈[0,2π]



 α 2 α α α (x, y) : aθα x 2 + 2h α θ x y + bθ y + 2gθ x + 2 f θ y + cθ = 0 .

The infinite tails  F P 1∞ and  F P 5∞ are determined by the membership functions  α   α if y 2 − y1θ μ (x, y)|  F P 1∞ = 0 elsewhere, and





μ (x, y)|  F P 5∞ =



2

α α if y 2 − y5θ 0 elsewhere.

α = 4(x − x1θ ), θ = 63.43◦

2

α = 4(x − x5θ ), θ = 60◦

6.2 Construction of Fuzzy Parabola

163

The angles θ = 63.43◦ and 60◦ are, √ respectively, the angles of the normals to the F P is core parabola at (5, −4) and (4, 2 3) with the positive x-axis. The core of  the parabola {(x, y) : y 2 = 4x − 4}.

6.2.2.1

Construction of Membership Function

In order to evaluate the membership function of  FP =  F P 1∞



 F P 1···5



 F P 5∞ ,

we require to evaluate, separately, the membership functions of the three parts: the F P 1∞ and  F P 5∞ . According to the defimiddle part  F P 1···5 and two infinite tails   nition of F P 1∞ , evaluation of the membership function of the infinite tails is trivial and given by 

  F P 1∞ = sup α :(x, y) lies on a crisp parabola F Pα that passes through μ (x, y)   1 (0) with membership value α a point on P and 

  F P 5∞ = sup α :(x, y) lies on a crisp parabola F Pα that passes through μ (x, y)   5 (0) with membership value α . a point on P However, the membership grade μ((x, y)|  F P 1···5 ) might not always be simple to evaluate. Furthermore, it is really a difficult task to obtain the closed form of the membership function of  F P 1···5 . Because, the membership value at a particular point is the supremum of a set of real numbers that is obtained by solving a set of nonlinear equations. We note that the definition of fuzzy parabola in Method 2 implies μ((x, y)|  F P 1···5 )= sup{α : where (x, y) lies in a parabola that passes through five same-points i , i = 1, 2, . . . , 5, with membership value α. Thus, to obtain μ((x, y)|  F P), we in P must first take five same-points with membership value α ∈ [0, 1]. Then, all possible values of α are identified for which (x, y) lies on the parabola that joins five same-points with membership values. The evaluation of α may require to solving a nonlinear equation. The supremum of all these α values is the membership value of μ((x, y)|  F P 1···5 ). We refer the parabola for which the supremum is attained as the adjoining parabola of the point (x, y). The following paragraph illustrates a systematic procedure to identify the memF P that passes through five bership value of a point (x0 , y0 ) in a fuzzy parabola  i , i = 1, 2, . . . , 5. fuzzy points P

164

6 Fuzzy Parabola

i ’s are For a given θ ∈ [0, 2π] and α ∈ [0, 1], we denote the same points on P α α α α α α α α α α , y1θ ), (x2θ , y2θ ), (x3θ , y3θ ), (x4θ , y4θ ) and (x5θ , y5θ ). (x1θ

In order to identify the set of possible α’s for which (x0 , y0 ) lies on the conic E θα , we fix a particular value of θ = θ0 ∈ [0, 2π] and identify the Eq. (6.4) of E θα . Accordingly, an equation on α will be obtained. We solve this equation. Thereby a set, Sθ0 say, of possible values of α so that (x0 , y0 ) lies on E θα0 will be obtained. We pick up the supremum value of Sθ0 as let it be sθ0 . Then, we vary θ0 across the set [0, 2π] and keep on capturing sθα0 ’s. According to the extension principle, the membership value of (x0 , y0 ) in the fuzzy parabola  F P is given by   F P = sup sθ0 . μ (x0 , y0 )|  θ

The following example illustrates the procedure numerically. Example 6.6 Take the fuzzy parabola considered on Example 6.5. We attempt to identify the membership value of the point (2, 1.78) on the fuzzy parabola  F P. In the first step, we determine the set of parabolas E θα ’s on which the point (2, 1.78) lies. As the equation of E θα is Eq. (6.3), we need to identify the possible values of α such that aθα (2)2 + 2h αθ (2)(1.78) + bθα (1.78)2 + 2gθα (2) + 2 f θα (1.78) + cθα = 0, which simplifies to 4aθα + 7.12h αθ + 3.1684bθα + 4gθα + 3.56 f θα + cθα = 0.

(6.4)

For instance, we put θ0 = 45◦ in Eq. (6.4). Then we obtain the following equation on α: −105.61 − 1462.04β + 813.94β 2 − 115.70β 3 + 3.91β 4 = 0. This equation gives following real values of α: α = −0.07, 2.96, 6.49, 20.19. Thus, for θ0 = 45◦ , the set Sθ0 of all possible value of α is an empty set. If, however, we put θ0 = 30◦ in Eq. (6.4), then we obtain the following equation on α: 638.35 − 2499.86α + 1263.57α2 − 290.39α3 + 22.83β 4 = 0. This equation gives the following real values of α: α = 0.30, 6.92.

6.2 Construction of Fuzzy Parabola

165

Thus, for θ0 = 30◦ , the set Sθ0 of all possible value of α is the singleton set {0.30}. Thus, as per the notation in the paragraph immediately before this example, s30◦ = 0.30. Now we vary θ0 across [0, 2π] and keep on identifying the value of sθ0 . Finally, we capture the supremum of all sθ0 ’s. One can easily verify that the supremum value for the considered point is 0.3 which is attained for the value of θ0 = 30◦ . Eventually, we note that the conic passing through to the same points, for θ = 30◦ and α = 0.30, 1 , (3.38, −2.61) ∈ P 2 , (1.61, 0.35) ∈ P 3 , (2.38, 2.22) ∈ P 4 , (4.61, 3.81) ∈ P 5 . (5.61, −3.65) ∈ P

is 6855.08 − 4654.28x + 255.42x 2 − 352.44y − 8.21x y + 732.27y 2 = 0 and this conic contains the point (2, 1.67). Thus, we have   μ (2, 1.67)|  F P = 0.30. alpha cut of fuzzy parabola and membership value of a crisp parabola in a fuzzy parabola.

6.2.2.2

Fuzzy Point Inside, On and Outside of a Fuzzy Parabola

Consider a parabola f (x, y) ≡ ax 2 + 2hx y + by 2 + 2gx + 2 f y + c = 0, h 2 = ab. Let I, O and C be inside, outside and the curve of the parabola. The presentation of the set C is {(x, y) : f (x, y) = 0}. The inside and outside regions are the sets I = {(x, y) : f (x, y) < 0} and O = {(x, y) : f (x, y) > 0}, respectively. In conventional geometry, a point (a, b) is called inside, outside or on the parabola if (a, b) ∈ I, (a, b) ∈ O or (a, b) ∈ C, respectively; that is, f (a, b) < 0, f (a, b) > 0 or f (a, b) = 0, respectively.  b) is fuzzily on a In fuzzy mathematics, analogously, we say a fuzzy point P(a, fuzzy parabola  F P ≡ (g(x, y)/ f (x, y)/ h(x, y)) L R = 0  ∈  must be fuzzily on  if P(1) F P(1). Then, the fuzzy point P F P with some membership value, βC say. This βC is defined by

166

6 Fuzzy Parabola

⎧ ⎪ 1 ⎪ ⎪ ⎪ ⎨β1 βC = β2  ⎪ ⎪  ⎪ ⎪ ⎩min β1 , β2 where β1 =

sup

(x,y):g(x,y)=0

⊆   ⊂ P F P or P(0) F P(0)   P(0) exceeds F P(0) on the side of g  exceeds  P(0) F P(0) on the side of h  exceeds  if P(0) F P(0) on the either sides of  F P(1), if if if

 and β2 = μ((x, y)| P)

sup

 μ((x, y)| P).

(x,y):h(x,y)=0

 ∈  cannot be fuzzily on  If P(1) / F P(1), then P F P and we define βC = 0 in this situation.  b) is said to be fuzzily inside a fuzzy parabola  A fuzzy point P(a, F P if  P(1) = (a, b) ∈ I(  F P(1)). Then, the membership grade of the measure of this fuzzily inside is defined by βI =

sup

 (x,y)∈O(  F P(1))∩ P(0)

   . min μ((x, y)|  F P), μ((x, y)| P)

 under the supremum finds the intersecting region of The set O(  F P(1)) ∩ P(0)  the exterior of the core parabola  F P(1) and the support set of the fuzzy point P.  that gets This intersecting set gives the region of the support of the fuzzy point P outside the core of  F P. The membership value βI essentially measures the height  ∩ O(  of the fuzzy set  FP ∩ P F P(1)).

The idea of fuzzily outside a fuzzy parabola is similarly defined. A fuzzy point  b) is said to be fuzzily outside the fuzzy parabola   = (a, b) ∈ P(a, F P if P(1) O(  F P(1)). Then, the membership grade of the measure of this fuzzily inside is defined by βO =

sup

 (x,y)∈I(  F P(1))∩ P(0)

   . min μ((x, y)|  F P), μ((x, y)| P)

The membership value βO essentially measures the height of the fuzzy set   ∩ I(  FP ∩ P F P(1)).

Example 6.7 Consider the fuzzy parabola    F P ≡ y − x 2 + 1/y − x 2 /y − x 2 − 1 = 0. We refer to Fig. 6.6 for a geometrical visualization. Let us identify the membership grade of belongingness of the following fuzzy point with regard to the fuzzy parabola

6.2 Construction of Fuzzy Parabola

167

 F P: 1 (1, 5) with membership function P 

1 − 2 (x − 1)2 + (y − 5)2 if (x − 1)2 + (y − 5)2 ≤ 14 , 1 (1, 5) = μ (x, y)| P 0 elsewhere, 

2 P



5 4

 , 58 with membership function

2 5 , μ (x, y)| P 4

9 16



⎧

⎨ 1 − 4 (x − 45 )2 + (y − = ⎩0

9 2 16 )



2  if x − 45 + y −

 9 2 16

≤

1 16 ,

elsewhere,

3 (−1, 2) with membership function P  1 2 2 2 2   3 (−1, 2) = 1 − 2 (x + 1) + (y − 2) if (x + 1) + (y − 2) ≤ 4 , μ (x, y)| P 0 elsewhere,

4 (−3, 1) with membership function P

     1 − 2 (x + 3)2 + (y − 1)2 if (x + 3)2 + (y − 1)2 ≤ 4 (−3, 1) = μ (x, y)| P 0 elsewhere,

1 2

4 (−2, 4) with membership function and P   1 − (x + 2)2 + (y − 4)2 if (x + 2)2 + (y − 4)2 ≤ 1  μ (x, y)| P5 (−3, 1) = 0 elsewhere. 

