Bipolar fuzzy circuits with applications

Abstract

We introduce the concept of bipolar fuzzy matroids and describe some properties of various types of bipolar fuzzy matroids. We apply the notion of bipolar fuzzy matroids to graph theory and linear algebra. We present closure of a bipolar fuzzy matroid, bipolar fuzzy rank function, fundamental sequence, circuit rectangles and put special emphasis on bipolar fuzzy circuits. We also describe certain applications of bipolar fuzzy matroids in decision support system and network analysis.

Keywords

Bipolar fuzzy matroid closure bipolar fuzzy rank function bipolar fuzzy circuit circuit rectangle

1 Introduction

Matroid theory laid down its foundations in 1935 after the work of Whitney [23]. This theory constitutes a useful approach for linking major ideas of linear algebra, graph theory, combinatorics and many other areas of Mathematics. Matroid theory have been a focus for an active research during the last few decades. But the crisp matroids do not discuss the uncertain behavior and degree of dependence of objects. To study this information loss, Goetschel [7] presented the approach of fuzzification to the class of independent subsets called fuzzy matroids. But in various absolute World decisions and problems, the data have double-sided information which cannot be dealt well by means of classical as well as fuzzy set theory. To overcome this issue, and to study two sided behavior of objects, we need the theory of bipolar fuzzy matroids. Bipolar fuzzy matroids can be applied to linear algebra, graph theory, combinatorics and many real World problems, e.g., network analysis, engineering, decision support systems, scheduling of tasks and selection of workers. Bipolar fuzzy matroids can also be used in secret sharing problem to share parts of secret information among different participants such that we have two sided information about each participant.

Fuzzy set theory owes its remarkable origin to the work of Zadeh [25] in 1965. Fuzzy set theory deals with real life data having non-linear uncertainty and vagueness. In 1994, Zhang [32] introduced bipolar fuzzy set theory as an extension of fuzzy set theory in which the membership degree range is [-1,1]. The membership degree 0 of an object, in a bipolar fuzzy set, means that the element is extraneous to the corresponding attribute, the membership value (0,1] of an object shows that the object somewhat satisfies the attribute, whereas the membership value [-1,0) of the object indicates that the object more or less satisfies the implicit counter-property. The main idea behind such characterization is linked with the existence of “bipolar information” (such as, positive information as well as negative information) about the given set. Negative information indicates what is voluntarily impossible, while positive information represents what is considered to be possible. Actually, a wide range of human thinking is based on two-sided or bipolar judgmental decisions on a negative side and a positive side. For illustration, profit and loss, hostility and friendship, competition and cooperation, conflicted interests and common interests, unlikelihood and likelihood, feedback and feedforward, and so on are generally two sides in coordination and decision making. Just like that, bipolar fuzzy set theory indeed have considerable impacts on many fields, including computer science, artificial intelligence, information science, decision science, cognitive science, economics, management science, neural science, medical science and social science. Recently, bipolar fuzzy set theory has been applied and studied a bit speedily and increasingly. Thus bipolar fuzzy sets and bipolar fuzzy graphs not only have applications in mathematical theories but also in real-world problems [6].

In 1988, Goetschel [7] presented the approach of fuzzification to matroid theory and discussed the uncertain behavior of matroids. The same authors [8 –10] introduced the concept of bases of fuzzy matroids, fuzzy matroid structures and greedy algorithm in fuzzy matroids. Fuzzy matroids can be used to study the uncertain behavior of objects but if the data have bipolar information to be dealt with, fuzzy matroids cannot give appropriate results. For this reason, we need the theory of bipolar fuzzy matroids to handle data and information having two sided uncertainties. The main idea of this research paper is to apply the useful notion of bipolar fuzzy set to matroid theory and check into thoroughly some basic properties of various types of bipolar fuzzy matroids. We apply the notion of bipolar fuzzy matroids to graph theory and linear algebra. We present closure of a bipolar fuzzy matroid, bipolar fuzzy rank function, fundamental sequence, circuit rectangles and put special emphasis on bipolar fuzzy circuits. We also describe certain applications of bipolar fuzzy matroids in decision support system and network analysis.

We have used basic ideas and terminologies throughout the paper. For different notations, concepts and terminologies which are not mentioned within the paper, the reader are directed to [1–5 , 28–31]. We will use bold symbols for the degree of membership of elements throughout the paper.

2 Preliminaries

The term crisp matroid has various equivalent definitions. We use here the simplest definition of matroid.

Definition 2.1. If Y is a non-empty universe and I is any subset of P (Y), the power set of Y, satisfying the following conditions,

If B₁ ∈ I and B₂ ⊂ B₁ then , B₂ ∈ I,

If B₁, B₂ ∈ I and |B₁| < |B₂| thenthereexists B₃ ∈ I suchthat B₁ ⊂ B₃ ⊆ B₁ ∪ B₂ .

Then the pair M = (Y, I) is called a matroid and I is known as the family of independent subsets of M.

Definition 2.2. [7] If M = (Y, I) is a matroid then the mapping R : P (Y) → {0, 1, 2, …, |Y|} defined by $R (B) = max {| E | : E \subseteq B, E \in I}$ is a rank function for M. If B ∈ P (Y), R is known as rank of B.

Definition 2.3. [25, 26] A fuzzy set τ in a non-empty universe Y is a mapping τ : Y → [0, 1]. A fuzzy relation on Y is a fuzzy subset δ in Y × Y. If τ is a fuzzy set on Y and δ is a fuzzy relation in Y then we can say that δ is a fuzzy relation on τ if δ (y, z)≤ min {τ (y), τ (z)} forall y, z ∈ Y.

Definition 2.4. [7] If $F (Y)$ is a power set of fuzzy subsets on Y and $I \subseteq F (Y)$ which satisfy the following conditions,

If $τ_{1} \in I$ and τ₂ ⊂ τ₁ then, $τ_{2} \in I$ ,

If $τ_{1}, τ_{2} \in I$ and |supp (τ₁) | < |supp (τ₂) | then there exists $τ_{3} \in I$ such that

τ₁ ⊂ τ₃ ⊆ τ₁ ∪ τ₂,

m (τ₃) ≥ min {m (τ₁), m (τ₂)} where, m (ν) = min {ν (y) : y ∈ supp (ν)}.

The pair $M = (X, I)$ is called a fuzzy matroid. $I$ is known as the collection of independent fuzzy subsets of M.

Definition 2.5. [32] A bipolar fuzzy set over a non-empty universe Y has the form $B = {(y, μ_{B}^{p} (y)$ , $μ_{B}^{n} (y) : y \in Y}$ where, $μ_{B}^{p} : Y \to [0, 1]$ and $μ_{B}^{n} : Y \to [- 1, 0]$ are mappings.

Definition 2.6. [1] Let Y be a non-empty universe. A bipolar fuzzy relation in Y is a mapping $D = (μ_{D}^{p}, μ_{D}^{n}) : Y \times Y \to [0, 1] \times [- 1, 0]$ such that $μ_{D}^{p} (yz) \in [0, 1]$ and $μ_{D}^{n} (yz) \in [- 1, 0]$ , for all y, z ∈ Y.

Definition 2.7. [1] A bipolar fuzzy graph on Y (non-empty universe) is a pair G = (B, D) where, $B = (μ_{B}^{p}, μ_{B}^{n})$ is a bipolar fuzzy set on Y and $D = (μ_{D}^{p}, μ_{D}^{n})$ is a bipolar fuzzy relation in Y such that, $\begin{matrix} μ_{D}^{p} (yz) & \leq & μ_{B}^{p} (y) \land μ_{B}^{p} (z), \\ μ_{D}^{n} (yz) & \geq & μ_{B}^{n} (y) \lor μ_{B}^{n} (z), for all y, z \in Y . \end{matrix}$

3 Bipolar fuzzy circuits

In this section, we present the notion of bipolar fuzzy vector spaces, bipolar fuzzy matroids, bipolar fuzzy circuits and study their properties.

