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A Direct Method to Compare Bipolar LR Fuzzy Numbers.

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 [-1, 1]. Also, Zhang [8] proposed a family of bipolar models. Zhou and Li [1] presented the concepts of bipolar fuzzy h-ideals and normal bipolar fuzzy h-ideals. Then, they investigated characterizations of bipolar fuzzy h-ideals by means of positive t-cut, negative s-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 [[alpha].sup.P]-cut and [[alpha].sup.N]-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 a-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 X be a nonempty set. A bipolar fuzzy set [??] in X is an object having the form

[??] = {(x, [[mu].sup.P.sub.[??]] (x), [[mu].sup.N.sub.[??]] (x)) | x [member of] X}, (1)

where [[mu].sup.P.sub.[??]](x): X [right arrow] [0, 1] and [[mu].sup.N.sub.[??]](x) : X [right arrow] [-1, 0].

Definition 2 (see [2]). Let [??] = ([[mu].sup.P.sub.[??]] (x), [[mu].sup.N.sub.[??]] (x)) be a bipolar-valued fuzzy set and (s, t) [member of] [-1, 0] x [0, 1]. The sets [A.sup.P.sub.t] = {x [member of] X | [[mu].sup.P.sub.[??]] (x) [greater than or equal to] t] and [A.sup.N.sub.s] = {x [member of] X | [[mu].sup.N.sub.[??]] (x) [less than or equal to] s} are, receptively, called the positive t-cut of [??] and the negative s-cut of [??]. For every k [member of] [0,1], the set

[A.sub.k] = [A.sup.P.sub.k] [intersection] [A.sup.N.sub.-k] (2)

is called the k-cut of [??].

We now define a bipolar triangular fuzzy number.

Definition 3. A bipolar triangular fuzzy number is defined as a quadruple [??] = ([a.sup.L], [a.sup.P], [a.sup.N], [a.sup.R]) with positive and negative membership functions [[mu].sup.P.sub.[??]](x) and [[mu].sup.N.sub.[??]] (x) as follows:

[mathematical expression not reproducible] (3)

where [[mu].sup.P.sub.[??]] and [[mu].sup.N.sub.[??]] are, respectively, the membership functions of positive and negative polars (see Figure 1).

Proposition 4. Let [??] = ([a.sup.L], [a.sup.P], [a.sup.N], [a.sup.R]) and [??] = ([b.sup.L], [b.sup.P], [b.sup.N], [b.sup.R]) be two bipolar fuzzy numbers. One has the following results:

[mathematical expression not reproducible]. (4)

Proof. The results are proved by using the extension principle.

Note 1. We denote the set of all bipolar triangular fuzzy numbers by F(R).

2.1. Kerre's Method to Compare Two Unipolar Fuzzy Numbers. In Kerre's method [13], first, using the extension principle or a-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 [mathematical expression not reproducible] if and only if

[mathematical expression not reproducible], (5)

where [mathematical expression not reproducible] and d(*, *) are defined as in Definitions 6 and 7.

Definition 6 (see [13]). The fuzzy max between two fuzzy numbers [??] and [??] is

[mathematical expression not reproducible]. (6)

Definition 7 (see [13]). The Hamming distance between two fuzzy numbers [??] and [??] is

[mathematical expression not reproducible]. (7)

It has been shown that Kerre's "[less than or equal to]" 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 [??] = [([m.sup.L], [m.sup.P], [m.sup.N], [m.sup.R]).sub.LR] and [??] = [([n.sup.L], [n.sup.P], [n.sup.N], [n.sup.R]).sub.LR] be two bipolar LR fuzzy numbers. Then,

[mathematical expression not reproducible], (8)

where [mathematical expression not reproducible] and [mathematical expression not reproducible] are, receptively, fuzzy maximum on positive and negative polars.

Proof. These are proved by using the extension principle directly.

