Duality in Bipolar Fuzzy Number Linear Programming Problem

We develop a linear ranking function for ordering bipolar fuzzy numbers and study its properties. Using this ranking function, we solve a bipolar fuzzy linear programing problem. Then, we present the dual of the problem and establish several duality results. Also, we presented an application of bipolar fuzzy number in real life problem.


Introduction
In a human decision making, there is a bipolar judgmental thinking on a negative side and a positive side; for instance, see [1]. In bipolar information, two types of information (as positive and negative) must be distinguished [2,3]. Positive information is given by observation or experimentation. But, negative information represents impossibility. This domain has recently invoked several interesting research areas such as psychology [4], image processing [5], human reasoning [6] and graph theory [7]. Zhang [8] initiated the concept of bipolar fuzzy set as a generalization of fuzzy set. He defined bipolar fuzzy set as an extension of fuzzy set whose range of membership degree is [−1, 1]. Akram [7,9] used the concept of bipolar fuzzy set in graph theory. Broumand [10] introduced the concept of bipolar-valued fuzzy sub-algebras of BCK/BCI-algebras and investigated some of their useful properties. 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. Some other works on bipolar fuzzy sets can be found in [11][12][13][14][15].
There are several methods for comparison of unipolar fuzzy numbers based on ranking functions [16,17] and most convenient methods for solving linear programing problems are based on the concept of ranking functions [16,18,19,20,21,22]. Inspired by ranking functions of unipolar fuzzy numbers, we develop a ranking function for bipolar fuzzy numbers. Then, we solve a bipolar fuzzy linear programing problem by using a certain ranking function and we give duality results for bipolar fuzzy linear programing problems.
In Section 2, we review the fundamental notions of bipolar fuzzy sets. Then, we propose a linear ranking function to order bipolar fuzzy numbers. In Section 3, we investigate and characterize several properties for the bipolar fuzzy number linear programing problem by using a linear ranking function. In Section 4, we introduce the dual of a bipolar fuzzy number linear programing problem. In Section 5, we present an application of bipolar fuzzy number in a real life problem. We give our concluding remarks in Section 6.

Definitions and Notations
Here, some necessary definitions and relevant results of bipolar fuzzy sets are given. Definition 2.1: [8] Let X be a nonempty set. A bipolar fuzzy setB in X is an object with the following formB tively called the positive t-cut ofB and the negative s-cut ofB, and for every k ∈ [0, 1], the set: is called the k-cut ofB.
We next define a bipolar triangular fuzzy number. A bipolar triangular fuzzy number is defined as a quadrupleÃ = (a L , a P , a N , a R ) with positive and negative membership functions µ P A (x) and µ Ñ A (x) as follows (see Figure 1): Otherwise. (2) Note 1: In a bipolar triangular fuzzy number, a P and a N can be equal (see Figure 2).

Note 2:
We denote the set of all bipolar triangular fuzzy numbers by F(R).  Using similar argument for fuzzy arithmetic, we now give the following proposition.
be two bipolar fuzzy numbers. Define:

Bipolar Ranking Function
There are several methods for solving unipolar fuzzy linear programing problems by using ranking function [16,18,19,20,21,22]. We can define ranking function on bipolar fuzzy numbers and use it for solving bipolar fuzzy number linear programing problems. A bipolar ranking function R b : F(R) → R maps bipolar fuzzy numbers into the real line, where a natural order exists. Orders on F(R) are defined as follows: Inspired by a special version of the ranking function on unipolar fuzzy numbers proposed by Yager [17], we define ranking function on bipolar fuzzy numbers as follows: By using Definition 2.2 we have: Then, for bipolar triangular fuzzy numbersã = (a L , a P , a N , (3) is linear.

Linear Programming Problem with Bipolar Triangular Fuzzy Numbers
A bipolar triangular fuzzy number linear programing problem (BTFNLPP) is defined to be given, x ∈ R n is to determined, and R b is a linear ranking function as defined by (3) The following result gives an alternative formulation of (4).