Definition 3.1. Let β be any bipolar fuzzy set on Y and t = (t^p, tⁿ), t^p ∈ [0, 1], tⁿ ∈ [-1, 0] then, $\begin{matrix} β_{t} & = & {y \in Y ∣ μ_{β}^{p} (y) \geq t^{p}, μ_{β}^{n} (y) \leq t^{n}}, \\ R^{p} (β) & = & {μ_{β}^{p} (y) ∣ μ_{β}^{p} (y) > 0}, \\ R^{n} (β) & = & {μ_{β}^{n} (y) ∣ μ_{β}^{n} (y) < 0} . \end{matrix}$

Definition 3.2. A bipolar fuzzy vector space over a field K is defined as a pair (Y, B) where, $B = (μ_{B}^{p}, μ_{B}^{n}) : Y \to [0, 1] \times [- 1, 0]$ is a mapping and Y is a vector space over K such that for all c, d ∈ K and y, z ∈ Y, $\begin{matrix} μ_{B}^{p} (cy + dz) \geq min {μ_{B}^{p} (y), μ_{B}^{p} (z)}, \\ μ_{B}^{n} (cy + dz) \leq max {μ_{B}^{n} (y), μ_{B}^{n} (z)} . \end{matrix}$

Example 3.1. Let Y be a vector space of 2 × 1 column vectors over $ℝ$ . Define a mapping B : Y → [0, 1] × [-1, 0] such that for each $z = {[\begin{matrix} x & y \end{matrix}]}^{t}$ , $B (z) = {\begin{matrix} (1, - 1), & z = {[\begin{matrix} 0 & 0 \end{matrix}]}^{t}, \\ (\frac{1}{3}, - \frac{2}{3}), & z = {[\begin{matrix} x & 0 \end{matrix}]}^{t} or z = {[\begin{matrix} 0 & y \end{matrix}]}^{t}, \\ (1, - 1), & x \neq 0 and y \neq 0 . \end{matrix}$

We now show that (Y, B) is a bipolar fuzzy vector space. For $z = {[\begin{matrix} 0 & 0 \end{matrix}]}^{t}$ , the case is trivial. So the following cases are to be discussed.

Case 1: Consider two column vectors $z = {[\begin{matrix} x & y \end{matrix}]}^{t}$ and $u = {[\begin{matrix} u & v \end{matrix}]}^{t}$ then, for any scalars c and d, $B (c z + d u) = B ([\begin{matrix} cx + du \\ cy + dv \end{matrix}]) .$

If either exactly one of c or d is zero or both are non-zero then, cx + du ≠ 0 and cy + dv ≠ 0 and so $μ_{B}^{p} (c z + d u) = 1 \geq min {μ_{B}^{p} (z), μ_{B}^{p} (u)}$ and $μ_{B}^{n} (c z + d u) = - 1 \leq max {μ_{B}^{n} (z), μ_{B}^{n} (u)}$ .

Case 2: If $z = {[\begin{matrix} x & 0 \end{matrix}]}^{t}$ and $u = {[\begin{matrix} 0 & v \end{matrix}]}^{t}$ then, $B (c z + d u) = B ({[\begin{matrix} cx & dv \end{matrix}]}^{t})$ . If both c and d are non-zero then, $\begin{matrix} μ_{B}^{p} (c z + d u) & = & 1 > min {μ_{B}^{p} (z), μ_{B}^{p} (u)} and \\ μ_{B}^{n} (c z + d u) & = & - 1 < max {μ_{B}^{n} (z), μ_{B}^{n} (u)} . \end{matrix}$

If exactly one of c or d is zero then, $\begin{matrix} μ_{B}^{p} (c z + d u) & = & \frac{1}{3} = min {μ_{B}^{p} (z), μ_{B}^{p} (u)} and \\ μ_{B}^{n} (c z + d u) & = & - \frac{2}{3} = max {μ_{B}^{n} (z), μ_{B}^{n} (u)} . \end{matrix}$

Hence (Y, B) is a bipolar fuzzy vector space.

Definition 3.3. Let (Y, B) be a bipolar fuzzy vector space over K. A set of vectors ${y_{k}}_{k = 1}^{n}$ is known as bipolar fuzzy linearly independent in (Y, B) if

${y_{k}}_{k = 1}^{n}$ is linearly independent,

$B (\sum_{k = 1}^{n} c_{k} y_{k}) = (⋀_{k = 1}^{n} μ_{B}^{p} (c_{k} y_{k}), ⋁_{k = 1}^{n} μ_{B}^{n} (c_{k} y_{k}))$ , for all ${c_{k}}_{k = 1}^{n} \subset K and distinct y_{i} \in Y$ , 1 ≤ i ≤ n.

Definition 3.4. A set of vectors $B = {y_{k}}_{k = 1}^{n}$ is said to be a bipolar fuzzy basis in (Y, B) if $B$ is a basis in Y and condition 2 of Definition 3.3 is satisfied.

Proposition 3.1. If (Y, B) is a bipolar fuzzy vector space then any set of vectors with distinct positive and negative degrees of membership is linearly independent and bipolar fuzzy linearly independent.

Proof. We prove this proposition by induction on the number of vectors, n, with distinct degree of membership. For n = 1, {y₁}, the statement is trivial. Assume that the statement is true for k vectors. Let ${y_{j}}_{j = 1}^{k + 1}$ be the set of vectors with distinct positive and negative degrees of membership. Suppose ${y_{j}}_{j = 1}^{k + 1}$ are not linearly independent and hence not bipolar fuzzy linearly independent. Then there exist scalars c₁, c₂, …, c_k such that $y_{k + 1} = \sum_{j = 1}^{k} c_{j} y_{j}$ and $\begin{matrix} B (y_{k + 1}) & = & (⋀_{j = 1}^{k} μ_{B}^{p} (c_{j} y_{j}), ⋁_{j = 1}^{k} μ_{B}^{n} (c_{j} y_{j})) \\ = & (⋀_{j = 1}^{k} μ_{B}^{p} (y_{j}), ⋁_{j = 1}^{k} μ_{B}^{n} (y_{j})) . \end{matrix}$

It clearly shows that $μ_{B}^{p} (y_{k + 1}) \in {μ_{B}^{p} (y_{j})}_{j = 1}^{k}$ and $μ_{B}^{n} (y_{k + 1}) \in {μ_{B}^{n} (y_{j})}_{j = 1}^{k}$ . A contradiction to the fact that y_k+1 has distinct positive and negative degrees of membership. Thus ${y_{j}}_{j = 1}^{k + 1}$ are linearly independent. The bipolar fuzzy linear independence of the vectors can be proved similarly. □

Proposition 3.2. Let (Y, B) be a bipolar fuzzy vector space then,

$B (0) = (max_{y \in Y} μ_{B}^{p} (y), min_{y \in Y} μ_{B}^{n} (y))$ ,

B (ay) = B (y) for all a ∈ K \ {0} and y ∈ Y,

If B (y) ≠ B (z) for some y, z ∈ Y then $B (y + z) = (μ_{B}^{p} (y) \land μ_{B}^{p} (z), μ_{B}^{n} (y) \lor μ_{B}^{n} (z))$ .

Proof. We prove only 3.2.1 and 3.2.3.

1. For any y ∈ Y, $B (0) = B (0 y) = (μ_{B}^{p} (0 y), μ_{B}^{n} (0 y))$ . By Definition 3.2, $μ_{B}^{p} (0) = μ_{B}^{p} (0 y) \geq μ_{B}^{p} (y)$ , for every y ∈ Y. Hence $μ_{B}^{p} (0) = max_{y \in Y} μ_{B}^{p} (y)$ . Similarly, it can be proved that $μ_{B}^{n} (0) = min_{y \in Y} μ_{B}^{n} (y)$ .