Let [??] = ([m.sup.L], [m.sup.P], [m.sup.N], [m.sup.R]) and [??] = ([n.sup.L], [n.sup.P], [n.sup.N], [n.sup.R]) be two arbitrary bipolar LR fuzzy numbers and let [mathematical expression not reproducible]. If [mathematical expression not reproducible].

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 [mathematical expression not reproducible], 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

[w.sub.z] = [w | w [less than or equal to] z},

[E.sub.z] = {(x, y) | x, y [member of] [w.sub.z], z = max (x, y)}. (9)

Lemma 9. Suppose [??] = [([m.sup.L], [m.sup.P], [m.sup.N], [m.sup.R]).sub.LR] and [??] = [([n.sup.L], [n.sup.P], [n.sup.N], [n.sup.R]).sub.LR] are two bipolar LR fuzzy numbers. Then, for positive polar and for all z [member of] R, one has

[mathematical expression not reproducible], (10)

and, for negative polar and for all z [member of] R, one has

[mathematical expression not reproducible]. (11)

Proof. First, we establish (10). Consider z [member of] R. Without loss of generality, suppose M(z) [greater than or equal to] N(z). Then, we need to show

[mathematical expression not reproducible]. (12)

There are two cases as described below.

(1) Case X < z. Since (x, y) [member of] [E.sub.z], y = z and so

[mathematical expression not reproducible]. (13)

(2) Case y < z. Since (x, y) [member of] [E.sub.z], x = z and so

[mathematical expression not reproducible]. (14)

Therefore, from (13) and (14), we have

[mathematical expression not reproducible], (15)

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 [??] = [([m.sup.L], [m.sup.P], [m.sup.N], [m.sup.R]).sub.LR] and [??] = [([n.sup.L], [n.sup.P], [n.sup.N], [n.sup.R]).sub.LR] are two arbitrary bipolar LR fuzzy numbers and [m.sup.P] [less than or equal to] [n.sup.P] and [m.sup.N] [less than or equal to] [n.sup.N]. For z [member of] R, one has

[mathematical expression not reproducible]. (16)

Proof. First, we prove [mathematical expression not reproducible]. Let z [member of] R. According to (6) and [m.sup.P] [less than or equal to] [n.sup.P], we consider three cases.

(1) Case [m.sup.P] < z < [n.sup.P]. Since the left side of the membership function of [??] is an increasing function, we have

[mathematical expression not reproducible]. (17)

Then, we conclude from [??](x) [less than or equal to] 1 = [??]([m.sup.P]) and [??](y) [less than or equal to] [??](z) that

[mathematical expression not reproducible]. (18)

Therefore,

[mathematical expression not reproducible]. (19)

But, min(1, [??](z)) = [??](z), and so we have

[mathematical expression not reproducible], (20)

to complete the proof for the case.

(2) Case z [greater than or equal to] [n.sup.P]. Without loss of generality, suppose that [mathematical expression not reproducible]. Then, according to Lemma 9, it is clear that

[mathematical expression not reproducible]. (21)

Now, we show

[mathematical expression not reproducible]. (22)

We know that (z, [n.sup.P]) [member of] [E.sub.z]. Then, we have

[mathematical expression not reproducible]. (23)

But

[mathematical expression not reproducible], (24)

and thus

[mathematical expression not reproducible]. (25)

Therefore, (22) is established and from (21) and (22), and the proof of the case is complete.

(3) Case z [less than or equal to] [m.sup.P] ([less than or equal to] [n.sup.P]). Since [??](x) and [??](y) are increasing functions on [min([m.sup.L], [n.sup.L]), [m.sup.P]], we have

[mathematical expression not reproducible]. (26)

Thus, we have

[mathematical expression not reproducible], (27)

and the proof is complete. Similarly, we can prove [mathematical expression not reproducible].

Definition 11. Suppose [??] = ([m.sup.L], [m.sup.P], [m.sup.N], [m.sup.R]) and [??] = ([n.sup.L], [n.sup.P], [n.sup.N], [n.sup.R]) are two arbitrary bipolar LR fuzzy numbers and let

[mathematical expression not reproducible]. (28)

If [mathematical expression not reproducible] then [mathematical expression not reproducible]; else [mathematical expression not reproducible].