In this problem, g(x, y) = y − x 2 + 1, f (x, y) = y − x 2 , h(x, y) = y − x 2 − 1. 2 ( 5 , 5 ), βI = 0, 1 (1, 5), βI = 1, βC = 0, and βO = 0, for P For the fuzzy point P 4 8 βC = 0, and βO =

sup

2 (0) (x,y)∈I (  F P(1))∩ P

  2 ) = μ(( 5 , 5 )|  min μ((x, y)|  F P), μ((x, y)| P 4 8 F P) =

1 16 ,

3 (−1, 2), βC = 0, and βO = 0, for P βI =

sup

 (x,y)∈O (  F P(1))∩ P(0)

  3 ) = μ((0.55, 0.31)| P 3 ) = 0.41. min μ((x, y)|  F P), μ((x, y)| P

168

6 Fuzzy Parabola

Fig. 6.6 Fuzzy points inside a fuzzy parabola

4 (−3, 1), βI = 0, βC = 0, and βO = 1, and for P 5 (−2, 4), βI = 0, βO = 0, for P and 5 ) = 0.26. βC = min{β1 , β2 } = β1 = β2 = μ((−1.75, 4.06)| P

6.2.3 Method 3: Symmetric Fuzzy Parabolas  b) is In this method, the core parabola, f (x, y) = 0 say, and a fuzzy point, P(a, given. The fuzzy parabola is generated by pulling the given fuzzy point on the core parabola. The pulling is done as follows. At first, we pick an arbitrary point, (u, v) say, on the core parabola. Then, we  b) to place it at (u, v). We, apply a rigid translation to the entire fuzzy point P(a,  hence, get a fuzzy point P(u, v). The complete structure and the distribution of the  v) thus obtained is just a photo-copy of membership value of the fuzzy point P(u,  b). P(a, Next, we vary the point (u, v) over the core parabola, apply the same rigid transla v). The collection of the fuzzy points tion, and generate the fuzzy point such as P(u,  v), for various (u, v) in the core parabola f (x, y) = 0, constitutes the such as P(u, fuzzy parabola,  F P 3 say, in this method. Thus,

6.2 Construction of Fuzzy Parabola

 F P3 =

169



 v). P(u,

(u,v): f (u,v)=0

 b) is In this method, precisely, we assume that the support of the fuzzy point P(a,  b) is given the neighborhood N (a, b). Further, let the membership function of P(a, by φ(x, y) for (x, y) in N (a, b), and zero otherwise.  b) at (u, v), we apply the rigid translation In order to place the fuzzy P(a, T (x, y) = (x + u − a, y + v − b) for all (x, y) ∈ N (a, b). Note that this translation neither twist nor shrink, nor expand the support set N (a, b)  b); but, it translates the entire neighborhood N (a, b) to a of the fuzzy point P(a, neighborhood of (u, v). Let the neighborhood of (u, v) thus obtained is N (u, v). Evidently, the neighborhoods N (a, b) and N (u, v) are photo-copy of each other where the centers are being altered. Hence, the membership function of the fuzzy  v) is given by φ(x + u − a, y + v − b) for (x, y) in N (a, b), and zero point P(u, otherwise. According to this definition, a symmetric fuzzy parabola can be observed as a fuzzy curve obtained by continuously pulling a fuzzy point on a crisp parabola. In the process of pulling, the core of the fuzzy point is retained on the core parabola. Figures 6.7, 6.8 and 6.9 illustrate the approach used to construct the symmetric fuzzy parabola. The given core parabola f (x, y) = 0 is the parabola A P Q. The given  b) is the grey shaded elliptical region N (a, b) centered at (a, b). A fuzzy point P(a,

 b) and the core parabola f (x, y) = 0. Further, for a given (u, v) Fig. 6.7 Given fuzzy point P(a,  v) through the rigid translation T (x, y) in the core, construction of P(u,

170

6 Fuzzy Parabola

 1 , v1 ), P(u  2 , v2 ), P(u  3 , v3 ), . . . Fig. 6.8 Varying (u, v) in the core parabola, identification of P(u

Fig. 6.9 Fuzzy parabola  F P 3 in Method 3

higher density of grey level implies higher membership value. Let the membership  b) is function of the fuzzy point P(a,    φ(x, y) if (x, y) ∈ N (a, b)  μ (x, y)| P = 0 elsewhere.

6.2 Construction of Fuzzy Parabola

171

Take a point (u, v) in the core parabola A P Q. With the help of the rigid translation T (x, y) = (x − a + u, y − b + v), the neighborhood N (a, b) is translated to N (u, v).

6.3 Conclusion The concept of fuzzy parabola has been initiated here. Few properties of fuzzy parabola have also been explored here. Other fuzzy conic sections can also be defined in this line. There are numerous applications of fuzzy parabola, like fountain spray water in the air, if an object is thrown up in the air, if we observe the movement of a flexible suspension bridge etc.

Chapter 7

Fuzzy Pareto Optimality

7.1 Introduction The decision making problems mostly involve optimization of conflicting multiple objectives. Multiobjective Optimization Problems (MOPs) essentially investigate trade-off between the objectives on a set of efficient solutions or on a set of satisficing solutions to the Decision Maker (DM). There have been numerous methods in the literature (Marler and Arora 2004; Deb 2001) to cope a classical MOP. However, existing classical techniques are collectively not enough to tackle all practical problems, since usually descriptions of the real-world situations are inherently imprecise (Lai and Hwang 1994). Thus fuzzy optimization problems have been studied in the literature since the pioneering work by Bellman and Zadeh (1970). Excellent survey on developments of fuzzy decision making can be obtained in Carlsson and Fullér (1996), Kahraman (2008). In fuzzy optimization problems, Ramík (2007) has mentioned that the main problem while dealing with Fuzzy Multiobjective Optimization Problems (FMOPs) is: unlike real numbers, fuzzy numbers are not linearly ordered. Thus a proper comparison of fuzzy number valued objective functions, with regard to FMOPs, still needs a significantly potential research. Finding a partial order relation for fuzzy vectors is also an important issue for FMOPs. Many researchers thus attempted to describe the concept of fuzzy dominance and fuzzy Pareto optimality. In the existing literature, it may be noted that several fuzzy dominance relations have been proposed for FMOPs. Methodologies proposed in the existing researches give a particular feasible solution or a part of fuzzy non-dominated set as solution of an FMOP. No research work is found yet to obtain the entire fuzzy non-dominated set. Here an attempt is made to capture the entire fuzzy non-dominated set of an FMOP (Ghosh and Chakraborty (2015a, b)). In this study a rigorous focus is laid on the geometric visualization of the complete fuzzy non-dominatedset and its formulation

© Springer Nature Switzerland AG 2019 D. Ghosh and D. Chakraborty, An Introduction to Analytical Fuzzy Plane Geometry, Studies in Fuzziness and Soft Computing 381, https://doi.org/10.1007/978-3-030-15722-7_7

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using fuzzy geometry. This chapter proposes a methodology to capture the formulated fuzzy non-dominated set. Thereafter, a proper non-dominated point for FMOPs is proposed for the final decision. The proposed method depends on a classical method (Ghosh and Chakraborty 2014) which generates complete Pareto set for a crisp MOP. The classical method is a non-gradient direction based technique. The direction is the only parameter of the method. The method searches for Pareto optimal solutions one after another by systematically changing this parameter, which is, obviously, independent of the DM’s preferences. The method essentially uses the cone of non-positiv e hyperoctant of the objective space to generate the complete fuzzy non-dominated set. Although the proposed classical method is a direction based method, it bears a necessary and sufficient condition for globally weak Pareto optimality. It is shown that a simple modification of the presented method can capture the D-Pareto optimal points of the problem, where D is any pointed convex cone. Thus, the formulated technique can not only generate the Pareto set but also can obtain the general DPareto set. To capture the entire non-dominated set of the objective feasible region of fuzzy multi objective optimization problem, the proposed classical method for MOPs is used (Ghosh and Chakraborty (2015a)). The method is demonstrated in the subsequent section.

7.2 Fuzzy Multi Objective Optimization Problem In practice, objectives and constraints for decision making problems in imprecise environment may involve many parameters whose possible values are assigned by DM. Usually, these imprecise parameters are represented by fuzzy numbers. The resulting FMOPs whose parameters are fuzzy appear more realistic than the conventional MOPs (Sakawa and Yano 1991). This chapter focuses on FMOPs whose parameters are fuzzy but decision variables are deterministic. A general model of an FMOP with fuzzy parameters is described by the following system: ⎧ min f (x, c) ⎪ ⎨   (7.1) subject to g(x, a) ≤ b, ⎪ ⎩ n x ∈C ⊆R , where  T f (x, c) = f 1 (x, c1 ), f 2 (x, c2 ), . . . , f k (x, ck ) , k ≥ 2,  T g(x, a ) = g1 (x, a1 ), g2 (x, a2 ), …, gm (x, am ) , m ≥ 1,  c j = ( c j1 , c j2 , . . . , c j p j )T , j = 1, 2, . . . , k,  ai = ( ai1 ,  ai2 , . . . , aiqi )T , i = 1, 2, . . . , m, ait are fuzzy numbers for each s = 1, 2, . . . , p j and t = 1, 2, . . . , qi .  c js and 

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175

, i.e., Denote the fuzzy constraint set of (7.1) by X  = X



  x ∈ C : g(x, a) ≤ b .

In the rest of the chapter, the notations  a and  b are used to represent the coefficient matrix ( a1 , a2 , . . . , am ) and the coefficient vector ( b1 ,  b2 , . . . ,  bm )t , respectively.    a2 , . . . , am are fuzzy vectors. However, b1 , b2 , . . . , bm are fuzzy numbers. Here  a1 ,

7.2.1 Assumptions for the Problem (i) For each point x in the set C, f j (x, c) and gi (x, a ) are continuous fuzzy numbers for all possible j and i. (1). c j (1)) has minimum value ‘zero’ on X (ii) For each j ∈ {1, 2, . . . , k}, f j (x, c j (1)) = This assumption is without any loss of generality. If min f j (x, (1) x∈X

c) will be redefined as f j (x, c) − m j . We m j = 0 for some j, then f j (x, c j (1)) = arg min x∈X(1) ( f j (x, c j (1)) − m j ). know that arg min x∈X(1) f j (x, (iii) Supports of decision feasible region and objective feasible region are compact. This assumption ensures that the minimizers of the objective functions are attainable by feasible points.  has non-empty core. (iv) The decision feasible region X Practically, in formulating a decision making problem, from the available data set the coefficients of the constraints are fitted. In the imprecise environment, the data sets are inherently imprecise. Thus, in the considered FMOP (7.1), the coefficients of the constraint set are taken as fuzzy. Again since the final decision for any decision making process is always crisp, decision variables in the problem will be considered as crisp. Under these assumptions for the FMOP (7.1) the fuzzy decision feasible region is constructed in the next section. In the next sections, the following definition of same and inverse points concept is used for fuzzy vectors. Definition 7.2.1 (Same and inverse points with respect to two vectors of a2 , fuzzy numbers) Let us consider two vectors of fuzzy numbers  a = ( a1 , b = ( b1 ,  b2 , . . . ,  bn ). The vectors a = (a1 , a2 , . . . , am ) ∈ ( a1 (0), …, am ) and  am (0)) and b = (b1 , b2 , . . . , bn ) ∈ ( b1 (0),  b2 (0), . . . ,  bn (0))  a2 (0), . . . ,  are said to be same points if each of a1 , a2 , . . . , am and b1 , b2 , . . . , bn has same membership value and they lie on the same side of the respective core points of the corresponding fuzzy numbers. The vectors a = (a1 , a2 , . . . , am ) and b = (b1 , b2 , . . . , bn ) are said to be inverse points if (a1 , a2 , . . . , am ) and a = ( a1 , a2 , . . . , am ) and (−b1 , −b2 , . . . , −bn ) are same points with respect to b2 , . . . , − bn ) − b = (− b1 , −

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7.2.2 Construction of Fuzzy Feasible Space Constraints of FMOP (7.1) are determined by the following set of m fuzzy inequalities: ⎫   g1 (x, a1 ) ≤ b1 , ⎪ ⎪ ⎪ ⎪   ⎬ a2 ) ≤ b2 , ⎪ g2 (x, (7.2) .. ⎪ ⎪ . ⎪ ⎪ ⎪ ⎭   am ) ≤ bm , gm (x, where x lies in the crisp set C and the fuzzy coefficient  ai is a fuzzy vecai2 , . . . , aiqi )t containing the fuzzy numbers  ai1 ,  ai2 , . . . ,  aiqi for all tor ( ai1 , i = 1, 2, . . . , m.