3. Suppose for y, z ∈ Y, B (y) ≠ B (z). Without loss of generality, assume that $μ_{B}^{p} (y) > μ_{B}^{p} (z)$ . By Definition 3.2, $μ_{B}^{p} (y + z) \geq μ_{B}^{p} (y) \land μ_{B}^{p} (z) = μ_{B}^{p} (z)$ . Also, $μ_{B}^{p} (z) = μ_{B}^{p} ((y + z) - y) \geq μ_{B}^{p} (y + z) \land μ_{B}^{p} (y) = μ_{B}^{p} (y + z)$ . Hence, $μ_{B}^{p} (y + z) = μ_{B}^{p} (z)$ . Other cases can be proved similarly. □

Remark 3.1. If $B$ is a bipolar fuzzy basis of (Y, B) then the membership value of every element of Y can be calculated from the membership values of basis elements, i.e., if $u = \sum_{k = 1}^{n} c_{k} u_{k}$ then, $\begin{matrix} B (u) & = & B (\sum_{k = 1}^{n} c_{k} u_{k}) \\ = & (⋀_{k = 1}^{n} μ_{B}^{p} (c_{k} u_{k}), ⋁_{k = 1}^{n} μ_{B}^{n} (c_{k} u_{k})) \\ = & (⋀_{k = 1}^{n} μ_{B}^{p} (u_{k}), ⋁_{k = 1}^{n} μ_{B}^{n} (u_{k})) . \end{matrix}$

We now come to the main idea of this research paper called bipolar fuzzy matroids and bipolar fuzzy circuits.

Definition 3.5. Let Y be a non-empty finite set of elements and $I \subseteq b F (Y)$ be a family of bipolar fuzzy subsets, $b F (Y)$ is a bipolar fuzzy power set of Y, satisfying the following the conditions,

If $β_{1} \in I$ , $β_{2} \in b F (Y)$ and β₂ ⊂ β₁ then, $β_{2} \in I$ ,

If $β_{1}, β_{2} \in I$ and |supp (β₁) | < |supp (β₂) | then there exists $β_{3} \in I$ such that

β₁ ⊂ β₃ ⊆ β₁ ∪ β₂,

m^p (β₃) ≥ min {m^p (β₁), m^p (β₂)}, $m^{p} (β_{i}) = min {μ_{β_{i}}^{p} (y) | y \in supp (β_{i})}$ , i = 1, 2, 3,

mⁿ (β₃) ≤ max {mⁿ (β₁), mⁿ (β₂)}, $m^{n} (β_{i}) = max {μ_{β_{i}}^{n} (y) | y \in supp (β_{i})}$ , i = 1, 2, 3.

Then the pair $M (Y) = (Y, I)$ is called a bipolar fuzzy matroid on Y, and $I$ is a family of independent bipolar fuzzy subsets of $M (Y)$ .

Note 3.1. ${α : α \in b F (Y), α \notin I}$ is the family of dependent bipolar fuzzy subsets in $M (Y)$ . A minimal dependent bipolar fuzzy set is called a bipolar fuzzy circuit. Equivalently, a bipolar fuzzy set $α \in b F (Y)$ is a bipolar fuzzy circuit in $M$ if $α \notin I$ and $β \in I$ for each β ⊂ α. The family of all bipolar fuzzy circuits is denoted by $C_{r} (M)$ . A bipolar fuzzy circuit having n number of elements is called a bipolar fuzzy n-circuit. A bipolar fuzzy matroid can be uniquely determined from $C_{r} (M)$ because the elements of $I$ are those members of $b F (Y)$ that contain no member of $C_{r} (M)$ .

Definition 3.6. Let Y be a non-empty universe. For any bipolar fuzzy matroid, the bipolar fuzzy rank function $λ_{r} : b F (Y) \to [0, \infty) \times (- \infty, 0]$ is defined as, $λ_{r} (β) = max {| γ | : γ \subseteq β and γ \in I}$ where, $| γ | = \sum_{y \in Y} \frac{μ_{γ}^{p} (y) + μ_{γ}^{n} (y) + 1}{2}$ . Clearly the bipolar fuzzy rank function of a bipolar fuzzy matroid possesses the following properties:

If $β_{1}, β_{2} \in b F (Y)$ and β₁ ⊆ β₂ then λ_r (β₁) ≤ λ_r (β₂),

If $β \in b F (Y)$ then, λ_r (β) ≤ |β|,

If $β \in I$ then, λ_r (β) = |β|.

We now describe the concept of various matroids and bipolar fuzzy matroids by examples.

Examples

A trivial example of a bipolar fuzzy matroid is known as an bipolar fuzzy uniform matroid which is defined as, $I = {β \in b F (Y) : | supp (β) | \leq l} .$

It is denoted by $U_{l, n} = (Y, I)$ where, l is any positive integer and |Y| = n. The bipolar fuzzy circuit of $U_{l, n}$ is a bipolar fuzzy subsets α such that |supp (α) | = l + 1.

Consider the example of a bipolar fuzzy uniform matroid $M = (Y, I)$ where, Y = {e₁, e₂, e₃} and $I = {β \in b F (Y) : | supp (β) | \leq 2}$ such that for any $β \in b F (Y)$ , β (y) = τ (y), for all y ∈ Y where, $τ (y) = {\begin{matrix} (0.2, - 0.3), & y = e_{1} \\ (0.4, - 0.5), & y = e_{2} \\ (0.1, - 0.3), & y = e_{3} \end{matrix} .$

$\begin{matrix} I & = & {\emptyset, {(e_{1}, 0.2, - 0.3)}, {(e_{2}, 0.4, - 0.5)}, \\ {(e_{3}, 0.1, - 0.3)}, {(e_{1}, 0.2, - 0.3), \\ (e_{2}, 0.4, - 0.5)}, {(e_{2}, 0.4, - 0.5), \\ (e_{3}, 0.1, - 0.3)}, {(e_{1}, 0.2, - 0.3), \\ (e_{3}, 0.1, - 0.3)}} . \end{matrix}$ (1)

The bipolar fuzzy circuit of $M$ is {(e₁, 0.2, - 0.3), (e₂, 0.4, - 0.5), (e₃, 0.1, - 0.3)} .

For β = {(e₂, 0.4, - 0.5), (e₁, 0.2, - 0.3)}, λ_r (β) =0.9.

A bipolar fuzzy linear matroid is derived from a bipolar fuzzy matrix. Assume that Y represents the column labels of a bipolar fuzzy matrix and β_x denotes a bipolar fuzzy submatrix having those columns labeled by Y. It is defined as, $I = {β_{x} \in b F (Y) :$ columns of β_x are bipolar fuzzy linearly independent}, $b F (Y)$ is the power set of all bipolar fuzzy submatrices labeled by elements in Y. For any $β_{x} \in b F (Y)$ , $| β_{x} | = \sum_{k = 1}^{r} ({sup}_{i = 1}^{c} \frac{μ_{β_{x}}^{p} (a_{ki}) + μ_{β_{x}}^{n} (a_{ki}) + 1}{2}), β_{x}^{*} = [a_{ij}]_{r \times c}$ .

Let Y = { 1 , 2 , 3 , 4 } be a set of column labels of bipolar fuzzy 2 × 1 vectors over $ℝ$ such that for any $β_{x} \in b F (Y)$ , β_x (y) = A (y) where, $\begin{matrix} A = \\ [\begin{matrix} 1 & 2 & 3 & 4 \\ (0.2, - 0.3) & (0.4, - 0.5) & (0.6, - 0.7) & (0.8, - 0.9) \\ (0.3, - 0.4) & (0.5, - 0.6) & (0.7, - 0.8) & (0.9, - 1.0) \end{matrix}] . \end{matrix}$

Take $I = {\emptyset, {1}, {2}, {4}, {1, 2}, {2, 4}}$ then, $M (A) = (A, I)$ is a bipolar fuzzy matroid on Y. The family of dependent bipolar fuzzy subsets of $M (A)$ is {{ 3 }, { 1 , 3 }, { 1 , 4 }, { 2 , 3 }, { 3 , 4 }} ∪ { η : η ⊆ A, |supp ( η ) |≥3}. For β = { 2 , 4 }, μ_r (β) =0.9 .