Theorem 12. Let [??] = [([m.sup.L], [m.sup.P], [m.sup.N], [m.sup.R]).sub.LR] and [??] = ([n.sup.L], [n.sup.P], [n.sup.N], [n.sup.R]) be two LR fuzzy numbers. If [m.sup.P] [less than or equal to] [n.sup.P] and [m.sup.N] [less than or equal to] [n.sup.N], then

[mathematical expression not reproducible], (29)

where [mathematical expression not reproducible] is defined as in (28).

Proof. From Theorem 10 and Definition 7, we have

[mathematical expression not reproducible]. (30)

Thus, we have

[mathematical expression not reproducible]. (31)

Also, for negative polar we have

[mathematical expression not reproducible]. (32)

Thus, we have

[mathematical expression not reproducible]. (33)

And therefore

[mathematical expression not reproducible]. (34)

We can rewrite (29) as

[mathematical expression not reproducible]. (35)

Note that when [??] = ([m.sup.L], [m.sup.P], [m.sup.N], [m.sup.R]) and [??] = ([n.sup.L], [n.sup.P], [n.sup.N], [n.sup.R]) are bipolar triangular fuzzy numbers, we can simplify (35), and then the computation of [mathematical expression not reproducible] can be simplified, if we can compute the intersection of [M.sub.R] and [N.sub.L]. Since each polar of [??] and [??] is a triangular fuzzy number, [[bar.x].sup.P] below,

[[bar.x].sup.P] = [m.sup.R][n.sup.P] - [n.sup.L][m.sup.P]/[n.sup.P] - [n.sup.L] - [m.sup.P] + [m.sup.R], (36)

is the length of the intersection point of [M.sub.R] and [N.sub.L] for the positive polar, and [[bar.x].sup.N] below,

[[bar.x].sup.N] = [m.sup.R][n.sup.N] - [n.sup.L][m.sup.N]/[n.sup.N] - [n.sup.L] - [m.sup.N] + [m.sup.R], (37)

is the length of the intersection point of [M.sub.R] and [N.sub.L] for the negative polar. We have the following proposition giving a reformulation of (29).

Proposition 13. Let [??] = ([m.sup.L], [m.sup.P], [m.sup.N], [m.sup.R]) and [??] = ([n.sup.L], [n.sup.P], [n.sup.N], [n.sup.R]) be two bipolar triangular fuzzy numbers with [m.sup.P] < [n.sup.P], [m.sup.N] < [n.sup.N], [[bar.x].sup.P] [member of] [[m.sup.P], [n.sup.P]], and [[bar.x].sup.N] [member of] [[m.sup.N], [n.sup.N]], where [[bar.x].sup.P] and [[bar.x].sup.N] are defined by (36) and (37). Then, one has

[mathematical expression not reproducible]. (38)

Note that the sign of [r.sup.b](*, *) is adequate to determine [mathematical expression not reproducible] or [mathematical expression not reproducible]. But, for bipolar LR fuzzy number linear programming problems, in some situations we need to compute the exact value of [r.sup.b](*, *).

Example 14. Let [??] = (2,4,6,10) and [??] = (2,5,8,12) be two bipolar triangular fuzzy numbers. Then, [mathematical expression not reproducible], and [[??].sup.N.sub.R] are

[mathematical expression not reproducible]. (39)

According to (38), we have [mathematical expression not reproducible] = 4.4889 and this means [mathematical expression not reproducible].

Next, we give some corollaries, the proofs of which are straightforward.