7.2.3 Evaluation of Fuzzy Inequality Region by Extension Principle  be the set rof points in C, satisfying (7.2). The points are generated from fuzzy Let X inequalities using sup-min composition of extension principle. Since the coefficients  must be a fuzzy set. Therefore, the points in C which  a and  b are imprecise, the set X . satisfy (7.2) will have different membership values in X  if there Direct use of extension principle states that a point x0 will belong to X exist a ∈  a (0) and b ∈  b(0) such that g(x0 , a) ≤ b. Therefore, = X



 x ∈ C : g(x, a) ≤ b, where a ∈  a (0) and b ∈  b(0) .

According to the extension principle, to find the membership value of a particular , sup-min composition will find the supremum of all the α’s in [0, 1] point x0 in X ai j (α) and bi in  bi (α), j = 1, 2, . . . , qi , i = 1, 2, . . . , m for which there exist ai j in  such that x0 satisfies ⎫ g1 (x0 , a11 , a12 , . . . , a1q1 ) ≤ b1 , ⎪ ⎪ ⎪ ⎬ g2 (x0 , a21 , a22 , . . . , a2q2 ) ≤ b2 , ⎪ .. ⎪ ⎪ . ⎪ ⎪ ⎭ gm (x0 , am1 , am2 , . . . , amqm ) ≤ bm ,

(7.3)

and

α = min μ(a11 | a11 ), μ(a12 | a12 ), . . . , μ(amqm | amqm ), μ(b1 | b1 ), . . . , μ(bm | bm ) . ). A computation procedure is suggested in the algorithm to find μ(x0 |X

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177

 if x0 satisfies each of g1 (x, a1 ) ≤ Note that a point x0 in C will belong to X ai (0) and bi in  bi (0), i = b1 , g2 (x, a2 ) ≤ b2 , …, gm (x, am ) ≤ bm for some ai in  1, 2, . . . , m. Therefore, if αi be the measure of belongingness of x0 in the fuzzy   region given by {x ∈ C : gi (x, ai ) ≤ bi }, i.e. ) = α = min{α1 , α2 , . . . , αm }. μ(x0 |X

(7.4)

Here it may be observed that for each i = 1, 2, . . . , m, the value of αi is the supremum of all the β’s (β ∈ [0, 1]) for which there exist ai = (ai1 , ai2 , . . . , aiqi )T ∈ bi (β) such that gi (x0 , ai ) ≤ bi .  ai (β) and bi ∈  Therefore, for each i = 1, 2, . . . , m,

sup min μ(ai1 | ai1 ), μ(ai2 | ai2 ), . . . , μ(aiqi | aiqi ), μ(bi | bi ) . αi = g(x0 ,ai1 ,ai2 ,...,aiqi )≤bi

For an algorithmic implementation, to search all possible sets of values of (ai1 , ai2 , . . . , aim )T and bi satisfying gi (x0 , ai1 , ai2 , . . . , aiqi ) ≤ bi , consider a uniform l l r r , ai1 ],  ai2 (0) = [ai2 , ai2 ], …,  aiqi (0) = discretization of the intervals  ai1 (0) = [ai1 l l r r  [aiqi , aiqi ] and bi (0) = [bi , bi ] in k equal number of grid points. If require, one may discretize in different numbers of grid points. Now fixing a particular bi in [bil , bir ], if search is made for ai1 , ai2 , . . . , aiqi with the requirement gi (x0 , ai1 , ai2 , . . . , aiqi ) ≤ bi , one can easily get the value of αi . The following Algorithm 1 gives the sequential procedure to find αi through direct use of extension principle. ) can be Once each αi (i = 1, 2, . . . , m) is known, the membership value μ(x0 |X obtained from (7.4).  can be In this way, the membership value of a particular point x0 ∈ C in X  evaluated. Now to find the membership function of X , construct the following set:   ai (α), bi ∈  bi (α), i = 1, 2, . . . , m Ω(α) = x ∈ Rn : gi (x, ai ) ≤ bi , ai ∈  for each α in [0, 1]. Thus according to the extension principle, membership function  is given by: of X ) = sup{α : x ∈ Ω(α)}. μ(x|X (7.5)  are determined by the following auxiliary result. The constituent α-cuts of X (α) = Ω(α) for all α in [0, 1]. Theorem 7.2.1 X Proof First let us prove the result for 0 < α ≤ 1. Let us take any α0 in (0, 1]. ), it Let x¯ be any element of Ω(α0 ). Then according to the definition of μ(.|X    follows that μ(x| ¯ X ) ≥ α0 . Thus, x¯ belongs to X (α0 ) and hence Ω(α0 ) ⊆ X (α0 ). (α0 ). Suppose μ(x|X ) = (α0 ) ⊆ Ω(α0 ), consider any element x in X To prove X β. Then β ≥ α0 > 0. It will be shown that x lies in Ω(β). Then the result will follow from Ω(β) ⊆ Ω(α0 ). ) = ) = β, there exists a sequence of numbers {βn } such that μ(x|X Since μ(x|X sup{βn : x ∈ Ω(βn ) for each n in N} = β. Here for all n ∈ N, βn ≤ β. Let us order the

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) through extension principle Algorithm 1 Algorithm to find μ(x0 |X Require: Give the value of k to discretize the support sets of  a11 ,  a12 , …,  amqm and  bi in k number of grid points. The positive integer k is a large number. It is assumed that  ai j (0) = [ail j , airj ] and  bi (0) = [bil , bir ] for j = 1, 2, . . . , qi and i = 1, 2, . . . , m. Final output α of the algorithm is the ). value μ(x0 |X 1: Initialize α1 ← 0, α2 ← 0, …, αm ← 0. 2: for i = 1 : 1 : m  %Comment: m is the number of fuzzy inequalities% do 3:

for bi = bil to bir with step length

bir −bil k

do

r −a l ai1 i1 do k r −a l ai2 length k i2 do

4:

l to a r with step length for ai1 = ai1 i1

5: 6:

l to a r with step for ai2 = ai2 i2 … … … … …

r −a l aiq iq

l to a r with step length i i 7: for aiqi = aiq do iqi k i 8: if gi (x0 , ai1 , ai2 , . . . , aiqi ) ≤ bi then 9: ti ← min{μ(ai1 | ai1 ), μ(ai2 | ai2 ), . . . , μ(aiqi | aiqi ), μ(bi | bi )} 10: if ti > αi then 11: αi ← ti  %Comment: Final value of αi is the membership value of x0  12: to satisfy g(x0 , ai ) ≤ bi % 13: end if 14: end if 15: end for 16: end for 17: end for 18: end for 19: end for 20: Return α = min{α1 , α2 , · · · αm }.

sequence β1 , β2 , β3 , . . . in non-decreasing order. Without loss of generality, let {βn } itself be an non-decreasing sequence. As β is supremum of the set {βn : n ∈ N}, the sequence {βn }, being a monotonic non-decreasing sequence, must be a convergent sequence and lim βn = β. One such sequence {βn } always exists, for example βn = n→∞

β − n1 for all n ∈ N. Thus, {βn } is a monotonic non-decreasing sequence converging to β and x ∈ Ω(βn ) for all n in N. It may be noted that Ω(β) ⊆ Ω(βn ) and Ω(βn+1 ) ⊆ Ω(βn ) for each n, because βn ≤ βn+1 ≤ β for all n ∈ N. For each n, since x ∈ Ω(βn ), there b(βn ) and an ∈  a (βn ) such that g(x, an ) ≤ bn . As a (0) and  b(0) are closed exist bn ∈  and bounded sets, the set  a (0) ×  b(0) is compact. Here {(an , bn )} is a sequence in  a (0) ×  b(0). Thus, there exists a convergent subsequence {(an p , bn p )} of {(an , bn )} in a (0) ×  b(0). Let lim (an p , bn p ) = (a0 , b0 ). Obviously, (a0 , b0 ) belongs to a (β) × p→∞

 b(β). It may be noted that g(x, an p ) ≤ bn p ∀ p ∈ N. This implies lim g(x, an p ) ≤ p→∞

lim bn p . Therefore, g(x, a0 ) ≤ b0 , since g is assumed as continuous. Thus, x must

p→∞

(α0 ) ⊆ Ω(α0 ). This shows X (α0 ) = Ω(α0 ). belong to Ω(β) ⊆ Ω(α0 ) and hence X

7.2 Fuzzy Multi Objective Optimization Problem

179

(α) = Ω(α) for all α in (0, 1]. Now As α0 is taken arbitrarily from (0, 1], X (α)) = closur e(∪α∈(0,1] Ω(α)) = Ω(0). Hence the result (0) = closur e(∪α∈(0,1] X X is proved.  It may be noted that for the FMOP (7.1), to find its decision feasible region X through extension principle, one needs to calculate membership value of each of the , a discretization points in the set C. For an effective computation to find the entire X of the set of elements in C will be considered. Then, for each discretized points in C, Algorithm 1 will be applied to get their individual membership values in the decision feasible region. However, here an important point is to notice is that for finding membership value for a point in C, a computational cost of m × O(k) × O(k q ) is required where q = max{q1 , q2 , . . . , qm }. Thus, if C is discretized by n number of grid points, extension-principle requires total computational complexity of n × m × O(k) × O(k q ) = O(n q+2 ), because usually n is much higher than k. The following subsection proposes an alternative procedure to find the decision feasible region of the FMOP (7.1) using the concept of inverse points. It is shown that employing inverse points in evaluation of the decision feasible region, only a computational cost of O(n 2 ) is needed and gives exactly same decision feasible region when all the functions gi (x, ai ) are increasing in each of their arguments (refer to Theorem 7.2.3). Thus the inverse points method needs much lesser computational cost compared to the extension principle method. The decision feasible region of the  . FMOP (7.1) when evaluated by the inverse point concept is denoted by X

7.2.4 Evaluation of Fuzzy Inequality Region by Inverse Points     Note that the constraint set of the FMOP (7.1) reads as x ∈ C : g(x, a) ≤ b . This set is a fuzzy set because the coefficients  a and  b are fuzzy. It may be observed   b may appear as equivalent to g(x,   that g(x, a) ≤ a) −  b ≤ 0. Corresponding to each x ∈ C, g(x, a ) is a fuzzy number (refer to the assumption (i), Sect. 7.2.1). The substraction g(x, a) −  b of fuzzy numbers can be done by the concept of inverse points. Thus, construct a set Ω (α) corresponding to each α in [0, 1] as follows: Ω (α) = {x ∈ C : g(x, a) − b ≤ 0, where g(x, a) and b are inverse points with membership value αong(x, a ) and  b

 Now define membership function of X by  ) = sup{α : x ∈ Ω (α)}. μ(x|X

(7.6)

 ), one may need its conIn order to obtain the mathematical formulation of μ(.|X

 stituent α-cuts X (α) for each α in [0, 1]. A natural question may arise whether there  (α) and Ω (α). The following theorem investigates is any interrelation between X