A bipolar fuzzy partition matroid in which the universe Y is partitioned into bipolar fuzzy sets δ₁, δ₂, …, δ_r such that

$I = {β \in b F (Y) : | supp (β) \cap supp (δ_{i}) | \leq l_{i}, for all 1 \leq i \leq r}$ for given positive integers l₁, l₂, …, l_r. The circuit of a bipolar fuzzy partition matroid is a bipolar fuzzy subset α such that |supp (α) ∩ supp (δ_i) | = l_i + 1 .

The very important class of bipolar fuzzy matroids are derived from bipolar fuzzy graphs. The detail is discussed in Proposition 3.3. The bipolar fuzzy matroid derived using this method is known as bipolar fuzzy cycle matroid, denoted by $M = (Y, I)$ where, Y = {y₁, y₂, …, y_n} is the set of edges and $I$ is defined as,

$I = {β \in b F (Y) ∣ β$ is a bipolar fuzzy set on X ⊆ Y} . Clearly $β \in I$ is an independent set of $M (G)$ if and only if supp (β) is not an edge set of any cycle. Equivalently, the members of $M$ are bipolar fuzzy graphs β such that supp (β) is a forest.

Consider the example of a bipolar fuzzy cycle matroid $(Y, I)$ where, Y = {y₁, y₂, y₃, y₄, y₅} and for any, $β \in I$ , β (y) = D (y), (C, D) is a bipolar fuzzy multigraph with edge set Y as shown in Fig. 1. By Proposition 3.3, $\begin{matrix} C_{r} (M) & = & {{(y_{5}, 0.3, - 0, 4)}, {(y_{2}, 0.2, - 0.3), \\ (y_{3}, 0.2, - 0.3)}, {(y_{1}, 0.2, - 0.3), \\ (y_{2}, 0.2, - 0.3), (y_{4}, 0.6, - 0.7)}, \\ {(y_{1}, 0.2, - 0.3), (y_{3}, 0.2, - 0.3)}} . \\ I & = & {\emptyset, {(y_{1}, 0.2, - 0.3)}, {(y_{2},, 0.2, - 0.3)}, \\ {(y_{3}, 0.2, - 0.3)}, {(y_{4}, 0.6, - 0.7)}, \\ {(y_{1}, 0.2, - 0.3), (y_{2}, 0.2, - 0.3)}, \\ {(y_{1}, 0.2, - 0.3), (y_{4}, 0.6, - 0.7)}, \\ {(y_{2}, 0.2, - 0.3), (y_{4}, 0.6, - 0.7)}, \\ {(y_{1}, 0.2, - 0.3), (y_{3}, 0.2, - 0.3)}, \\ {(y_{3}, 0.2, - 0.3), (y_{4}, 0.6, - 0.7)}} . \end{matrix}$

For β = {(y₂, 0.2, - 0.3), (y₄, 0.6, - 0.7)}, λ_r (β) =0.9.

Fig.1

3-polar fuzzy multigraph.

Proposition 3.3. For any any bipolar fuzzy matroid $(Y, I)$ , if $C_{r} (M)$ is the family of bipolar fuzzy edge sets α such that supp (α) is the edge set of a cycle. Then $C_{r} (M)$ is the family of bipolar fuzzy circuits on $M$ .

The proof of Proposition 3.3 follows from Example 4.

Example 3.2. For any bipolar fuzzy graph G = (C, D) and t = (t^p, tⁿ), 0 ≤ t^p ≤ 1, -1 ≤ tⁿ ≤ 0 define, $E_{t} = {yz \in supp (D) | μ_{D}^{p} (yz) > t^{p}, μ_{D}^{n} (yz) < t^{n}}$ , F_t = {F|F isaforestinthecrispgraph (Y, E_t)}, $I_{t} = {E (F) | F \in F_{t}}$ where, E (F) is the edge set of F.

Then $(E_{t}, I_{t})$ is a matroid for each t . Define $J =$ {β∈ $b F (Y) |$ β_t∈ $I_{t},$ for each t = (t^p, tⁿ), 0 ≤ t^p ≤ 1, -1 ≤ tⁿ ≤ 0} then, $(Y, J)$ is a bipolar fuzzy cycle matroid.

Theorem 3.1. Let $M = (Y, I)$ be a bipolar fuzzy matroid and, for each t = (t^p, tⁿ), 0 ≤ t^p ≤ 1, - 1 ≤ tⁿ ≤ 0, define $φ_{t} = {β_{t} | β \in I}$ . Then (Y, φ_t) is a matroid on Y.

Proof. We prove conditions 1 and 2 of Definition 2.1. Assume that β_{1
t} ∈ φ_t and α ⊆ β_{1
t}. Define a bipolar fuzzy set $β_{2} \in b F (Y)$ by $β_{2} (y) = {\begin{matrix} t & y \in α, \\ (0, 0) & otherwise . \end{matrix}$

Clearly β₂ ⊆ β₁, $β_{2} \in I$ and β_{2
t} = α therefore, α ∈ φ_t. To prove condition 2, let α₁, α₂ ∈ φ_t and |α₁| < |α₂|. Then there exist β₁ and β₂ such that β_{1
t} = α₁ and β_{2
t} = α₂. Define ${\hat{β}}_{1}$ and ${\hat{β}}_{2}$ by $\begin{matrix} {\hat{β}}_{1} (y) & = & {\begin{matrix} t & y \in α_{1}, \\ (0, 0) & otherwise . \end{matrix} \\ {\hat{β}}_{2} (y) & = & {\begin{matrix} t & y \in α_{2}, \\ (0, 0) & otherwise . \end{matrix} \end{matrix}$

It is clear that $supp ({\hat{β}}_{1}) < supp ({\hat{β}}_{2})$ . Since $M$ is a bipolar fuzzy matroid there exists β₃ such that ${\hat{β}}_{1} \subseteq β_{3} \subseteq {\hat{β}}_{1} \cup {\hat{β}}_{2}$ . Since ${\hat{β}}_{1} \cup {\hat{β}}_{2} (y) = {\begin{matrix} t & y \in α_{1} \cup α_{2}, \\ (0, 0) & otherwise . \end{matrix}$

Therefore, there exists a set α₃ such that $β_{3} (y) = {\begin{matrix} t & y \in α_{3}, \\ (0, 0) & otherwise . \end{matrix}$

Also, α₁ ⊆ α₃ ⊆ α₁ ∪ α₂, α₃ ∈ φ_t. Hence $M_{t}$ is a matroid on Y. □

Remark 3.2. Let $M = (Y, I)$ be a bipolar fuzzy matroid and, for each t = (t^p, tⁿ), 0 ≤ t^p ≤ 1, -1 ≤ tⁿ ≤ 0, $M_{t} = (Y, I_{t})$ be a matroid on a finite set Y as given in Theorem 3.1. As Y is finite, so there is a finite sequence t ₀, t ₁, t ₂, …, t _m, $0 < t_{1}^{p} < t_{2}^{p} < \dots < t_{m}^{p}$ , $0 > t_{1}^{n} > t_{2}^{n} > \dots > t_{m}^{n} > - 1$ such that $M_{t_{i}} = (Y, I_{t_{i}})$ is a crisp matroid, for each 1 ≤ i ≤ m, and

$t_{0} = (0, 0), t_{m}^{p} \leq 1 and t_{m}^{n} \geq - 1$ ,

$I_{w} \neq \emptyset$ if w = (w^p, wⁿ), $0 \leq w^{p} \leq t_{m}^{p}$ and $t_{m}^{n} \leq w^{n} \leq 0$ ,

If for s = (s^p, sⁿ), $t_{i}^{p} < w^{p}, s^{p} < t_{i + 1}^{p}$ and $t_{i}^{n} > w^{n}, s^{n} > t_{i + 1}^{n}$ then, $I_{w} = I_{s}$ , 0 ≤ i ≤ m - 1,

If $t_{i}^{p} < w < t_{i + 1}^{p} < s^{p} < t_{i + 2}^{p}$ and $t_{i}^{n} > w > t_{i + 1}^{n} > s^{n} > t_{i + 2}^{n}$ then, $I_{w} \supset I_{s}$ , 0 ≤ i ≤ m - 2.