Corollary 15. Let [??] = ([m.sup.L], [m.sup.P], [m.sup.N], [m.sup.R]) and [??] = ([n.sup.L], [n.sup.P], [n.sup.N], [n.sup.R]) be two bipolar triangular fuzzy numbers. If [m.sup.R] [less than or equal to] [n.sup.L], then

[mathematical expression not reproducible]. (40)

Corollary 16. Let [??] = ([m.sup.L], [m.sup.P], [m.sup.N], [m.sup.R]) and [??] = ([n.sup.L], [n.sup.P], [n.sup.N], [n.sup.R]) be two bipolar triangular fuzzy numbers. If [m.sup.P] < [n.sup.P] and [m.sup.N] = [n.sup.N], where [mathematical expression not reproducible], with [[bar.x].sup.P] as defined by (36), then

[mathematical expression not reproducible]. (41)

Corollary 17. Let [??] = ([m.sup.L], [m.sup.P], [m.sup.N], [m.sup.R]) and [??] = ([n.sup.L], [n.sup.P], [n.sup.N], [n.sup.R]) be two bipolar triangular fuzzy numbers. If [m.sup.P] < [n.sup.P] and [m.sup.N] < [n.sup.N], where [mathematical expression not reproducible], with [[bar.x].sup.P] as defined by (36), and [mathematical expression not reproducible], with [[bar.x].sup.N] as defined by (37), then

[mathematical expression not reproducible]. (42)

A property of (42) is that for two bipolar triangular fuzzy numbers such as [??] and [??] we have [mathematical expression not reproducible], where [lambda] [greater than or equal to] 0.

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 G = (V, E) is an arbitrary weighted graph, where V = {1, ..., n} is the vertex set of G and E [subset or equal to] V x V is the edge set of G. The maximum weighted matching problem is

[mathematical expression not reproducible], (43)

where x(e) = 1, if two persons u and v are matched to each other, and x(e) = 0, otherwise, and [??](e) is the weight of edge e (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 O(1) operations, making the process useful for optimization algorithms. Also, we presented an application of bipolar fuzzy number in a real life problem.

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

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.

References

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[8] W.-R. Zhang, "NPN fuzzy sets and NPN qualitative algebra: A computational framework for bipolar cognitive modeling and multiagent decision analysis," IEEE Transactions on Systems, Man, and Cybernetics, PartB: Cybernetics, vol. 26, no. 4, pp. 561-574, 1996.

[9] M. Akram, "Bipolar fuzzy graphs," Information Sciences, vol. 181, no. 24, pp. 5548-5564, 2011.

[10] M. Akram, "Bipolar fuzzy graphs with applications," Knowledge-Based Systems, vol. 39, pp. 1-8, 2013.

[11] A. A. Talebi and H. Rashmanlou, "Complement and isomorphism on bipolar fuzzy graphs," Fuzzy Information and Engineering, vol. 6, no. 4, pp. 505-522, 2014.

[12] A. Tahmasbpour and R. A. Borzooei, "Chromatic number of bipolar fuzzy graphs," Journal of Applied Mathematics & Informatics, vol. 34, no. 1-2, pp. 49-60, 2016.

[13] J. J. Buckely and L. J. Jowers, "Monte carlo method in fuzzy optimization," in Studies in Fuzziness and Soft Computing, 2007

[14] C. E. Bell, "Weighted matching with vertex weights: An application to scheduling training sessions in NASA space shuttle cockpit simulators," European Journal of Operational Research, vol. 73, no. 3, pp. 443-449,1994.

[15] X. Yao, D. Gong, P. Wang, and L. Chen, "Multi-objective optimization model and evolutional solution of network node matching problem," Physica A: Statistical Mechanics and its Applications, vol. 483, pp. 495-502, 2017

Reza Ghanbari (iD), (1) Khatere Ghorbani-Moghadam, (2) and Nezam Mahdavi-Amiri (2)

(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; rghanbari@um.ac.ir

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

Academic Editor: Mehmet Onder Efe

Caption: Figure 1: Triangular bipolar fuzzy number [a.sup.P] [not equal to] [a.sup.N].
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Title Annotation:Research Article
Author:Ghanbari, Reza; Ghorbani-Moghadam, Khatere; Mahdavi-Amiri, Nezam
Publication:Advances in Fuzzy Systems
Date:Jan 1, 2018
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