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the same. Prior to the theorem following lemma is given which will be useful to the theorem. Lemma 7.2.1 For any 0 ≤ α1 ≤ α2 ≤ 1, the set Ω (α2 ) is a subset of Ω (α1 ). Proof Take an element x0 from Ω (α2 ). Due to the definition of Ω (α2 ), there exist a ) and  b, respecinverse points g(x, aα2 ) and bα2 of membership value α2 on g(x, tively, such that x0 lies in {x ∈ C : g(x, aα2 ) ≤ bα2 }. Therefore, g(x0 , aα2 ) ≤ bα2 . a) It may be noted that due to the assumption (i) for the problem (7.1), each gi (x0 , is a continuous fuzzy number. As α2 ≥ α1 , one element g(x0 , aα1 ) can be found with a ) for which g(x0 , aα1 ) ≤ g(x0 , aα2 ). Similarly, there membership value α1 on g(x0 , b such that bα2 ≤ bα1 . exists bα1 with membership value α1 on  Therefore, g(x0 , aα1 ) ≤ g(x0 , aα2 ) ≤ bα2 ≤ bα1 . Hence, x0 lies in {x ∈ C : g(x, aα1 ) ≤ bα1 }. It is easy to see that g(x, aα1 ) and bα1 are inverse points of mema ) and  b, respectively. Thus, x0 belongs to Ω (α1 ). Hence, bership value α1 on g(x,

Ω (α2 ) ⊆ Ω (α1 ).  (α) = Ω (α) for all α in [0, 1]. Theorem 7.2.2 X  (α) is a subset of Ω (α) for any α ∈ (0, 1], take an Proof In order to argue that X

  ) ≥ α. Let μ(x0 |X  ) = γ. It will be shown element x0 from X (α). Then, μ(x0 |X

that x0 lies in Ω (α).  ) = sup{γ : x0 ∈ Ω(γ )} = γ. Suppose γ1 , γ2 , γ3 , …be a monotonic Here μ(x0 |X increasing sequence of numbers in (0, 1] such that x0 ∈ Ω (γ1 ), x0 ∈ Ω (γ2 ), x0 ∈ Ω (γ3 ), …. Since {γn } is a monotonic increasing sequence and γ being supremum of {γ1 , γ2 , γ3 , …}, the sequence {γn } must be convergent and converges to γ. It may be noted here that γn ≤ γ and γn+1 ≥ γn for all n ∈ N. Therefore, from Lemma 7.2.1, Ω (γ) ⊆ Ω (γn ) and Ω (γn+1 ) ⊆ Ω (γn ) for each n. As x0 ∈ Ω (γn ), there exist a ) and  b inverse points g(x0 , aγn ) and bγn with respect to the fuzzy numbers g(x0 , a) such that g(x0 , aγn ) ≤ bγn . Due to the assumption (i) for the problem (7.1), g(x0 , a )(0) ×  b(0) is a compact set. is a continuous fuzzy number. Therefore, g(x0 , a )(0) ×  b(0). Thus, there exists a The sequence {(g(x0 , aγn ), bγn )} lies in g(x0 , a) ×  b(0). convergent subsequence {(g(x0 , aγnk ), bγnk )} of {(g(x0 , aγn ), bγn )} in g(x0 , Let lim (g(x0 , aγnk ), bγnk ) = (g(x0 , a0 ), b0 ). Obviously, (g(x0 , a0 ), b0 ) belongs k→∞

to g(x0 , a )(γ) ×  b(γ). It may be noted that g(x0 , aγnk ) ≤ bγnk for all k ∈ N. This implies lim g(x0 , aγnk ) ≤ lim bγnk . Therefore, g(x0 , a0 ) ≤ b0 . Thus, x0 must belong k→∞

k→∞

 (α) ⊆ Ω (α). to Ω (γ). Again Ω (γ) ⊆ Ω (α), since γ ≥ α. Hence X  (α) take an element x¯ ∈ Ω (α). Then according to (7.6), To prove Ω (α) ⊆ X  (α).  ) ≥ α. Thus, x¯ belongs to X μ(x| ¯X  (α) = Ω (α) for any α ∈ (0, 1]. This shows that X  (α)) = closur e(∪α∈(0,1] Ω (α)) = Ω (0). Hence  (0) = closur e(∪α∈(0,1] X Now X the theorem is proved.

7.2 Fuzzy Multi Objective Optimization Problem

181

Example 7.1 Let the constraint set of an FMOP be  b,  a1 x 1 +  a2 x 2 ≤ x ∈ C = R2  = (9/12/15). Here the where  a1 =  4 = (3/4/5),  a2 =  3 = (2/3/6) and  b = 12  is to be constructed.  . The set X  12 4x1 +  3x2 ≤ decision feasible region is x ∈ R2  a) =  4x1 +  3x2 = (3x1 + 2x2 /4x1 + 3x2 / Corresponding to each x ∈ R2 , g(x, a) 5x1 + 6x2 ). For each α in [0, 1], inverse points with membership value α in g(x, and  b = (9/12/15) are (3x1 + 2x2 ) + α(x1 + x2 ) and 15 − 3α, or (5x1 + 6x2 ) − α(x1 + 3x2 ) and 9 + 3α, respectively. From the definition of Ω (α), for each α ∈ [0, 1],  Ω (α) = x ∈ R2 : g((x1 , x2 ), a) − b ≤ 0, where g((x1 , x2 ), a) and b  are inverse points with respect to g(x, a ) and  b   = {x ∈ R2 : (3x1 + 2x2 ) + γ(x1 + x2 ) − (15 − 3γ) ≤ 0} γ∈[α,1]

∪ {x ∈ R2 : (5x1 + 6x2 ) − γ(x1 + 3x2 ) − (9 + 3γ) ≤ 0}   {x ∈ R2 : (3 + γ)x1 + (2 + γ)x2 ≤ 15 − 3γ} =



γ∈[α,1]

 ∪ {x ∈ R2 : (5 − γ)x1 + (6 − 3γ)x2 ≤ 9 + 3γ} .

Due to the Theorem 7.2.2,  (α) = Ω (α) X   = {x ∈ R2 : (3 + γ)x1 + (2 + γ)x2 ≤ 15 − 3γ} γ∈[α,1]

 ∪ {x ∈ R2 : (5 − γ)x1 + (6 − 3γ)x2 ≤ 9 + 3γ} .

   is x ∈ R2 : 4x1 + 3x2 ≤ 12 . Core of X     .  is depicted in the Fig. 7.1. It will be tried to find μ (2, 4)X The constraint set X It may be noted that (2, 4) lies on (3 + γ)x1 + (2 + γ)x2 ≤ 15 − 3γ for γ ≤ 19 . However, lie on (5 − γ)x1 + (6 − 3γ)x2 ≤ 9 + 3γ for any γ in [0, 1].  cannot   (2, 4)  = 1 . Thus, μ (2, 4)X 9  , To find membership value of a particular point, x0 ∈ C say, on the fuzzy set X α α one can fix a value of α ∈ [0, 1] and then search inverse points (ai1 , . . . , aiqi ) and biα α α satisfying gi (x, ai1 , . . . , aiq ) ≤ biα . Among all possible such α’s, to find the highest α, i computational procedure will start from α = 1 and run sequentially with decreasing α α , . . . , aiq ) and biα is found values up to α = 0 until a pair of inverse points aiα = (ai1 i α α α , . . . , aiq ) ≤ b . Supremum value of such α’s is the membership such that gi (x, ai1 i i

 value μ(x0 |X ). The following Algorithm 2 gives details of the procedure.

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7 Fuzzy Pareto Optimality

Fig. 7.1 Fuzzy constraint  of the Example 7.1 set X

 ) through inverse points Algorithm 2 Algorithm to obtain μ(x0 |X Require: Give k, the number of grid points to discretize the interval of membership values [0, 1].  ). Final output α of the algorithm is the value μ(x0 |X 1: for α = 1 to 0 with step length k1 do 2: for i = 1 : 1 : m do 3: Find inverse points aiα and biα with membership value α on  ai and  bi 4: if gi (x, aiα ) ≤ biα then 5: Go to Step 9 6: end if 7: end for 8: end for 9: Return α

Through computing membership value of each of the points in C, the fuzzy set  can be obtained. Practically, through computing membership values of a huge X  can be captured. number of grid points in C, the fuzzy region corresponding to X  through evaluComputational cost of the Algorithm 2 is m × O(k). To find X ation of membership values of n number of grid points in C, Algorithm 2 requires computational cost of n × m × O(k) = O(n 2 ), as k is much lesser than n. Recall here that to find the decision feasible region, the extension principle method required the computational cost of O(n q+2 ) where q = max{q1 , q2 , . . . , qm }. As q is usually much higher than 1, to evaluate the decision feasible region, the computational

7.2 Fuzzy Multi Objective Optimization Problem

183

complexity of the inverse points method is vastly lesser than the extension principle method.

 and X  7.2.5 Relation Between X It may be noted that in the decision feasible set of the FMOP (7.1), corresponding to ai ) is a continuous fuzzy number for each i = 1, 2, . . . , m. every x in C, each gi (x, ai ) is a fuzzy number, there Let x1 ∈ C. As gi (x1 ,    exist  two reference functions ai ) = gil (x1 ) gim (x1 ) gir (x1 ) L 1 R1 . L 1 and R1 such that gi (x1 , , there exist Similarly, for x2 ∈ C,x2 = x1  two reference functions L 2 and R2 such ai ) = gil (x2 ) gim (x2 ) gir (x2 ) L 2 R2 . that gi (x2 , Apparently, L 1 = L 2 and R1 = R2 , since formulation of the reference functions ai and L 1 , L 2 , R1 and R2 depends on the reference functions of the components of  does not depend on the points in C. ai ) = (gil (x)/gim (x)/gir (x)) L R . Indeed, the left spread gim (x) − gil (x) Let gi (x, and the right spread gir (x) − gim (x) vary point-to-point on C. It may be noted that though the functions gil (x), gim (x) and gir (x) seem to be dependent on x alone, they implicitly depend on the constants in  ai (0). Following formulation shows how to ai ). evaluate membership value of any point lying in the support of gi (x, ai ) is evaluated in the following way: For each i = 1, 2, . . . , m and x ∈ C, gi (x,    ai ) = μ y gi (x,

sup

y=gi (x,ai )

π(ai | ai ) where π(ai | ai ) =

min

j=1,2,...,qi

μ(ai j | ai j ).

Theorem 7.2.3 For each α ∈ [0, 1], (1) In general, Ω (α) is subset of Ω(α). b(0) = [bl , br ]. Let Ψi and Φi (2) Let for each i = 1, 2, . . . m,  ai (0) = [ail , ai r ] and  l be the restrictions of μ(.| ai ) on [ai , a (1)] and [ ai (1), air ], respectively; Ψm+1 and l   b(1), br ], respectively. If Ψi Φm+1 be the restrictions of μ(.|b) on [b , b(1)] and [ and Φi are continuous and strictly increasing functions and each component of g is strictly increasing function,1 then Ω(α) = Ω (α). Proof Both the results will be proved when g is a scalar-valued function. If g is a vector-valued function, the result can be proved by componentwise generalization. (1) Let x0 ∈ Ω (α). Therefore, there exist a ∈  a (0) and b ∈  b(0) such that g(x0 , a) ≤ a) b, where g(x, a) and b are inverse points with membership value α on g(x0 , and  b, respectively. If possible let μ(a| a ) < α. Then according to the evaluation    a ), it can be obtained that μ g(x0 , a)g(x0 , a) of membership function of g(x0 , < α. A contradiction arises. This shows that x0 lies in Ω(α). Thus, Ω (α) ⊆ Ω(α). function h(x1 , x2 , . . . , xn ) is said to be strictly increasing when x1 > y1 , x2 > y2 , …, xn > yn implies h(x1 , x2 , . . . , xn ) > h(y1 , y2 , . . . , yn ). 1A

184

7 Fuzzy Pareto Optimality

(2) To prove Ω(α) ⊆ Ω (α), take any element x0 ∈ Ω(α). As x0 ∈ Ω(α), there a1 (α), a2 (α), . . . , am (α))t =  a (α) and b0 ∈  b(α) exist a 0 = (a10 , a20 , . . . , am0 )t ∈ ( 0 0 such that g(x0 , a ) ≤ b . Let φ(a1 , a2 , . . . , am , b) = g(x0 , a1 , a2 , . . . , am ) − b. As g is a strictly increasing function with respect to each ai , the function φ is increasing with respect to each ai and decreasing for b. Suppose −b = am+1 , 0 . Define a function h by − b = am+1 and −b0 = −am+1 h(a1 , a2 , . . . , am , am+1 ) = φ(a1 , a2 , . . . , am , −am+1 ) = g(x0 , a1 , a2 , . . . , am ) + am+1 .