The sequence t ₀, t ₁, t ₂, …, t _m is known as fundamental sequence of $M$ . Let ${\bar{t}}_{i} = \frac{1}{2} (t_{i - 1} + t_{i})$ for 1 ≤ i ≤ m. The decreasing sequence of crisp matroids $M_{{\bar{t}}_{1}} \supset M_{{\bar{t}}_{2}} \supset \dots \supset M_{{\bar{t}}_{m}}$ is known as $M -$ indeced matroid sequence.

Definition 3.7. Let t ₀, t ₁, …, t _m be the fundamental sequence of a bipolar fuzzy matroid. For any t = (t^p, tⁿ), 0 < t^p ≤ 1 and -1 ≤ tⁿ < 0, define ${\bar{I}}_{t} = I_{{\bar{t}}_{i}}$ where, ${\bar{t}}_{i} = \frac{1}{2} (t_{i - 1} + t_{i})$ and $t_{i - 1}^{p} < t^{p} \leq t_{i}^{p}$ , $t_{i - 1}^{n} > t^{n} \geq t_{i}^{n}$ . If $t^{p} > t_{m}^{p}$ and $t^{n} < t_{m}^{n}$ $take {\bar{I}}_{t} = I_{t}$ . Define

$\bar{I} = {β \in b F (Y) : β_{t} \in {\bar{I}}_{t},$ for each t = (t^p, tⁿ), 0 < t^p ≤ 1 and - 1 ≤ tⁿ < 0} .

Then $\bar{M} = (Y, \bar{I})$ is known as closure of $M = (Y, I)$ .

Example 3.3. We now explain the concept of closure by an example of a bipolar fuzzy uniform matroid $M = (Y, I)$ where, Y = {y₁, y₂, y₃} and $I = {β \in b F (Y) : | supp (η) | \leq 1}$ such that for any $β \in b F (Y)$ , β (y) = τ (y), for all y ∈ Y where, $τ (y) = {\begin{matrix} (0.2, - 0.3), & y = y_{1} \\ (0.3, - 0.4), & y = y_{2} \\ (0.4, - 0.5), & y = y_{3} \end{matrix} .$

$I = {\emptyset,$ {(y₁, 0.2, - 0.3)} , {(y₂, 0.3, - 0.4)} , {(y₃, 0.4, -0.5)}} .

The fundamental sequence of $M$ is { t ₀ = (0, 0), t ₁ = (0.2, - 0.3), t ₂ = (0.3, - 0.4), t ₃ = (0.4, - 0.5)}. From routine calculations, ${\bar{t}}_{1} = (0.1, - 0.15)$ , ${\bar{t}}_{2} = (0.25, - 0.35)$ , ${\bar{t}}_{3} = (0.35, - 0.45)$ . Now $I_{{\bar{t}}_{1}} = {{y_{1}}, {y_{2}}, {y_{3}}}$ , $I_{{\bar{t}}_{2}} = {{y_{2}}, {y_{3}}}$ , $I_{{\bar{t}}_{3}} = {{y_{3}}}$ . Hence the closure of $I$ can be defined as, $\begin{matrix} \bar{I} & = & {\emptyset, {(y_{1}, 0.2, - 0.3)}, {(y_{2}, 0.3, - 0.4)}, \\ {(y_{1}, 0.2, - 0.3), (y_{2}, 0.3, - 0.4)}, \\ {(y_{1}, 0.2, - 0.3), (y_{3}, 0.4, - 0.5)}, \\ {(y_{2}, 0.3, - 0.4), (y_{3}, 0.4, - 0.5)}, \\ {(y_{3}, 0.4, - 0.5)}} . \end{matrix}$

Theorem 3.2. The closure $\bar{M} = (Y, \bar{I})$ of a bipolar fuzzy matroid $M = (Y, I)$ is also a bipolar fuzzy matroid.

The proof of Theorem 3.2 is a clear consequence of Theorem 3.1.

Definition 3.8. A bipolar fuzzy matroid with fundamental sequence t ₀, t ₁, …, t _m is known as a closed bipolar fuzzy matroid if for each t = (t^p, tⁿ), $t_{i - 1}^{p} < t^{p} \leq t_{i}^{p}$ and $t_{i - 1}^{n} > t^{n} \geq t_{i}^{n}$ , $I_{t} = I_{t_{i}}$ .

Remark 3.3. Note that the closure of a bipolar fuzzy matroid is closed and that it is the smallest closed bipolar fuzzy matroid containing $M$ . Also the fundamental sequence of $M$ and $\bar{M}$ is same.

Theorem 3.3. Let $M = (Y, I)$ be a bipolar fuzzy matroid and suppose $β \in b F (Y)$ then, $β \in I$ if and only if for each t = (t^p, tⁿ), β_t ∈ φ_t where, t^p ∈ R^p (β), tⁿ ∈ Rⁿ (β).

Theorem 3.3 follows from the proof of Theorem 3.1.

Theorem 3.4. If $M = (Y, I)$ is a closed fuzzy matroid, then each dependent bipolar fuzzy set in $M$ properly contains a dependent bipolar fuzzy subset.

Proof. Assume that α is a dependent bipolar fuzzy set in $M$ and $R^{p} (α) = {t_{1}^{p}, t_{2}^{p}, \dots, t_{m}^{p}}$ , $R^{n} (α) = {t_{1}^{n}, t_{2}^{n}, \dots, t_{m}^{n}}$ where, $t_{1}^{p} < t_{2}^{p} < \dots < t_{m}^{p}$ and $t_{1}^{n} > t_{2}^{n} > \dots > t_{m}^{n}$ . By Theorem 3.3, there exists t _i, 1 ≤ i ≤ m, such that α_{t

_i} ∉ φ_{t

_i}. Since $M$ is a closed bipolar fuzzy matroid therefore, there is an ɛ = (ɛ^p, ɛⁿ), ɛ^p > 0, ɛⁿ < 0 such that for each t = (t^p, tⁿ), $t_{i}^{p} - ɛ^{p} \leq t^{p} \leq t_{i}^{p}$ , $t_{i}^{n} - ɛ^{n} \geq t^{n} \geq t_{i}^{n}$ , $α_{t} = α_{t_{i}} and φ_{t} = φ_{t_{i}} .$

Let γ be a proper bipolar fuzzy subset of α defined by $γ (y) = {\begin{matrix} t_{i} - ɛ & μ_{α}^{p} (y) \geq t_{i}^{p} and μ_{α}^{n} (y) \leq t_{i}^{n}, \\ (0, 0) & otherwise . \end{matrix}$

Note that γ_{t
_i-ɛ} ∉ φ_{t
_i-ɛ} and hence by Theorem 3.3, $γ \notin I$ . □

As a consequence of Theorem 3.4, we adopt the following definition of a bipolar fuzzy circuit.

