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).
Proposition 4. Let and be two bipolar fuzzy numbers. One has the following results:
Proof. The results are proved by using the extension principle.
Note 1. We denote the set of all bipolar triangular fuzzy numbers by .
2.1. Kerre’s Method to Compare Two Unipolar Fuzzy Numbers
In Kerre’s method [13], 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.
Definition 5. Based on Kerre’s method, one says if and only if where and are defined as in Definitions 6 and 7.
Definition 6 (see [13]). The fuzzy max between two fuzzy numbers and is
Definition 7 (see [13]). The Hamming distance between two fuzzy numbers and is
It has been shown that Kerre’s “” is transitive [13].
3. Proposed Method for Comparison of Two Bipolar Fuzzy Numbers
Here, we at first intend to extend Kerre’s method [13] for comparison of two bipolar LR fuzzy numbers. We need to find fuzzy maximum for positive and negative polars.
Proposition 8. Let and be two bipolar LR fuzzy numbers. Then,where and are, receptively, fuzzy maximum on positive and negative polars.
Proof. These are proved by using the extension principle directly.
Let and be two arbitrary bipolar LR fuzzy numbers and let and . If then ; else .
3.1. Modified Kerre’s Method for Comparison of Two Bipolar Fuzzy Numbers
To compute the fuzzy maximum of two bipolar LR fuzzy numbers, we need to compute fuzzy maximum for each polar as given by (8). Here, we first give a result that leads to a direct and efficient formula to compute the fuzzy maximum of two arbitrary bipolar LR fuzzy numbers. Then, applying the direct formula for , we modify Kerre’s method to compare two bipolar LR fuzzy numbers. Next, using our modified Kerre’s method for comparing of two bipolar LR fuzzy numbers, we establish some simple formulas for comparison of bipolar triangular fuzzy numbers.
Define
Lemma 9. Suppose and are two bipolar LR fuzzy numbers. Then, for positive polar and for all , one hasand, for negative polar and for all , one has
Proof. First, we establish (10). Consider . Without loss of generality, suppose . Then, we need to show There are two cases as described below.
(1) Case . Since , and so(2) Case . Since , and soTherefore, from (13) and (14), we have and the proof is complete. In a similar manner, we can establish (11).
Next, in Theorem 10, we give a direct formula to compute the maximum of two arbitrary bipolar LR fuzzy numbers.
Theorem 10. Suppose and are two arbitrary bipolar LR fuzzy numbers and and . For , one has
Proof. First, we prove . Let . According to (6) and , we consider three cases.
(1) Case . Since the left side of the membership function of is an increasing function, we have Then, we conclude from and that Therefore, But, , and so we have to complete the proof for the case.
(2) Case . Without loss of generality, suppose that . Then, according to Lemma 9, it is clear thatNow, we showWe know that . Then, we have But and thus Therefore, (22) is established and from (21) and (22), and the proof of the case is complete.
(3) Case . Since and are increasing functions on , we have Thus, we have and the proof is complete. Similarly, we can prove .
Definition 11. Suppose and are two arbitrary bipolar LR fuzzy numbers and letIf then ; else .
Theorem 12. Let and be two LR fuzzy numbers. If and , thenwhere is defined as in (28).
Proof. From Theorem 10 and Definition 7, we have Thus, we have Also, for negative polar we have Thus, we have And therefore
We can rewrite (29) asNote that when and are bipolar triangular fuzzy numbers, we can simplify (35), and then the computation of can be simplified, if we can compute the intersection of and . Since each polar of and is a triangular fuzzy number, below,is the length of the intersection point of and for the positive polar, and below,is the length of the intersection point of and for the negative polar. We have the following proposition giving a reformulation of (29).
Proposition 13. Let and be two bipolar triangular fuzzy numbers with , , , and , where and are defined by (36) and (37). Then, one has
Note that the sign of is adequate to determine or . But, for bipolar LR fuzzy number linear programming problems, in some situations we need to compute the exact value of .
Example 14. Let and be two bipolar triangular fuzzy numbers. Then, , , , , , , , and are According to (38), we have and this means .
Next, we give some corollaries, the proofs of which are straightforward.
Corollary 15. Let and be two bipolar triangular fuzzy numbers. If , then
Corollary 16. Let and be two bipolar triangular fuzzy numbers. If and , where , with as defined by (36), then
Corollary 17. Let and be two bipolar triangular fuzzy numbers. If and , where , with as defined by (36), and , with as defined by (37), then
A property of (42) is that for two bipolar triangular fuzzy numbers such as and we have , where .
4. Application of Proposed Method in a Real Life Problem
Akram [10] studied an application of bipolar fuzzy sets in graph theory. He used bipolar fuzzy set for a social group. Here, we demonstrate an application of bipolar fuzzy number in maximum weighted matching problem; matching problem has some applications in various fields such as scheduling [14] and network [15] problems. We consider each vertex to be person and weight of each edge between two vertices be the influence of each person (vertex) to another person. In general, influence can be positive or negative. Suppose is an arbitrary weighted graph, where is the vertex set of and is the edge set of . The maximum weighted matching problem is where , if two persons and are matched to each other, and , otherwise, and is the weight of edge (giving the influence of one person to another person), considered as a bipolar fuzzy number, since influence of a person cannot always be positive. The aim is to match every person to another person so that they have a stable relation.
5. Conclusions
We proposed a new efficient method for ordering bipolar fuzzy numbers. In this method, for comparison of bipolar LR fuzzy numbers, we used an extension of Kerre’s method used in ordering of unipolar fuzzy numbers. In our proposed method, we provided a formula to compare two bipolar triangular fuzzy numbers in operations, making the process useful for optimization algorithms. Also, we presented an application of bipolar fuzzy number in a real life problem.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The first author acknowledges the Research Council of Ferdowsi University of Mashhad, the second author would like to thank Dr. Reza Ghanbari and acknowledge the Department of Applied Mathematics at Ferdowsi University of Mashhad for the hospitality during her sabbatical leave there, and the third author acknowledges the Research Council of Sharif University of Technology for supporting this work.