Advances in Fuzzy Systems

Volume 2018, Article ID 9578270, 7 pages

https://doi.org/10.1155/2018/9578270

## A Direct Method to Compare Bipolar LR Fuzzy Numbers

^{1}Faculty of Mathematical Sciences, Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran^{2}Faculty of Mathematical Sciences, Sharif University of Technology, Tehran, Iran

Correspondence should be addressed to Reza Ghanbari; ri.ca.mu@irabnahgr

Received 12 November 2017; Accepted 13 February 2018; Published 14 March 2018

Academic Editor: Mehmet Onder Efe

Copyright © 2018 Reza Ghanbari et al. 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 propose a new method for ordering bipolar fuzzy numbers. In this method, for comparison of bipolar LR fuzzy numbers, we use an extension of Kerre’s method being used in ordering of unipolar fuzzy numbers. We give a direct formula to compare two bipolar triangular fuzzy numbers in operations, making the process useful for many optimization algorithms. Also, we present an application of bipolar fuzzy number in a real life problem.

#### 1. Introduction

Fuzzy sets are useful mathematical structures to represent a collection of objects whose boundary is vague. There is a bipolar judgmental thinking on a negative side as well as a positive side in a human decision making (see [1]). This domain has recently invoked many interesting research topics such as algebra [2, 3], psychology [4], image processing [5], and human reasoning [6].

Zhang [7] initiated the concept of bipolar fuzzy sets as a generalization of fuzzy sets. He defined bipolar fuzzy sets as an extension of fuzzy sets whose membership degree range is . Also, Zhang [8] proposed a family of bipolar models. Zhou and Li [1] presented the concepts of bipolar fuzzy -ideals and normal bipolar fuzzy -ideals. Then, they investigated characterizations of bipolar fuzzy -ideals by means of positive -cut, negative -cut, homomorphism, and equivalence relation.

Akram [9, 10] used the concept of bipolar fuzzy sets in graph theory. Talebi and Rashmanlou [11] presented some properties of the self-complement and self-weak complement bipolar fuzzy graphs. Tahmasbpour and Borzooei [12] introduced two different approaches corresponding to chromatic number of a bipolar fuzzy graph. They computed total chromatic number based on -cut and -cut of a bipolar fuzzy graph with the edges and vertices both being bipolar fuzzy sets.

Comparison of two fuzzy numbers is a major computational task in various algorithms. Kerre’s method [13] for comparison of two unipolar fuzzy numbers is a well-known method in ordering unipolar fuzzy numbers. In this method, first, using the extension principle or -cut computations, the fuzzy maximum of two fuzzy numbers is computed and then, using the Hamming distance, the comparison is carried out. Inspired by Kerre’s method for comparison of two unipolar fuzzy numbers, we develop a method for comparison of two bipolar fuzzy numbers.

Here, we review the fundamental notions of bipolar fuzzy sets and Kerre’s method for unipolar numbers in Section 2. In Section 3, we propose an extension of Kerre’s method for bipolar fuzzy numbers and give a direct formula to compare two bipolar triangular fuzzy numbers. In Section 4, we present an application of bipolar fuzzy numbers in a real life problem. We conclude in Section 5.

#### 2. Preliminaries

Here, we give some necessary definitions and new results on bipolar fuzzy set theory.

*Definition 1 (see [7]). *Let be a nonempty set. A bipolar fuzzy set in is an object having the form where and .

*Definition 2 (see [2]). *Let be a bipolar-valued fuzzy set and . The sets and are, receptively, called the positive -cut of and the negative -cut of . For every , the set is called the -cut of .

We now define a bipolar triangular fuzzy number.

*Definition 3. *A bipolar triangular fuzzy number is defined as a quadruple with positive and negative membership functions and as follows:

where and are, respectively, the membership functions of positive and negative polars (see Figure 1).