Definition 3.9. Let $M = (Y, I)$ be a bipolar fuzzy matroid. Then $α \in b F (Y)$ is bipolar fuzzy circuit in $M$ if $α \notin I$ and $α \ \ z \in I$ for each z ∈ supp (α), where α ∖∖ z is defined as, $(α \ \ z) (y) = {\begin{matrix} α (y) & y \neg = z, \\ (0, 0) & y = z . \end{matrix}$

We now characterize the properties of bipolar fuzzy circuits in the form of crisp circuits.

Theorem 3.5. (Characterization Theorem) Assume that $M = (Y, I)$ is a bipolar fuzzy matroid and $α \in b F (Y)$ . Let $R^{p} (α) = {t_{1}^{p}, t_{2}^{p}, \dots, t_{m}^{p}}$ and $R^{n} (α) = {t_{1}^{n}, t_{2}^{n}, \dots, t_{m}^{n}}$ where, $t_{1}^{p} < t_{2}^{p} < \dots < t_{m}^{p}$ and $t_{1}^{n} > t_{2}^{n} > \dots > t_{m}^{n}$ . Then α is a bipolar fuzzy circuit in $M$ if and only if

α_t

₁ is a crisp circuit in (Y, φ_t

₁),

α_{t

_i} ∈ φ_{t

_i}, for 2 ≤ i ≤ m.

Proof. Let α be a bipolar fuzzy circuit then, $α \notin I$ . By Theorem 3.3, there exists at least one i, 1 ≤ i ≤ m, (say, i = i′) such that α_{t

_i′} ∉ φ_{t

_i′}. We now show that i′ = 1. On contrary, assume that 2 ≤ i′ ≤ m. Since $t_{1}^{p} < t_{i^{'}}^{p}$ and $t_{1}^{n} > t_{i^{'}}^{n}$ therefore, α_{t

_i′} ⊂ α_{t

₁}. Then there exists, z ∈ α_{t

₁} \ α_{t

_i′}. Clearly, (α \\ z) _{t

_i′} = α_{t

_i′} ∉ φ_{t

_i′}.

It follows from Theorem 3.3 that $α \ \ z \notin I$ which is a contradiction to the supposition that α is a bipolar fuzzy circuit. Thus, for 2 ≤ i ≤ m, α_{t

_i′} ∈ φ_{t

_i′} and it establishes 3.5.2.

We now prove 3.5.1. It is clear that α_{t

₁} ∉ φ_{t

₁}. If α_{t

₁} is not a circuit then, there exist B ⊂ α_{t

₁} such that B ∉ φ_{t

₁}. If x ∈ α_{t

₁} \ B then, B ⊆ (α \\ x) _{t

₁} ⇒ (α \\ x) _{t

₁} ∉ φ_{t

₁}. So, by Theorem 3.3, $α \ \ x \notin I$ . A contradiction to the fact that α is a bipolar fuzzy circuit and it establishes 3.5.1.

Conversely, let $α \in b F (Y)$ satisfies 3.5.1 and 3.5.2. By Theorem 3.3, $α \notin I$ . Suppose that y ∈ supp (α) then, $μ_{α}^{p} (y) \geq t_{1}^{p}$ and $μ_{α}^{n} (y) \leq t_{1}^{n}$ . Hence (α \\ y) _{t

₁} = α_{t

₁} \ y ∈ φ_{t

₁} and for 2 ≤ i ≤ m α_{t

_i} ⊇ (α \\ y) _{t

_i} ∈ φ_{t

_i}. Clearly, by Theorem 3.3, $α \ \ y \in I$ and so α is a bipolar fuzzy circuit. □

We can say that a bipolar fuzzy circuit α is elementary if it is an elementary bipolar fuzzy set, i.e., α is single valued on supp (α).

Corollary 3.1. Let $M = (Y, I)$ be a bipolar fuzzy matroid then, α is a bipolar fuzzy circuit in $M$ if and only if α = λ ∪ γ where, λ is an elementary bipolar fuzzy circuit, $γ \in I$ and supp (γ) ⊂ supp (λ).

Proof. Assume that α is a bipolar fuzzy circuit in $M$ , $R^{p} (α) = {t_{1}^{p}, t_{2}^{p}, \dots, t_{m}^{p}}$ and $R^{n} (α) = {t_{1}^{n}, t_{2}^{n}, \dots, t_{m}^{n}}$ where, $t_{1}^{p} < t_{2}^{p} < \dots < t_{m}^{p}$ and $t_{1}^{n} > t_{2}^{n} > \dots > t_{m}^{n}$ . Define λ and γ as, $\begin{matrix} λ (y) & = & {\begin{matrix} t_{1} & y \in supp (α) \\ (0, 0) & otherwise, \end{matrix} \\ γ (y) & = & {\begin{matrix} α (y) & if y \in α_{t_{2}} \\ (0, 0) & otherwise . \end{matrix} \end{matrix}$

Consequently, λ is an elementary bipolar fuzzy circuit, $γ \in I$ , supp (γ) ⊂ supp (λ) and α = λ ∪ γ.

Conversely, let α = λ ∪ γ, λ is elementary, $γ \in I$ and supp (γ) ⊂ supp (λ). If $R^{p} (α) = {t_{1}^{p}, t_{2}^{p}, \dots, t_{m}^{p}}$ and $R^{n} (α) = {t_{1}^{n}, t_{2}^{n}, \dots, t_{m}^{n}}$ where, $t_{1}^{p} < t_{2}^{p} < \dots < t_{m}^{p}$ and $t_{1}^{n} > t_{2}^{n} > \dots > t_{m}^{n}$ then, $R^{p} (λ) = {t_{1}^{p}}$ , $R^{n} (λ) = {t_{1}^{n}}$ , $R^{p} (γ) = {t_{2}^{p}, \dots, t_{m}^{p}}$ and $R^{n} (γ) = {t_{2}^{n}, \dots, t_{m}^{n}}$ . It follows from Theorem 3.5 that,

α_t

₁ is a circuit in φ_t

₁,

For 2 ≤ i ≤ m, α_{t

_i} ∈ φ_{t

_i}.

Thus, by Theorem 3.5, α is a bipolar fuzzy circuit. □

Note 3.2. Corollary 3.1 implies that every bipolar subset of a bipolar fuzzy circuit is either a bipolar fuzzy circuit or an independent bipolar fuzzy set.

Corollary 3.2. Suppose that $M = (Y, I)$ is a bipolar fuzzy matroid and $γ \in b F (Y)$ . If $γ \notin I$ then, there exists a bipolar fuzzy circuit α such that α ⊆ γ.

Proof. Let $R^{p} (γ) = {t_{1}^{p}, t_{2}^{p}, \dots, t_{m}^{p}}$ and $R^{n} (γ) = {t_{1}^{n}, t_{2}^{n}, \dots, t_{m}^{n}}$ . By Theorem 3.3, there exists t_i such that γ_{t

_i} ∉ φ_{t

_i}. Let B ⊆ γ_{t

_i} be a circuit in (Y, φ_{t

_i}) and define $α \in b F (Y)$ by $α (y) = {\begin{matrix} t_{i} & if y \in B \\ (0, 0) & otherwise . \end{matrix}$

Then α ⊆ γ and $R^{p} (α) = {t_{i}^{p}}$ , $R^{n} (α) = {t_{i}^{n}}$ , supp (α) = B and hence α is a bipolar fuzzy circuit. □

Theorem 3.6. If α₁ and α₂ are bipolar fuzzy circuits in a bipolar fuzzy matroid $M$ and α₁ ⊆ α₂ then, supp (α₁) = supp (α₂).

Proof. Let α₁ and α₂ be two bipolar fuzzy circuits in $M = (Y, I)$ . On contrary, assume that supp (α₁) ⊂ supp (α₂) then, there exists y ∈ Y such that y ∈ supp (α₂) \ supp (α₁). Moreover, $α_{1} \subseteq α_{2} \ \ y \in I$ and hence, $α_{1} \in I$ . A contradiction to the supposition that α₁ is a bipolar fuzzy circuit. □

3.1 Circuit rectangles

We now introduce and discuss the notion of circuit rectangles and circuit function.

