Advances in Fuzzy Systems

Volume 2018, Article ID 8623465, 9 pages

https://doi.org/10.1155/2018/8623465

## Gaussian Qualitative Trigonometric Functions in a Fuzzy Circle

Department of Mathematics, Hindustan Institute of Technology and Science (Deemed to be University), Chennai 603103, India

Correspondence should be addressed to M. Clement Joe Anand; moc.oohay@imeojra

Received 15 November 2017; Revised 26 February 2018; Accepted 11 April 2018; Published 3 June 2018

Academic Editor: Erich Peter Klement

Copyright © 2018 M. Clement Joe Anand and Janani Bharatraj. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We build a bridge between qualitative representation and quantitative representation using fuzzy qualitative trigonometry. A unit circle obtained from fuzzy qualitative representation replaces the quantitative unit circle. Namely, we have developed the concept of a qualitative unit circle from the view of fuzzy theory using Gaussian membership functions, which play a key role in shaping the fuzzy circle and help in obtaining sharper boundaries. We have also developed the trigonometric identities based on qualitative representation by defining trigonometric functions qualitatively and applied the concept to fuzzy particle swarm optimization using -cuts.

#### 1. Introduction

The term “Trigonometry” was first coined as the title of a book* (Trigonometria)*, which translates to “triangle’s measurement.” Although trigonometry is now taught with an emphasis on right triangles, its origin goes back to an era where it was used to determine the positions of celestial bodies and the distances between them and to understand the concept of chords in circles. Euclidean geometry, that is, planar geometry, deals with two-dimensional figures. The study of planar geometry provides constructions for planar figures, their properties, and the relationships between points, lines, and figures. The planar figure formed when a point traces a path at a fixed distance with respect to another fixed point called a circle, where the circle divides a plane into interior and exterior regions. Various theorems and properties on circles have been developed since time immemorial.

Gradually, researchers have developed theories on intersecting circles, which led to divergent properties between circular triangles and spherical triangles. The introduction of fuzzy sets and systems by Zadeh [1] changed the face of research in trigonometry. Fuzzy trigonometry was introduced by Buckley and Eslami [2], wherein continuous fuzzy numbers and sets were defined using the principle of extension. This method formed the basis of fuzzy trigonometry but failed to satisfy many criteria and identities. Furthermore, Ress [3] developed an approach for mapping standard trigonometric functions into the fuzzy realm. Using these modified fuzzy trigonometric functions, the proofs of a few inverse trigonometric identities, in addition to the standard identities, were given. A breakthrough in the study of fuzzy trigonometry was achieved by Liu et al. [4], with an aim of connecting symbolic cognitive functions to qualitative functions. The basic identities were satisfied, but a few properties could not be achieved. Ghosh and Chakraborty [5] proposed two methodologies for describing a fuzzy circle. The first methodology defines a circle as a set of points which are equidistant from a fixed point. The second methodology describes a circle using three fuzzy points. The definitions using both methods are as follows.(1)Considering fuzzy numbers plotted along various line segments which pass through a specific point which is fuzzy and located at a distance of from the fixed fuzzy point, then the fuzzy circle is defined as where the distance between and a random point on the support is always fixed.(2)Considering three fuzzy points, the fuzzy circle is drawn by passing the circle through these three points, and the supremum of the membership values defines the fuzzy circle which was proposed.

Technology in today’s world needs advanced level procedures or heuristics which involve few or zero assumptions about the problem being optimized. The need for metaheuristics has many advantages over other algorithms, with one advantage being its ability to search very large candidate solutions spaces. Particle swarm optimization (PSO) is one such metaheuristic method which allows optimization based on iterations, which in turn helps in improving the candidate solution. The solution is improved by creating a population of candidate solutions (i.e., particles) and making them move around in the search-space, taking into consideration their positions and velocities. The movement of the particle is influenced by its local best known position, simultaneously being guided towards the best known position in the search area. These are updated time-to-time for all other particles to form a swarm. PSO was first introduced by Kennedy et al. with an intention of simulating the social behaviour in a flock of birds or a school of fish. Shi and Eberhart [6] developed an optimization technique for fuzzy systems by dynamically adjusting the inertia weight, improving PSO’s performance. Pang et al. [7] utilized fuzzy discrete PSO to solve the Travelling salesman problem., Abdelbar et al. [8] compared the behaviour of the Gaussian and Cauchy membership functions in fuzzy PSO and concluded that the Cauchy membership functions are best suited for fuzzy PSO. It was noted that traditional PSO converges prematurely when applied as a global optimization technique. To prevent this downfall, Anantathanavit and Munlin [9] proposed radius (R-PSO) as an extension of standard PSO. R-PSO regroups the particles into a circle of the given radius and determines the best agent particle of the group. The experiment results proved that R-PSO performed better than traditional PSO when solving multimodal complex problems.

To produce wholesome proofs for identities based on fuzzy trigonometric functions, we introduce the Gaussian qualitative trigonometric functions (GQTFs) on a fuzzy circle. The unit fuzzy circle is defined using Gaussian membership function (GMF) partitions. Using the GMF circumference, fuzzy centre, and fuzzy point on the circle, trigonometric functions and their identities are successfully developed. In this paper, we use the advantages of R-PSO and GMFs to define a fuzzy R-PSO based on a fuzzy qualitative circle. Section 2 gives the preliminaries and introduces the concept of GQTFs on the fuzzy circle. Section 3 provides insight on the formation of the fuzzy centre and fuzzy point on the circle and provides a broad perspective of the redefined trigonometric functions, their ratio identities, their Pythagorean identities, their Taylor’s series expansions, their differentials, and laws for obtaining the solutions of triangles. Section 4 describes the properties of intersecting fuzzy circles, and Section 5 describes the PSO with fuzzy matrices. Section 6 describes fuzzy R-PSO with GMF, which makes an efficient model for finding the best agent. Finally, Section 7 concludes the paper.

#### 2. Prerequisites

The prerequisites required for the developed concept have been taken from various research articles which are cited in the references. Slightly modified versions of the concepts have been given in this section.

##### 2.1. Fuzzy Subset

Considering a set which is either finite or infinite, for every element in , the set consisting of all ordered pairs of the form is called a fuzzy subset, denoted by , in . Here, is called the membership function. The membership function gives a mapping of the elements in set to the membership set .

##### 2.2. Zadeh’s Extension Principle

Considering a fuzzy subset of a universal set , if is a function, then the extension principle generates a function whose membership function is defined over the supremum of all , that is,Here, is the preimage of in .

##### 2.3. Gaussian Membership Function (GMF)

The GMF is given bywhere and are the centre and width of the fuzzy set, respectively. An example of this function is shown in Figure 1. This membership function is also defined in terms of the interval as follows:The GMF with fuzzification factor* m* is given by