Then, h is a strictly increasing function with respect to each of its variable. 0 0 ) = g(x0 , a10 , a20 , . . . , am0 ) + am+1 = g(x0 , a 0 ) − b0 . Let k = h(a10 , a20 , . . . , am+1 Therefore, k ≤ 0. Here two cases may arise: (2.a) g(x0 , a 0 ) − b0 ≤ g(x0 , a (1)) −  b(1) or a (1)) −  b(1). (2.b) g(x0 , a 0 ) − b0 > g(x0 , 0 ) ≤ h( a1 (1), a2 (1), . . . , am+1 (1)). Define (2.a) In this case, h(a10 , a20 , . . . , am+1 a function ψ as −1 ). ψ = h(Ψ1−1 , Ψ2−1 , . . . , Ψm+1

As each Ψi−1 is continuous and strictly increasing, so is ψ on its domain. Also ψ is an one-to-one function. Let β = ψ −1 (k) and ai∗ = Ψi−1 (β), i = 1, 2, . . . , m + 1. ∗ 0 ) = ψ(β) = k = h(a10 , a20 , . . . , am+1 ). MoreTherefore, h(a1∗ , a2∗ , . . . , am+1 ai (1) and β = μ(k|h( a1 , a2 , . . . , am+1 )) ≥ α. over, ai∗ ≤  a (1)) and b∗ ≥  b(1) where a ∗ = (a1∗ , a2∗ , . . . , Therefore, g(x0 , a ∗ ) ≤ g(x0 , am∗ ). Also μ(g(x0 , a ∗ )|g(x0 , a )) = β = μ(b∗ | b) ≥ α. Thus g(x0 , a ∗ ) and b∗ are inverse points with membership value β on a ) and  b, respectively. Again g(x0 , a ∗ ) − b∗ = k ≤ 0. Hence x0 lies g(x0 , in Ω (β) ⊆ Ω (α). (2.b) This case is similar to the case (2.a) with Φ1 , Φ2 , . . . , Φm+1 in place of Ψ1 , Ψ2 , . . . , Ψm+1 , respectively. Hence under the conditions stated in the theorem, Ω (α) = Ω(α) is obtained. Note 7.1 Proof of the Theorem 7.2.3 follows that under the stated conditions in the theorem, corresponding to each points x0 in Ω (α), there exist inverse points a1 (α), a2 (α), . . . , am (α)) and b∗ ∈  b(α) such that a ∗ = (a1∗ , a2∗ , . . . , am∗ ) ∈ ( ∗ ∗ g(x0 , a ) ≤ b . Note 7.2 Under the assumptions of the Theorem 7.2.3,

7.2 Fuzzy Multi Objective Optimization Problem

 (α) = Ω (α) = X

m   

185

x ∈ C : gi (x, ai1 , . . . , aiqi ) ≤ bi , where the numbers

i=1 α∈[0,1] α α ai1 , . . . , aiq , biα are inverse points of membership α with respect to i   ai1 , . . . , aiqi ,  bi .

 of the FMOP (7.1) is determined by a fuzzy Corollary 7.2.1 If the constraint set X  b where x = (x1 , x2 , . . . , xn )t ∈ C ⊆ a2 x 2 + · · · +  an x n ≤ linear inequality a1 x 1 +  n R , then  (α) = {x ∈ C : a1α x1 + a2α x2 + · · · + anα xn ≤ bα : where a1α , a2α , . . . , anα and bα X are inverse points with respect to  a1 , a2 , . . . , an and  b}. Theorem 7.2.3 and its proof inform that under the stated conditions in the theorem, if ai and bi are inverse points in  ai and  bi , then gi (x, ai ) and bi are inverse points  (α) it ai ) and  bi . Therefore, to compute X with respect to the fuzzy numbers gi (x, is only needed to take union of all x ∈ C for which gi (x, ai1 , . . . , aiqi ) ≤ bi where α α , . . . , aiq ) and biα are inverse points of membership value α with respect to (ai1 i ( ai1 , . . . , aiqi ) and  bi for all i = 1, 2, . . . , m. Through this consideration, finally = X  are obtained. Ω(α) = Ω (α) and X  on the decision space Rn , it will After the construction of fuzzy feasible region X be tried to formulate fuzzy feasible region on the objective space Rk . Denote the  objective feasible region of the FMOP (7.1) by Y.

7.2.6 Fuzzy Objective Function Space  in the previous section, μ(x|X ) is known to us According to the formulation of X  ) = α. corresponding to each x ∈ C. Take any particular x0 from X (0). Let μ(x0 |X c j ) is a continuous fuzzy number for all j = 1, 2, . . . , k. For any y ∈ Rk , Now f j (x0 , c)) by y = (y1 , y2 , . . . , yk )t , let us define membership function of μ(y| f (x0 ,   c)) = min μ(y1 | f 1 (x0 , c1 )), μ(y2 | f 2 (x0 , c2 )), . . . , μ(yk | f k (x0 , ck )) . μ(y| f (x0 , jm

jl

jr

c j ) be c0 and [c0 , c0 ], respecLet core and support of the fuzzy number f j (x0 , c j )) is continuous, the memtively, for each j = 1, 2, . . . , k. Since each μ(.| f j (x0 , c)) must be continuous. Therefore, μ(y| f (x0 , c)) will bership function μ(.| f (x0 , determine a continuous fuzzy point on Rk with support [c01l , c01r ] × [c02l , c02r ] × · · · × [c0kl , c0kr ] and core at (c01m , c02m , . . . , c0km ). (0), a fuzzy point given by μ(.| f (x0 , c)) will be Thus, corresponding to x0 ∈ X ) = α, membership value of each point on the support of obtained. Since μ(x0 |X

186

7 Fuzzy Pareto Optimality

 Define a fuzzy set the fuzzy point f (x0 , c) may not be expected to exceed α on Y. c) by fˆ(x0 , cj ) on the support of f (x0 ,            , μ y  f (x0 , μ y  fˆ(x0 , c) = min μ x0 X c) . It may be observed here that graph of μ(y| fˆ(x0 , c)) will be obtained by truncating , c )) where μ(y| f (x0 , c)) is exceeding α = the upper part of the graph of μ(y| f (x 0    ). The graph of μ y  fˆ(x0 , c) has been pictorially presented in the Fig. 7.2. μ(x0 |X (0) has been taken arbitrarily, a fuzzy set fˆ(x, c) will be obtained As x0 ∈ X (0). Now define Y  as corresponding to each x ∈ X = Y



fˆ(x, c).

(7.7)

(0) x∈X

Example 7.2 Consider the following minimization problem with two fuzzy objectives:

 through computing μ(y| fˆ(x0 , Fig. 7.2 Construction of Y c))

7.2 Fuzzy Multi Objective Optimization Problem

187

     x1 + (− 21 /0/ 14 ) f1 =  x2 + (− 21 /0/ 14 ) f2   12, subject to  3x1 +  4x2 ≤ x1 ≥ 0, x2 ≥ 0, min

= X  . The constraint  = (9/12/15). Here X where  3 = (2/3/6),  4 = (3/4/5) and 12

 set X has been demonstrated in the Example 7.1 (Fig. 7.1). , constituent α-cuts of the fuzzy Here corresponding to each x0 = (x10 , x20 ) ∈ X c) are given by point f (x0 ,  α−1 1 − α  α−1 1 − α f (x0 , c)(α) = x10 + , x10 + × x20 + , x20 + = S (α), say. 2 4 2 4

For instance, for x0 = (1, 1), c)(α) = f (x0 ,

α + 1 5 − α 1 + α 5 − α , × , . 2 4 2 4

(0) has membership value γ on X , then fˆ((1, 1), If x0 ∈ X c) is given by: fˆ((1, 1), c)(α) =



S(α) S(β)

if 0 ≤ α ≤ γ if α ≥ γ.

c)’s will determine the objective feasible region. The union of all possible fˆ(x0 ,  is the interior and boundary Support of the complete objective feasible region Y of the pentagon ABC D E A in the Fig. 7.3. The coordinates of the vertices of the  is the triangle O F G. pentagon are given in the figure. Core of Y  it will now be attempted to After evaluation of the objective feasible region Y,  formulate and capture the complete non-dominated set of Y.

7.2.7 Fuzzy Non-dominance  We note that Y(1) is a crisp set in the criteria feasible region. According to the classical definition of nondominated set through Pareto dominance we get Ehrgott  is nondominated (Pareto point) if (2005): a point y ∈ Y(1)  ∩ (y − Rk = {y}. Y(1)  is Therefore complete nondominated set of Y(1)  N = Y(1)

   y∈Y(1)

 ∩ (y − Rk ) = {y} y : Y(1)



188

7 Fuzzy Pareto Optimality

Fig. 7.3 Objective feasible region of the Example 7.2

 (1) is said If a point y in Y(1) is nondominated, then the point f −1 (y) in X as efficient point on decision feasible region. Through efficient points, complete nondominated set can be defined as    N =  ∩ ( f (x, c) − Rk ) = { f (x, c)} . Y(1) { f (x, c) : Y(1) (0) x∈X c∈ c(0)

Analogous to the classical concept of non-dominated set, let us define fuzzy nondominated set as follows.  and c ∈  Definition 7.2.2 (Fuzzy non-dominated set) Let us consider any x ∈ X c. k  If the fuzzy region Y ∩ ( f (x, c) − R ) is a normal fuzzy set with singleton core  { f (x, c)}, then this intersection region may be said as a non-dominated region of Y.   Fuzzy non-dominated set, Y N say, of Y may be defined by

7.2 Fuzzy Multi Objective Optimization Problem

N = Y

  (0) x∈X c∈ c(0)

189

 ∩ ( f (x, c) − Rk ) : Y  ∩ ( f (x, c) − Rk ) is a Y

 normal fuzzy set with singleton core { f (x, c)} .