Definition 3.10. The truncation of a bipolar fuzzy set β at level t = (t^p, tⁿ) is a bipolar fuzzy set $β_{t}^{*}$ defined as, $β_{t}^{*} (y) = {\begin{matrix} β (y) & if μ_{β}^{p} (y) \leq t^{p}, μ_{β}^{n} (y) \geq t^{n} \\ (0, 0) & μ_{β}^{p} (y) > t^{p}, μ_{β}^{n} (y) < t^{n} . \end{matrix}$

If $M = (Y, I)$ is a bipolar fuzzy matroid then, define $η = (η^{p}, η^{n}) : b F (Y) \to [0, 1] \times [- 1, 0]$ by $\begin{matrix} η^{p} (β) & = & max {t^{p} ∣ β_{t}^{*} is not a bipolar fuzzy \\ circuit in M}, \\ η^{n} (β) & = & min {t^{n} ∣ β_{t}^{*} is not a bipolar fuzzy \\ circuit in M} . \end{matrix}$

The function η is known as circuit function for β.

Example 3.4. Consider the example of a bipolar fuzzy uniform matroid as given in Example 1. Take β = {(e₁, 0.2, - 0.3), (e₂, 0.4, - 0.5), (e₃, 0.1, - 0.3)}.

For t = (0.2, - 0.3), $β_{t}^{*} = {(e_{1}, 0.2, - 0.3), (e_{3}, 0.1, - 0.3)}$ .

For t = (0.1, - 0.3), $β_{t}^{*} = {(e_{3}, 0.1, - 0.3)}$ .

For t = (0.4, - 0.5), $β_{t}^{*} = β$ .

Clearly, η (β) = (0.2, - 0, 3). If we take β = {(e₁, 0.2, - 0.3), (e₂, 0.4, - 0.5)} then, η (β) = (1, - 1).

Theorem 3.7. Let $M = (Y, I)$ be a bipolar fuzzy matroid and η is a circuit function for β. Then the following conditions hold.

If $β \in b F (Y)$ is not a bipolar fuzzy circuit then, η (β) = (1, - 1).

If β is a bipolar fuzzy circuit then, η^p (β) ≥ m^p (β) and ηⁿ (β) ≤ mⁿ (β).

Proof. 1 . Since β is not a bipolar fuzzy circuit therefore, for t = (1, - 1), $β_{t}^{*} = β$ . Consequently, η (β) = (1, - 1).

2 . Let $R^{p} (β) = {t_{1}^{p}, t_{2}^{p}, \dots, t_{m}^{p}}$ , $R^{n} (β) = {t_{1}^{n}, t_{2}^{n}, \dots, t_{m}^{n}}$ and $t_{i} = (t_{i}^{p}, t_{i}^{n})$ , 1 ≤ i ≤ m. For t ∈ { t ₁, t ₂, …, t _m-1}, $β_{t}^{*}$ is not a bipolar fuzzy circuit and for $t \in {t ∣ t^{p} \geq t_{m}^{p}, t_{n} \leq t_{m}^{n}}$ , $β_{t}^{*}$ is a bipolar fuzzy circuit. For $t^{p} < t_{1}^{p}$ and $t^{n} > t_{1}^{n}$ , $β_{t}^{*} = \emptyset$ . Hence, η^p (β) ≥ m^p (β) and ηⁿ (β) ≤ mⁿ (β). □

Definition 3.11. Let $M = (Y, I)$ be a closed bipolar fuzzy matroid and α be a bipolar fuzzy circuit in $M$ . Let I (α) = [mⁿ (α), 0) × (0, m^p (α)] is called circuit rectangle for α.

Theorem 3.8. Let α be a bipolar fuzzy circuit in a closed bipolar fuzzy matroid $M = (Y, I)$ . Let I (α) = [mⁿ (α), 0) × (0, m^p (α)] be a circuit rectangle for α then,

If t ∈ I (α) then, α_t is a circuit in $M_{t} = (Y, I_{t})$ .

If t ∉ I (α) then, α_t ∈ φ_t.

Proof. By Theorem 3.1, α = δ ∪ γ, where δ is an elementary bipolar fuzzy circuit, $γ \in I$ , and supp (γ) ⊂ supp (δ). Clearly, $R^{p} (δ) = {m^{p} (α)}$ and Rⁿ (α) = {mⁿ (α)}. If t^p > m^p (α) and tⁿ < mⁿ (α) then, α_t = γ_t ∈ φ_t. If 0 < t^p ≤ m^p (α) and 0 > tⁿ ≥ mⁿ (α) then, α_t = supp (δ). Hence α_t is a circuit in $M_{t}$ if t ∈ I (α). □

Applications: Bipolar fuzzy matroids have interesting applications in graph theory, combinatorics, algebra and many applications in addition to Mathematics. Bipolar fuzzy matroids are used to discuss the uncertain behavior of objects if the data have incomplete information. We now discuss applications of bipolar fuzzy matroids in decision support system and network analysis.

1. Decision support system: Bipolar fuzzy matroids can be used in decision support systems to find the ordering of n tasks to be fulfilled in q days if each task requires full day attention. All tasks are available at 0 time and each task has a profit p^p associated with it, and a penalty pⁿ to be paid if it is not completed at deadline d. The profit $p_{j}^{p}$ can be gained if each task j is completed at the deadline d_j. The problem is to find a bipolar fuzzy ordering of tasks to maximize the total profit. It doesn’t look like a bipolar fuzzy matroid problem because bipolar fuzzy matroid problem asks to find an optimal bipolar fuzzy subset, but this problem requires to find an optimal schedule. However, this is a bipolar fuzzy matroid problem. The profit and penalty of any ordering can be determined by a bipolar fuzzy subset of tasks that are on or before time.

For a bipolar fuzzy subset S of deadlines {d₁, d₂, …, d_n} corresponding to tasks T = {t₁, t₂, …, t_n} if there is a ordering such that every task in S is on or before time, and all tasks out of S are late. The procedure for the selection of tasks is given in Algorithm 1 whose time complexity is O(n2ⁿ).

Algorithm 1

Input the deadline d_i corresponding to each task t_i and set Y = {(t_i, d_i) ∣1 ≤ i ≤ n}.

Input the profit $p_{i}^{p}$ and penalty $p_{i}^{n}$ corresponding to each task t_i.

do i from 1 → 2ⁿ

supp (β_i) ∈ P (Y)

do j from 1 → |supp (β_i) |

y_j ∈ supp (β_i)

$cost (β_{i}) = cost (β_{i}) + \frac{μ_{β_{i}}^{p} (y_{j}) + μ_{β_{i}}^{n} (y_{j}) + 1}{2}$

end do

Calculate cost= max {cost (β_i) ∣1 ≤ i ≤ 2ⁿ}

Choose a subset β_i such that cost = cost (β_i).

Construct a bipolar fuzzy matroid $M = (Y, I)$ of tasks and deadlines such that, $I = {β ∣ supp (β) \subseteq supp (β_{i})} .$

To perform only |supp (β) | number of tasks on time, any bipolar fuzzy subset of |supp (β) | elements can be chosen from the bipolar fuzzy matroid to gain maximum profit.