Proposition 3.1: Problem (4) is equivalent to
Proof: Since R b is a linear ranking function, we have: So, if we denote the i-th row ofÃ byã i , we have

Duality
Similar to the duality theory in linear optimization (see, for example, Luenberger and Ye [23]), for every BTFNLPP there is an associated dual BTFNLPP (DBTFNLPP) which satisfies some important properties.
Definition 4.1: Using the notation of (4) define Indeed, (DBTFNLPP) is the dual of (BTFNLPP) and appropriate duality results can be established. Next, we show the weak duality result.  Proof: MultiplyingÃx 0 = R bb on the left by y 0 T , we have y 0 TÃ x 0 = R b y 0 Tb and multiplying y 0 TÃ ≤ R bc T on the right by x 0 ≥ 0, we have y 0 TÃ x 0 ≤ R bc T x 0 . So, we get y 0 Tb ≤ R bc T x 0 .
Note 3: Similar to some duality results for unipolar fuzzy number linear programing problem, the value of the ranking function for the bipolar fuzzy value of the objective function at any feasible solution to BTFNLPP is always bigger than or equal to the value of the ranking function for the bipolar fuzzy value of the objective function for any feasible solution to DBTFNLPP. Also, if x 0 and y 0 are feasible solutions to BTFNLP and DBTFNLP problems, respectively, and y 0 Tb = R bc T x 0 , then x 0 and y 0 are optimal solutions to their respective problems and if any one of the BTFNLPP or DBTFNLPP is unbounded, then the other problem has no feasible solution. Next, we define a basic solution and then establish the strong duality result. Assume rank(A) = m, and partition A as [B, N] where B, m × m, is nonsingular. It is obvious that rank(B) = m and B = R b (B), whereB is the fuzzy matrix inÃ corresponding to B. Let y j be a solution of By = a j . A basic solution is . . . ,b m ) = (a B 1 , . . . , a B m ) with B i being the index corresponding to the i-th column of B, that is,b i = a B i and b = R b (b). If x B ≥ 0, then the basic solution is feasible and the corresponding fuzzy objective value isz= R bc T B x B , wherẽ c B = (c B1 , . . . ,c Bm ) T . Now, corresponding to every nonbasic variable x j , 1 ≤ j ≤ n, j = B i , i = 1, . . . , m definẽ

Definition 4.2: Let
Theorem 4.2: Assume the BTFNLPP is non-degenerate (see [23]). A basic feasible solution x B = B −1 b, x N = 0 is optimal to (4) if and only ifz j ≤ R bcj, for all j, 1 ≤ j ≤ n.
Proof: Let x = (x T B , x T N ) T be an optimal solution to (4), where x B = B −1 b, x N = 0, corresponding to an optimal basis B. Then, the corresponding optimal objective value is Now, since x is feasible, we have x ≥ 0, and based on Definition 4.2, Hence, we can rewrite (9) as follows: Substituting (10) in (8), we obtaiñ Thus,z Now, from (11) it is obvious that if there is a nonbasic variable x j withz j > R bcj, then we can enter x j into the basis and obtainz * > R bz (since the problem is non-degenerate and x j > 0 in the new basis). This is contrary toz * being optimal and hence we must havez j ≤ R bcj for all j, 1 ≤ j ≤ n. The converse of the theorem is similar to Theorem 3.1 in [19].
where by letting y * = R b (c T B B −1 ) = c T B B −1 , we can write y * TÃ ≤ R bc T . Thus, y * is a feasible solution to DBTFNLPP. Based on Definition 4.2, we have Hence, Due to Lemma 3.1, the converse of the theorem follows similarly.

Theorem 4.4:
For any BTFNLPP and its corresponding DBTFNLPP, exactly one of the following statements is true.
(1) Both have optimal solutions x * and y * withc T x * = R b y * Tb .
(2) One problem is unbounded and the other is infeasible.
Proof: 1 and 2 are proven in Theorem 4.2 and 3.1 respectively and we give an example for 3.

Application of Proposed Method in Real Life Problems
Akram [9] studied an application of bipolar fuzzy set in graph theory. He used bipolar fuzzy set in a social group. Here, we demonstrate an application of bipolar fuzzy number in maximum weighted matching problem; matching problem has some applications in different fields such as scheduling [24] and network [25] problems. We consider each vertex as a person and weight of each edge between two vertices shows the influence of each person (vertex) to another person. In general, influence can be positive or negative. Suppose where x(e) = 1 if two persons u and $v are matched to each other, and x(e) = 0, otherwise, andw(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. We want to match every person to another person so that they have a stable relation.

Conclusions and Future Work
We presented a linear ranking function for ordering bipolar fuzzy numbers and studied some of its properties. We defined bipolar triangular fuzzy number linear programing problems and established its dual. We then proved the common duality results for the primal and dual problems. Similar to fuzzy primal simplex algorithms (see [21,26]), simplex, dual simplex [21], VNS algorithm [27,28] and primal-dual algorithms can be developed for solving bipolar fuzzy linear programing problems.