 ∩ (y − Rk )). N = ∪ y∈Y(1) Theorem 7.2.4 Y  N (Y  and A N be set of all non-dominated points of A. Proof Let A denotes Y(1)  ∩ (y − Rk )) ⊆ Y N , let us take any element y¯ in A N . Then To prove ∪ y∈A N (Y  and c¯ in   ∩ ( y¯ − Rk ) there exist x¯ in X c such that y¯ = f (x, ¯ c). ¯ We note that Y k  = 1. Here core of  ∩ ( y¯ − R ) = μ(y|Y) is a normalized fuzzy set, since μ( y¯ | Y  ∩ ( y¯ − Rk ) contains only the point y¯ , since otherwise if this set contains any other Y  ∩ ( f (x, ¯ c) ¯ − element except y¯ , then it violets the assumption that y¯ ∈ A N . Thus, Y k ¯ c)}. ¯ Arbitrariness of y¯ ∈ A N R ) is a normal fuzzy set with singleton core { f (x,  ∩ (y − Rk )) ⊆ Y N . proves that ∪ y∈A N (Y  Let T be an element of Y N . Then due to Definition 7.2.2 there must exist x¯ in (0) and c¯ in  ∩ ( f (x, ¯ c)}. ¯ X c such that T = Y ¯ c) ¯ − Rk ) and T has singleton core { f (x, We will show that y¯ = f (x, ¯ c) ¯ must belong to A N . Let us note that T be a normal  ∩ (y − Rk ))(1) = fuzzy set with singleton core { y¯ }. Since y¯ − Rk is a crisp set, ((Y k  Y(1) ∩ ( y¯ − R ) which is the singleton set { y¯ }. Thus, y¯ ∈ A N and hence T belongs  ∩ (y − Rk )). Therefore, Y N ⊆ ∪ y∈A N (Y  ∩ (y − Rk )). Hence the result to ∪ y∈A N (Y is proved. N , Therefore, to capture entire non-dominated set of criteria feasible region, Y  ∩ (y − Rk ) for all possible non-dominated point y of we need to take union of Y  Y(1). Thus, for obtaining entire fuzzy non-dominated set of FMOP (7.1) we have  to first capture complete non-dominated set of the classical set Y(1). In the next section, a method Ghosh and Chakraborty (2012) to generate complete Pareto set of a classical multi-criteria optimization problem is demonstrated. Thereafter a techinique  ∩ (y − Rk ) which is the set Y N . is propped to get the union ∪ y∈Y(1)  N (Y

7.3 A Direction Based Scalarization Technique to Generate  N Y(1)  (1) by the funcWe note that Y(1), essentially, is obtained from the image of X   tion f (x, c(1)), i.e., Y(1) = f (X (1), c(1)). According to Ghosh and Chakraborty  (2012), to obtain a non-dominated point of Y(1), one may solve the following min-

190

7 Fuzzy Pareto Optimality

imization problem corresponding to a particular βˆ ∈ Sk−1 = Sk−1 ∩ Rk (here Sk−1 represents unit ball in Rk ): ⎧ ⎪ ⎨ ⎪ ⎩

min z subject to z βˆ ≥ f (x, c(1)), (1). x ∈X

(7.8)

Geometrically, z βˆ for z ≥ 0 represents points on the line which is directed along βˆ and passing through origin. For a particular z ≥ 0, constraints of (7.8) corresponding (1) : z βˆ ≥ f (x, (1), to βˆ presents the set {x ∈ X c(1))} = (z βˆ − Rk ) ∩ f (X c(1)) = k ˆ  (z β − R ) ∩ Y(1). When z 2 > z 1 ≥ 0, we note that z 2 βˆ > z 1 βˆ and thus z 1 βˆ − Rk ⊂ z 2 βˆ − Rk ,  ⊂ (z 2 βˆ − Rk ) ∩ Y(1).  which implies (z 1 βˆ − Rk ) ∩ Y(1) Figure 7.4 depicts this phenomena for a bi-criteria problem where ||O A1 || = z 1 and ||O A2 || = z 2 . ‘Min z’ of (7.8) corresponding to βˆ will be attained for such a z, equals to z β say, for which  will about to be null set. If at extremum the intersecting region (z β βˆ − Rk ) ∩ Y(1) k ˆ ˆ  situation, (z β β − R ) ∩ Y(1) = {z β β}, then trivially z β βˆ is a non-dominated point

Fig. 7.4 Illustration of Problem (7.8) corresponding to βˆ for a bi-criteria problem

(1) N 7.3 A Direction Based Scalarization Technique to Generate Y

191

Pascoletti and Serafini (1984), since otherwise there must exist another z 0 βˆ for which ˆ is z β . z 0 βˆ < z β βˆ and this contradicts to ‘min z’ of IC(β)  On the other hand, for any non-dominated point y¯ in Y(1), setting sˆ = y¯ , let || y¯ ||

us note that solution of (7.8) corresponding to sˆ will be z s for which z s sˆ = y¯ . Here  ⊂ Rk . Thus, entire non-dominated set of Y(1)  can be captured sˆ ∈ Sk since Y(1) k−1 ˆ ˆ through solving (7.8) corresponding to β for each β in S . Let us observe that in the extremum situation, at which ‘min z’ of (7.8) for βˆ is  may contain some other point, y0 obtained as z β , the intersection (z β βˆ − Rk ) ∩ Y(1) y0 ˆ ˆ ˆ say, except z β β. Obviously, if β0 = ||y0 || = β then solution of (7.8) corresponding  to βˆ0 is also y0 . Thus defining a set S¯ k−1 = βˆ : Problem (7.8) corresponding to βˆ  c(1))/|| f (xβ , c(1))|| = βˆ , let us observe that to has solution xβ such that f (xβ ,  obtain complete non-dominated set of Y(1), we only need to solve Problem (7.8) k−1 k−1 ˆ ˆ ¯  N denotes complete instead of for all β ∈ S . Therefore, if Y(1) for all β ∈ S  non-dominated set of Y(1), then  N = Y(1)



ˆ {z β βˆ : z β is ‘min z of IC(β)}.

ˆ S¯ k−1 β∈

N for fuzzy MOP (7.1). Let us now attempt to capture entire Y

N 7.4 On Generation of Entire Y The classical method presented in the above subsection can capture complete non dominated set of Y(1). Due to this method and Theorem 7.2.4,  N =  ∩ (y − Rk )) Y (Y  N y∈Y(1)

=



 ∩ (z β βˆ − Rk )), (Y

ˆ S¯ k−1 β∈

ˆ To compute Y(0)  corresponding to each βˆ ∈ S¯ k−1 , where z β is ‘min z’ of (7.8) for β.  ∩ (z β βˆ − Rk ) can be obtained by restricting Y  on the set Aβ = {y ∈ Y(0)  : the set Y k ˆ ˆ   y ≤ z β β}. Now if we take any y ∈ Aβ , then μ(y|Y ∩ (z β β − R )) = μ(y|Y), since  ∩ (z β βˆ − Rk ) as A β . z β βˆ − Rk is a crisp set. For any βˆ ∈ S¯ k−1 , let us denote Y  when y in Aβ β ) = μ(y|Y) β is given by μ(y| A Obviously, membership function of A N can be obtained by and ‘0’ otherwise. Thus, entire Y N = Y



β A

ˆ S¯ k−1 β∈

ˆ β for three different values of β. Figure 7.5 portrays the sets A

192 Fig. 7.5 Illustration of β s N through A computing Y

7 Fuzzy Pareto Optimality

N 7.4 On Generation of Entire Y

193

Above discussion and results show that the problem (7.8) is to be solved for each unit vectors βˆ ∈ S¯ k−1 to obtain complete fuzzy non-dominated set of FMOP (7.1). Determination of S¯ k−1 may not be a easy task due to its very definition. Thus we consider a superset of S¯ k−1 to ensure complete non-dominated set generation. In practice and algorithmic implementation of the proposed method, a discretization ˆ Solving (7.8), for for technique of Sk−1 would be considered to get required βs. ˆ β will be increased more and more β, number of obtained nondominated element A and gradually the entire fuzzy non-dominated set will be captured. In the following, an algorithmic implementation of the proposed methodology is given. The algorithm finds a uniform discretization of Sk−1 and then apply (7.8) for seeking a nondominated β . element A

7.4.1 Algorithmic Implementation of the Proposed Method Let us note that any βˆ ∈ Sk−1 can be expressed by 

cos φ1 , cos φ2 sin φ1 , cos φ3 sin φ2 sin φ1 , . . . , cos φk−1

k−2  i=1

sin φi ,

k−1 

 sin φi ,

i=1

for φi ∈ [0, π2 ], i = 1, 2, . . . , (k − 1). This is well know spherical discretization technique. However, if we discretize each φi to equal number of sub intervals, then set of discretized points will be much congested near the point (1, 0, . . . , 0). Thus, to get a k−1 uniform discretized points i on S , let us attempt to divide φ1 by m number of points and φi by r ound(m l=1 sin φi ) number of points, for i = 2, 3, . . . , k − 1. Here

Fig. 7.6 Spherical discretization of S2≥

194

7 Fuzzy Pareto Optimality

round is the rounding function to the nearest integer. Spherical discretized points and uniform discretized points for S2 are shown in the following Figs. 7.6 and 7.7. Following Algorithm 3 provides a sequential procedure to obtain complete Pareto set of a tri-criteria problem. In tri-criteria problem, we need to run 3 for loops for each φi , i = 1, 2, 3. For k-criteria problem, we only have to run k for loops for each φi , i = 1, 2, . . . , k. Algorithm 3 Algorithm to generate complete fuzzy non-dominated set Require: Given fuzzy MOP:

⎧ ⎪ ⎨ ⎪ ⎩

min f (x, c)  subject to g(x, a) ≤ b, x ∈ C ⊆ Rn .

N of the algorithm is the complete fuzzy non-dominated set of the problem. Final output Y 1: Initialize φ1 ← 0, φ2 ← 0, and φ3 ← 0. N ← ∅. Initialize Y Give m (total number of grid points for φ1 ). π 2: for φ1 = 0 to π2 with step length 2m do 3: m 2 ← r ound(m sin φ1 ). 4: for φ2 = 0 to π2 with step length 2mπ 2 do 5: m 3 ← r ound(m sin φ1 sin φ2 ) 6: for φ3 = 0 to π2 with step length 2mπ 3 do 7:

ˆ Find z β which is solution of the following problem for β: ⎧ ⎪ ⎨

min z

subject to z βˆ ≥ f (x, c(1)), ⎪ ⎩ (1). x ∈X  3 ˆ β = Y  8: Evaluate A  (z β β − R ).    Aβ . 9: Set Y N ← Y N 10: end for 11: end for 12: end for

In the next, an numerical example has been presented to elaborate the proposed N . method to capture Y

7.4.2 An Illustrative Example Example 7.3 Let us consider the following fuzzy bi-criteria minimization problem:

N 7.4 On Generation of Entire Y

195

Fig. 7.7 Uniform spherical discretization of S2≥

   x1 + (− 41 /0/ 21 ) f min 1 = f2 x2 + (− 1 /0/ 1 ) 4

2

 2, 2(x2 − 1)2 ≤ subject to  4(x1 − 1)2 +  1 1 − ≤ x1 ≤ 1, − ≤ x2 ≤ 1, 2 2 where  4 = (2/4/6) and  2 = (1/2/3). Same points with respect to  4,  2 and − 2 are 2 + 2α, 1 + α and − (3 − α)

or

6 − 2α, 3 − α and − (1 + α).

Therefore according to decision feasible region construction through inverse  is given by points (Sect. 1.8), the set X   = X {(x1 , x2 ) ∈ [− 21 , 1] × [− 21 , 1] : α∈[0,1]

(2 + 2α)(x1 − 1)2 + (1 + α)(x2 − 1)2 ≤ 3 − α}. ∪{(x1 , x2 ) ∈ [− 21 , 1] × [− 21 , 1] : (6 − 2α)(x1 − 1)2 + (3 − α)(x2 − 1)2 ≤  1 + α} .  is determined by For each α ∈ [0, 1], α-cut of X (α) = Ω(α) = (x1 , x2 ) ∈ [− 1 , 1] × [− 1 , 1] : 2(x1 − 1)2 + (x2 − 1)2 ≤ X 2 2

3−α 1+α

.