As an example consider a set of eight tasks t₁, t₂, …, t₈ with deadlines d₁, d₂, …, d₈. If a task t_i is completed on or before deadline, it gains a profit $p_{i}^{p}$ , and if it is late the penalty to be paid is $p_{i}^{n}$ . The percentage profit and penalty associated with each task is given in Table 1. The family of a particular number of bipolar fuzzy tasks that maximize the profit is a bipolar fuzzy matroid. Any bipolar fuzzy subset S of tasks will maximize the profit if $\sum_{(t_{i}, d_{i}) \in S} \frac{p_{i}^{p} + p_{i}^{n} + 1}{2}$ is maximum. The bipolar fuzzy matroid is given as, $\begin{matrix} {\emptyset, {((t_{4}, d_{4}), 0.9, - 0.5)}, {((t_{6}, d_{6}), 0.7, - 0.3)}, \\ {((t_{7}, d_{7}), 0.8, - 0.4)}, {((t_{8}, d_{8}), 0.5, - 0.1)}, \\ {((t_{4}, d_{4}), 0.9, - 0.5), ((t_{6}, d_{6}), 0.7, - 0.3)}, \\ {((t_{4}, d_{4}), 0.9, - 0.5), ((t_{7}, d_{7}), 0.8, - 0.4)}, \\ {((t_{4}, d_{4}), 0.9, - 0.5), ((t_{8}, d_{8}), 0.5, - 0.1)}, \\ {((t_{6}, d_{6}), 0.7, - 0.3), ((t_{7}, d_{7}), 0.8, - 0.4)}, \\ {((t_{6}, d_{6}), 0.7, - 0.3), ((t_{8}, d_{8}), 0.5, - 0.1)}, \\ {((t_{7}, d_{7}), 0.8, - 0.4), ((t_{8}, d_{8}), 0.5, - 0.1)}, \\ {((t_{4}, d_{4}), 0.9, - 0.5), ((t_{6}, d_{6}), 0.7, - 0.3), \\ ((t_{7}, d_{7}), 0.8, - 0.4)}, {((t_{4}, d_{4}), 0.9, - 0.5), \\ ((t_{6}, d_{6}), 0.7, - 0.3), ((t_{8}, d_{8}), 0.5, - 0.1)}, \\ {((t_{4}, d_{4}), 0.9, - 0.5), ((t_{7}, d_{7}), 0.8, - 0.4), \\ ((t_{8}, d_{8}), 0.5, - 0.1)}, {((t_{6}, d_{6}), 0.7, - 0.3), \\ ((t_{7}, d_{7}), 0.8, - 0.4), ((t_{8}, d_{8}), 0.5, - 0.1)}, \\ {((t_{4}, d_{4}), 0.9, - 0.5), ((t_{6}, d_{6}), 0.7, - 0.3), \\ ((t_{7}, d_{7}), 0.8, - 0.4), ((t_{8}, d_{8}), 0.5, - 0.1)}} \end{matrix}$

Table 1
Decision Support System

Task Deadline Profit Penalty Task Deadline Profit Penalty

t ₁ d ₁ 0.8 0.7 t ₅ d ₅ 0.6 0.8

t ₂ d ₂ 0.5 0.6 t ₆ d ₆ 0.7 0.3

t ₃ d ₃ 0.6 0.8 t ₇ d ₇ 0.8 0.4

t ₄ d ₄ 0.9 0.5 t ₈ d ₈ 0.5 0.1

Task	Deadline	Profit	Penalty	Task	Deadline	Profit	Penalty
t ₁	d ₁	0.8	0.7	t ₅	d ₅	0.6	0.8
t ₂	d ₂	0.5	0.6	t ₆	d ₆	0.7	0.3
t ₃	d ₃	0.6	0.8	t ₇	d ₇	0.8	0.4
t ₄	d ₄	0.9	0.5	t ₈	d ₈	0.5	0.1

If d₁ < d₂ < … < d₈ then, the subset supp (S) = {d₄, d₆, d₇, d₈} will maximize the profit. If we want to perform only 3 tasks on time then any bipolar fuzzy subset of three elements can be chosen from the bipolar fuzzy matroid and similarly.

2. Network analysis: Bipolar fuzzy matroids can be used in network analysis problems to determine the minimum number of connections for wireless communication. The procedure for the selection of minimum number of locations from a wireless connection is explained in the following steps.

Input the n number of locations L₁, L₂, …, L_n of wireless communication network.

Input the adjacency matrix ξ = [L_ij] _{n
²} of locations.

From this adjacency matrix, arrange the positive membership values in increasing order and negative membership values in decreasing order.

Select an edge having minimum positive membership value and maximum negative membership value.

Repeat Step 4 so that the selected edge does not create any circuit with previous selected edges.

Stop the procedure if the connection between every pair of locations is set up.

Here we explain the use of bipolar fuzzy matroids in network analysis. Bipolar fuzzy graph in Fig. 2 represents a wireless communication between five locations L₁, L₂, L₃, L₄, L₅. The positive degree of each edge shows the signal damage through this path, and negative degree indicates the strong signal strength in sending a message from one location to the other. Each pair of vertices is connected by an edge. But, in general, we do not need connections among all the vertices because the vertices linked indirectly will also have a message service between them, i.e., if there is a connection from L₂ to L₃ and L₃ to L₄ then we can send a message from L₂ to L₄ even if there is no edge between L₂ and L₄. The problem is to find a set of edges such that we are able to send message between every two vertices under the condition that the signal disturbance is minimum. The procedure is as follows. Arrange the positive membership values of edges in increasing order and negative membership values in decreasing order as, {(0.5, - 0.28), (0.6, - 0.33), (0.6, - 0.37), (0.7, - 0.41), (0.7, - 0.44), (0.7, - 0.46), (0.7, - 0.48), (0.8, - 0.5), (0.8, - 0.51), (0.8, - 0.53)}. At each step, select an edge having minimum positive membership value and maximum negative membership so that it does not create any circuit with previous selected edges. The bipolar fuzzy set of selected edges is, $\begin{matrix} {(L_{3} L_{4}, 0.5, - 0.28), (L_{3} L_{5}, 0.6, - 0.33), \\ (L_{1} L_{5}, 0.6, - 0.37), (L_{2} L_{4}, 0.7, - 0.41), \\ (L_{1} L_{4}, 0.7, - 0.46)} . \end{matrix}$

Fig.2

Wireless communication.

4 Conclusions

A matroid is one of the most useful mathematical tools which is used in linear algebra, graph theory and various branches of Mathematics. Bipolar fuzzy models constitute a generalization of Zadeh’s fuzzy models. In this research paper, we have introduced the concept of bipolar fuzzy matroids and bipolar fuzzy circuits. We also have presented certain applications of bipolar fuzzy matroids in decision support system and network analysis. If we have multiple parameters having uncertain behavior then these applications can also be discussed with generalized soft networks. We are extending our work to (1) Decision support systems based on intuitionistic fuzzy soft circuits, (2) Fuzzy rough soft circuits, (3) and Neutrosophic soft circuits.

Competing Interests: The authors declare that there is no conflict of interest regarding the publication of this article.

Footnotes

Acknowledgments

The authors are highly thankful to the Associate Editor, and the anonymous referees for their valuable comments and suggestions.

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Bipolar fuzzy circuits with applications

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Bipolar fuzzy circuits

Table 1 Decision Support System Task Deadline Profit Penalty Task Deadline Profit Penalty t 1 d 1 0.8 0.7 t 5 d 5 0.6 0.8 t 2 d 2 0.5 0.6 t 6 d 6 0.7 0.3 t 3 d 3 0.6 0.8 t 7 d 7 0.8 0.4 t 4 d 4 0.9 0.5 t 8 d 8 0.5 0.1

Footnotes

Acknowledgments

References

Table 1
Decision Support System

Task Deadline Profit Penalty Task Deadline Profit Penalty

t ₁ d ₁ 0.8 0.7 t ₅ d ₅ 0.6 0.8

t ₂ d ₂ 0.5 0.6 t ₆ d ₆ 0.7 0.3

t ₃ d ₃ 0.6 0.8 t ₇ d ₇ 0.8 0.4

t ₄ d ₄ 0.9 0.5 t ₈ d ₈ 0.5 0.1