 For any (x1 , x2 ) ∈ [− 21 , 1] × [− 21 , 1], closed form of membership function of X is as follows: ⎧ ⎪ if 2(x1 − 1)2 + (x2 − 1)2 ≤ 1 ⎨1 4 ) = μ((x1 , x2 )|X −1 if 1 ≤ 2(x1 − 1)2 + (x2 − 1)2 ≤ 3 1+2(x1 −1)2 +(x2 −1)2 ⎪ ⎩ 0 elsewhere.

196

7 Fuzzy Pareto Optimality

Fig. 7.8 Fuzzy decision  of the feasible region X Example 7.3

X2

2

E (1

3 2

,11)

B (1

1 2

,1 )

A (1, 1)

1 (1, 0) C O

F ( 1,1  3 )

X1

2( x1  1) 2  ( x2  1) 2  13

2( x1  1)  ( x2  1)  1 2

2

2( x1  1) 2  ( x2  1) 2  3

 is depicted in the Fig. 7.8. Any point on the part The decision feasible region X } has memof ellipse {(x1 , x2 ) ∈ [− 21 , 1] × [− 21 , 1] : 2(x1 − 1)2 + (x2 − 1)2 ≤ 3−α 1+α  bership value α on X . (0), f (x0 , Corresponding to each point x = (x1 , x2 ) ∈ X c) determines a fuzzy 1 1 1 point with support [x1 + 4 , x1 + 2 ] × [x2 − 4 , x2 + 21 ]. For each α ∈ [0, 1], α-cut of f (x0 , c) is given by  α−1 1 − α  α−1 1 − α , x1 + × x2 + , x2 + = S(α) say . f (x0 , c)(α) = x1 + 4 2 4 2

(0) (with μ((x1 , x2 )|X ) = γ say) the Here corresponding to any (x1 , x2 ) in X c) is given by its α-cuts for each α in [0, 1] as follows: fuzzy set fˆ((x1 , x2 ), !

S(α) fˆ((x1 , x2 ), c)(α) = S(γ)

if α ∈ [0, γ] if α ∈ [γ, 1].

(0), then μ(( 1 , 1 )|X ) = For example if we consider (x1 , x2 ) = ( 15 , 21 ) ∈ X 5 2 1 = 0.58 and for each α ∈ [0, 1]:

400 253

−

N 7.4 On Generation of Entire Y

197

Fig. 7.9 Criteria feasible  and non-dominated region Y N of the Example 7.3 set Y

! 1 1 S(α) c)(α) = fˆ(( , ), 5 2 [0.1, 0.4] × [0.4, 0.7]

if α ∈ [0, 0.58] if α ∈ [0.58, 1],

where S(α) = [ 5α−1 , 7−5α ] × [ 1+α , 2−α ]. 10 4 2  20  Criˆ c) determines the criteria feasible region Y. The union x∈X(0) f ((x1 , x2 ), teria feasible region is shown in the Fig. 7.9.  that We obtain from the proposed method on Y(1)    ( f − 3 )2 

√ ( f 2 − 23 )2 1 2  N = ( f 1 , f 2 ) ∈ 3 − √1 , 3 × 3 − 3, 3 : Y(1) + =1 3 1 7 2 2 4 4 2 2 2 (4 + √ ) (4) 2

N of the and S¯ 1 = {βˆ = (cos θβ , sin θβ ) : 0 ≤ θβ ≤ π2 }. Entire non-dominated set Y  considered problem is the fuzzy region, on the support of Y, lying inside and boundary N is the set of the region Q N RS Q in the Fig. 7.9. For each α ∈ [0, 1], α-cut of Y 3

( f 2 − 23 )2 ( f 1 − 23 )2 1 3 3 √ 3 " " + =1 . ( f1 , f2 ) ∈ −√ , × − 3, : 3 3−α 4 2 4 2 2 2 ( 4 + 2(1+α) )2 ( 43 + 3−α 1+α )

198

7 Fuzzy Pareto Optimality

7.5 Final Selection of Solution After generating complete fuzzy non-dominated set, one may search for preferable solutions for decision maker. To guess what solutions might be most preferable for a decision maker, a concept of so-called ‘knee’ of criteria Pareto set of classical MOPs has been studied by Das (1999), Branke et al. (2004), Rachmawati and Srinivasan (2006) and Deb and Gupta (2011). Knee points are appeared to be proper non-dominated point. The locations on the non-dominated set where criteria have bounded trade-offs are roughly known as properly non-dominated points. There are various definition of properly non-dominated points (Ehrgott 2005). Details of those definitions and their inter relations can be found in Ehrgott (2005). For fuzzy MOPs, no methodology is found yet for final decision making. In the following, a definition of proper non-dominated set is given below. In the definition ˜ b˜ is said to be dominated by a, ˜ if for two fuzzy numbers a˜ and b, ˜ written as a˜ ≺ b, ˜ ˜ = [a L , a R ] and b(0) = [b L , b R ]. For each x ∈ X˜ , α-cut (α ∈ a R < b L where a(0) [0, 1]) of f˜i (x) is denoted by f˜i (x)(α) = [ f˜iαL (x), f˜iαR (x)]. One can easily note that at the core level of Y˜ N the following definition gives the well-known Geoffrion definition of proper Pareto points. Definition 7.5.1 (Proper non-dominated points). Let yˆ be a point point in a nondominated region of Y˜ N . Then f −1 ( yˆ ) = xˆ (say) must belong to the decision feasible region X˜ . The point xˆ ∈ X˜ is called properly efficient with membership value β if (i) μ(x|X˜ ) = β and (ii) there is a real number M > 0 such that for all i and x ∈ X˜ (β) satisfying f˜i (x) ≺ ˆ there exists an index j satisfying f˜j (x) ˆ ≺ f˜j (x) such that f˜i (x) f˜iαR (x) ˆ − f˜iαL (x) ≤ M for all α ∈ [0, 1]. R L (x) − f˜jα (x) ˆ f˜jα Example 7.4 Let us consider the following bi-criteria optimization problem:  min

   f˜1 (x1 , x2 ) (0/1/2)x1 = (0/1/3)x2 f˜2 (x1 , x2 )

1 1  (0/ /1), subject to (0/ /2)(x1 − 1)2 + (0/1/3)(x2 − 1)2 ≤ 4 4 (x1 , x2 ) ∈ [0, 1] × [0, 1]. The decision feasible region is given by = X



(x1 , x2 ) : a(x1 − 1)2 + b(x2 − 1)2 ≤ c,

α∈[0,1]

1 7 1 1

where a ∈ [ α, 2 − α], b ∈ [α, 3 − 2α], c ∈ [ α, 1 − α] . 4 4 4 4

7.5 Final Selection of Solution

199

Therefore,

2 (1) = (x1 , x2 ) : (x1 − 1)2 + (x2 − 1) ≤ 1 . X 1/4 L R For the objective functions, f˜1α (x1 , x2 ) = αx1 , f˜1α (x1 , x2 ) = (2 − α)x1 , L R ˜ ˜ f 2α (x1 , x2 ) = αx2 and f 2α (x1 , x2 ) = (3 − 2α)x2 . Let us take the point xˆ = (1, 21 ). Membership value of xˆ in the feasible region X˜ √ is β = 1. Thus, we take a point x = (1 − , 1 − 21 1 − 2 ) ∈ X˜ (1) for any fixed ∈ (0, 1). √ Let us note that f˜1 (1 − , 1 − 21 1 − 2 ) = (1 − )(0/1/2) ≺ (0/1/2) = f˜1 (1, 21 ). Thus, let us take i = 1. √ √ Again f˜2 (1 − , 1 − 21 1 − 2 ) = (1 − 21 1 − 2 )(0/1/3)  21 (0/1/3) = f˜2 (1, 21 ). Therefore, j = 2. Let us now notice that for α = 1,

ˆ − f˜iαL (x ) f˜iαR (x) 2 = → ∞ as → 0 + . √ R L ˜ ˜ ˆ f jα (x ) − f jα (x) 1 − 1 − 2 Hence, the point xˆ = (1, 21 ) is not a properly efficient point for the considered problem. A sequential procedure to find complete proper non-dominated set for a FMOP is given in the following Algorithm 4. The following Algorithm 4 assumes that the decision feasible region X˜ is priory evaluated (through Algorithm 2) and for a given (β) is known to us. β ∈ [0, 1] the set X  (β) by Suppose X (β) contains n discretized points and we represent the set X  the array X (β)[k] for k = 1, 2, · · · , n. The algorithm will return an array, FPES say, containing fuzzy proper efficient points. FPES is abbreviation of ‘fuzzy proper efficient set’. The algorithm also returns a f (x) ¯ which is smallest distance away from the ideal point. This outcome point x¯ can be a decision maker’s preferable solution. Here x¯ is a proper efficient point. An application of the procedure presented in the Algorithm is given in the following example. Example 7.5 Let us consider the following fuzzy tri-objective optimization problem: ⎞ ⎞ ⎛ ˜ ˜ 22 2x1 + 3x f˜1 (x1 , x2 ) ⎜ ˜ 2 − 3x ˜ 2 cos x2 ⎟ min ⎝ f˜2 (x1 , x2 )⎠ = ⎝1x 1 ⎠ ˜ 1 x2 −3˜ log |x2 | 2x f˜3 (x1 , x2 ) ˜ 4 ˜ x ⎛

1x1 +3e2

 2, 2(x2 − 1)2 ≤ subject to  4(x1 − 1)2 +  1 1 − ≤ x1 ≤ 1, − ≤ x2 ≤ 1, 2 2

200

7 Fuzzy Pareto Optimality

Algorithm 4 Finding set of all fuzzy proper efficient points Require: 1: Give a satisfaction level β ∈ [0, 1] and a bound, M > 0 say, of trade-off between the objectives. 2: Give min1 , min2 , …minm , minimum possible value on the support of the objectives f˜1 , f˜2 , …, f˜m . 3: Set i ← 1, t ← 1 and FPES[1] ← 0  i is for i-th objective,  t is for t-th proper efficient point 4: for k = 1 : 1 : n (n points) do  k is for k-th element in X˜ (β) (β)[k] 5: xˆ ← X 6: for α = β to 1 with step length 1−β p do  p ∈ N being a large integer representing number of grid points 7: l ← 1, but l = k. If l happens to be k, then set l ← k + 1.  l is for l-th element in X˜ (β) (β)[l] 8: x ←X R (x) < f˜ L ( x) L ( x), R ( x)], L (x), f˜ R (x)] 9: if f˜iα  f˜i (x)(α) ˆ = [ f˜iα ˆ f˜iα ˆ f˜i (x)(α) = [ f˜iα iα ˆ then iα 10: for j ← 1 : 1 : n, but j = i. If j = i, then set j ← i + 1. do  j is for j-th objective L (x) ˆ f˜iα f˜ R (x)− 11: if f˜ R (x) ˆ < f˜L (x) and iα ≤ M then R L jα

12:

jα

ˆ f˜jα (x)− f˜jα (x)

FPES[t] ← xˆ and t ← t + 1.

 xˆ is a properly efficient point 13: k ← k + 1 and go to the Line 3. 14: end if 15: end for 16: end if 17: l ← l + 1, but l = k. If l = k, then set l ← k + 1. 18: end for 19: end for 20: min ← V a large number 21: for p = 1 : 1 : (t − 1) do (β)[ p]), (min1 , min2 , . . . , minm )) then 22: if min < d( f (X  d(x, y) is the usual distance metric on Rm (β)[ p]), (min1 , min2 , . . . , minm ) 23: c ← p and min ← d( f (X 24: end if 25: end for (β)[c] 26: Return X  This point is the final solution

where 1˜ = (0/1/2), 2˜ = (1/2/3) and 3˜ = (2/3/5). In this example, for a particular β ∈ [0, 1] the constraint set X˜ (β) is given by 1 1 3−β

X˜ (β) = (x1 , x2 ) ∈ [− , 1] × [− , 1] : 2(x1 − 1)2 + (x2 − 1)2 ≤ 2 2 1+β   3π

r cos θ 3−β ,π ≤ θ ≤ . = 1 + √ , 1 + r sin θ : 0 ≤ r ≤ 1+β 2 2 Thus X˜ (β) can be easily discretized in n = k 2 grid points for a discretization of ] (interval for r ) and [ π2 , 3π ] (interval for θ). k number of grid points of [0, 3−β 1+β 2

7.5 Final Selection of Solution Table 7.1 Final decision points Exp. no. (n) Satisfaction level (β) 1 2 3 4 5 6 7 8 9 10

0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.00

201

Trade-off bound (M)

Final solution (x1∗ , x2∗ )

5 5 1 1 1 1 1 1 2 10

(0.3157, 0.2493) (0.3486, 0.2471) (0.3746, 0.2466) (0.3985, 0.2442) (0.4301, 0.2432) (0.4605, 0.2434) (0.4911, 0.2336) (0.5128, 0.2316) (0.5493, 0.2295) (0.5493, 0.2295)

The above Table 7.1 explores final preferable solution (given by Algorithm 4) of the considered FMOP for 10 different trade-off values (M) and satisfaction values (β).

7.6 Conclusion Here a methodology has been suggested to obtain entire non-dominated or Pareto frontier of Fuzzy Multi Objective decision making problem. Pareto set reflects the compromise combination as decision maker may not be willing to compromise a huge loss of one objective for unit improvement of another. Geometrical and algorithimic illustration have been detailed. Final selection from the Pareto set is entirely decision maker’s choice.

References Bellman, R.E., Zadeh, L.A.: Decision making in a fuzzy environment. Manag. Sci. 17(4), B141– B164 (1970) Branke, J., Deb, K., Dierolf, H., Osswald, M.: Finding knees in multi-objective optimization. In: Yao, X., Burke, E.K., Lozano, J.A., Smith, J., Merelo-Guervós, J.J., Bullinaria, J.A., Rowe, J.E., Tino, P., Kabán, A., Schwefel, H.P. (eds.) Parallel Problem Solving from Nature - PPSN VIII. Lecture Notes in Computer Science, vol. 3242, pp. 722–731. Springer, Berlin, Heidelberg (2004) Carlsson, C., Fullér, R.: Fuzzy multiple criteria decision making: recent developments. Fuzzy Sets Syst. 78, 139–153 (1996) Das, I.: On characterizing the “knee” of the pareto curve based on normal-boundary intersection. Struct. Optim. 18(2), 107–115 (1999) Deb, K.: Multi-objective Optimization Using Evolutionary Algorithms. Wiley, New York (2001) Deb, K., Gupta, S.: Understanding knee points in bicriteria problems and their implications as preferred solution principles. Eng. Optim. 43(11), 1175–1204 (2011)

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Ehrgott, E.: Multicriteria Optimization, vol. 491, 2nd edn. Lecture Notes in Economics and Mathematical Systems, Springer, Berlin (2005) Ghosh, D., Chakraborty, D.: On solving fuzzy multi-criteria quadratic programming problems. In: Proceedings 6th International Conference of IMBIC, MSAST 2012, Kolkata, pp. 82–94 (2012) Ghosh, D., Chakraborty, D.: Ideal Cone: A New Method to Generate Complete Pareto Set of Multi-criteria Optimization Problems. In: Mohapatra R., Giri D., Saxena P., Srivastava P. (eds.) Mathematics and Computing 2013. Springer Proceedings in Mathematics & Statistics, vol. 91. Springer, New Delhi Ghosh, D., Chakraborty, D.: A method for capturing the entire fuzzy non-dominated set of a fuzzy multi-criteria optimization problem. Fuzzy Sets Syst. 272, 1–29 (2015a) Ghosh, D., Chakraborty, D.: On Fuzzy Ideal Cone Method to Capture Entire Fuzzy Nondominated Set of Fuzzy Multi-criteria Optimization Problems with Fuzzy Parameters. In: Facets of Uncertainties and Applications, pp. 249–260. Springer, New Delhi (2015b) Kahraman, C.: Fuzzy Multi-criteria Decision Making–Theory and Applications With Recent Developments, vol. 16. Springer Optimization and its Applications. Springer, Berlin (2008) Lai, Y.-J., Hwang, C.-L.: Fuzzy Multiple Objective Decision Making: Methods And Applications, vol. 404. Lecture Notes in Economics and Mathematical Systems, Springer, New York (1994) Marler, R.T., Arora, J.S.: Survey of multi-objective optimization methods for engineering. Struct. Multidiscip. Optim. 26, 369–395 (2004) Pascoletti, A., Serafini, P.: Scalarizing vector optimization problemss. J. Optim. Theor. Appl. 42(4), 499–524 (1984) Rachmawati, L., Srinivasan, D.: A multi-objective evolutionary algorithm with weighted-sum niching for convergence on knee regions. In: Proceedings of the 8th Annual Conference on Genetic and Evolutionary Computation, ACM, Springer, pp. 749–750 (2006) Ramík, J.: Optimal solutions in optimization problem with objective function depending on fuzzy parameters. Fuzzy Sets Syst. 158, 1873–1881 (2007) Sakawa, M., Yano, H.: Feasibility and pareto optimality for multiobjective nonlinear programming problems with fuzzy parameters. Fuzzy Sets Syst. 43, 1–15 (1991)

Chapter 8

Concluding Remarks and Future Directions

8.1 Chapter Summary and Conclusion The following are the main contributions of this book. In Chap. 1 some preliminaries on fuzzy set theory have been given here. Relevant literature on fuzzy geometry is detailed here. Decomposition principle and the extension principle on fuzzy set are revisited with some illustrations on the operations of fuzzy numbers. A new concept, same and inverse points of two fuzzy numbers have been introduced here. Chapter 2 starts with the basics of fuzzy geometry. Fuzzy point is the basic building block of fuzzy geometry. After defining fuzzy point the new concept, same and inverse point with respect to two fuzzy points has been introduced. This concept is essential in describing the nature and shape of a fuzzy point. If the fuzzy point is known with its entirety then any unary or binary operations with fuzzy point/points are possible. In this chapter linear combination of two fuzzy points and hence fuzzy line segment has been defined geometrically and algebraically. The basic properties of fuzzy distance, the ideas about the containment of a fuzzy point on a fuzzy line segment and the coincidence of two fuzzy points are also described. Fuzzy line segment has been extended bi-infinitely to get the fuzzy line in a twodimensional space. A fuzzy line passing through several fuzzy points whose cores are colinear is constructed. Chapter 3 mainly concentrates on fuzzy line. Consecutively, four different forms for fuzzy lines: a two-point form, a point-slope form, a slopeintercept form, and an intercept form are formulated. The inter relationship of these forms are also discussed here. In Chap. 4, new concepts about fuzzy triangle, fuzzy triangular properties and some basics of fuzzy trigonometry are proposed. After defining a fuzzy triangle, its area and perimeter have been studied. All the proposed concepts introduced here depend on the newly defined concepts of same and inverse points.

© Springer Nature Switzerland AG 2019 D. Ghosh and D. Chakraborty, An Introduction to Analytical Fuzzy Plane Geometry, Studies in Fuzziness and Soft Computing 381, https://doi.org/10.1007/978-3-030-15722-7_8

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204

8 Concluding Remarks and Future Directions

In Chap. 5, a comprehensive study on finding a fuzzy number with a fixed fuzzy distance from a given fuzzy number is presented. Accordingly, a fuzzy circle corresponding to a given fuzzy point and a fuzzy radius are formulated. A fuzzy circle passing through three fuzzy points is also introduced. Chapter 6 details the construction of fuzzy parabola. Geometric interpretation is given here with some numerical examples. In Chap. 7, through the developed fuzzy geometrical ideas, a rigorous study on representation of fuzzy decision feasible region and fuzzy objective feasible region of fuzzy multi-objective optimization problem are presented. This chapter also introduces a fuzzy dominance structure for fuzzy MOPs, analogous to the classical Pareto dominance concept. A new method named fuzzy ideal cone is presented to capture fuzzy non-dominated set of fuzzy MOP. For the final decision making, a properly fuzzy non-dominated solution is introduced.

8.2 Future Scopes There are several dimensions where the developed fuzzy geometry may be extended and applied. In this book only the fuzzy parabola has been introduced. But there are more conic sections which are developed by intersection of a plane and fuzzy circular cone, e.g. fuzzy ellipse, fuzzy hyperbola. Properties of these along with the ideas of corresponding eccentricity, focus and directrix need to develop. It will be more challenging task to look into the situation when a fuzzy plane intersects fuzzy circular cone. One can look at other fuzzy objects and their properties in three dimension e.g. fuzzy plane, fuzzy sphere, fuzzy ellipsoid, fuzzy circular cylinder, fuzzy parabolic cylinder, paraboloid etc. A higher dimensional concepts can also be developed thereafter. Fuzzy calculus is a field of study of many researchers. It is necessary to revisit the derivative and integral of fuzzy valued functions in fuzzy geometrical sense. For a general FMOP a construction procedure of fuzzy decision space and fuzzy objective feasible region has been suggested. No sensitivity analysis or scaling analysis was performed on the proposed technique. Future research may be pursued on this topic. There are a few application areas may be mentioned in this regard—Fuzzy interpolation or extrapolation, fuzzy medical imaging etc. In medical imaging it is seen that the aura boundaries of the cells, organs etc. are hazy in nature. So along with digital geometry, fuzzy geometry may give a new dimension to the study of medical image analysis.

Index

A Alpha-cut, 8 properties, 9

B Bijective transformation, 28

C Coincidence, 49 Containment, 46

D Decomposition principle, 9

E Effective combination, 39 Extension principle, 10

F Fuzzy circle, 115 center, 136 radius, 138 Fuzzy distance, 48 Fuzzy feasible space, 176 Fuzzy geometrical plane, 23 Fuzzy line, 53 general form, 84 intercept form, 74, 76 point-slope form, 64 construction, 67 y-intercept, 68

slope-intercept form, 69 two-point form, 56 construction, 59 slope, 60 y-intercept, 62 Fuzzy line segment, 44 Fuzzy Multi Objective Optimization Problem (FMOP), 174 Proper non-dominated points, 198 Fuzzy non-dominace, 187 Fuzzy number, 10 LR-type, 11 triangular, 11 Fuzzy number along a line, 27, 116 Fuzzy parabola, 145 Fuzzy point, 24 addition, 32 with elliptical base, 24 with rectangular base, 24 with rhomboidal base, 26 Fuzzy set, 8 Fuzzy sine function, 108 Fuzzy triangle, 93 perimeter, 99 Fuzzy trigonometry, 107

I Inverse points, 43 with respect to fuzzy points, 43

M Membership function, 8

© Springer Nature Switzerland AG 2019 D. Ghosh and D. Chakraborty, An Introduction to Analytical Fuzzy Plane Geometry, Studies in Fuzziness and Soft Computing 381, https://doi.org/10.1007/978-3-030-15722-7

205

206 R Redundant combination, 39

Index S Same points, 40 with respect to fuzzy points, 40