Complex bipolar fuzzy sets: An application in a transport’s company

Abstract

Zadeh’s fuzzy sets are very useful tool to handle imprecision and uncertainty, but they are unable to characterize the negative characteristics in a certain problem. This problem was solved by Zhang et al. as they introduced the concept of bipolar fuzzy sets. Thus, fuzzy set generalizes the classical set and bipolar fuzzy set generalize the fuzzy set. These theories are based on the set of real numbers. On the other hand, the set of complex numbers is the generalization of the set of real numbers so, complex fuzzy sets generalize the fuzzy sets, with wide range of values to handle the imprecision and uncertainty. So, in this article, we study complex bipolar fuzzy sets which is the generalization of bipolar fuzzy set and complex fuzzy set with wide range of values by adding positive membership function and negative membership function to unit circle in the complex plane, where one can handle vagueness in a much better way as compared to bipolar fuzzy sets. Thus this paper leads us towards a new direction of research, which has many applications in different directions. We develop the notions of union, intersection, complement, Cartesian product and De-Morgan’s laws of complex bipolar fuzzy sets. Furthermore, we develop the complex bipolar fuzzy relations, fundamental operations on complex bipolar fuzzy matrices and some operators of complex bipolar fuzzy matrices. We also discuss the distance measure on complex bipolar fuzzy sets and complex bipolar fuzzy aggregation operators. Finally, we apply the developed approach to a numerical problem with the algorithm.

Keywords

Complex fuzzy sets complex bipolar fuzzy relations complex bipolar fuzzy matrices distance measure

1 Introduction

Fuzzy sets and their extensions: In 1965, Zadeh [1] presented the idea of fuzzy sets. to solve difficulties in dealing with uncertainties. Since then, the theory of fuzzy sets and fuzzy logic have been examined by many researchers to solve many real life problems involving ambiguous and uncertain environment. Atanassov [2, 3], initiated the notion of intuitionistic fuzzy set, which is the generalization of fuzzy set and a more meaningful as well as intensive due to the presence of degree of truth and falsity membership. Later, many researcher have done work on IFSs and add valuable contribution to IFS literature. Smarandache [4] in 1999, define the theme of ņeutrosophic sets, which is the most generalize version of fuzzy sets. The concept of bipolar set was first initiated by Zhang et al. [5, 6], which is the generalization of fuzzy set by adding negative membership term. This is familiar that the scope of each positive membership and negative membership functions are restricted to [-1, 1]. The notion of fuzzy matrix was first introduced by Kim et al. [24]. By the use of IFSs, first time the notion of intuitionistic fuzzy soft matrix space presented by Mondal et al. [7], where he introduced some basic operator on the basis of weights. Also Adak et al. [8] introduced intuitionistic fuzzy block matrix and its determinant. Some operations on IFSs presented by Fathi [9]. For applications in decision making problems we refer the reader [10 –15].

Complex fuzzy sets and their extensions: There are several researchers who worked at complex number and fuzzy sets, for instance Buckly [19], Nguyen et al. [18] and Zhang et al. [20]. On the other hand Ramot et al. [16] presented an presented the idea that is entirely different from other researchers, wherever they extensive the variety of membership function to unit circle in the complex plane. Further to solve enigma they added an extra terms which is called phase term in translating human language to complex valued functions on physical terms and vice versa. Rikarour et al. [17], studied different aspect of complex fuzzy sets. The notion of complex intuitionistic fuzzy sets was introduced by Abdulazeez et al. [21]. The author introduced CIFS lies in its capability for membership and non-membership capacities to get excess scope of qualities. Husban et al. [22] worked on complex intuitionistic fuzzy groups which depends on the notion of complex intuitionistic fuzzy space from the genuine rang of membership function and non membership function [0, 1] . Distance measure in complex Atanassove’s intuitionistic fuzzy sets was given by Abdulazeez et al. [23]. Kim et al. [24], studied generalized fuzzy matrices. More about complex fuzzy sets can be seen in [25 –40]. For more about the complex version of neutrosophic sets, we refer the reader [41 –43].

Complex bipolar fuzzy sets: Being motivated from the concept of complex fuzzy set which is the generalization of fuzzy sets, we extended the idea of bipolar fuzzy sets in the complex plan and introduce the idea of complex bipolar fuzzy sets, which is the generality of complex fuzzy sets and bipolar fuzzy sets. Thus we combine two sets in order to develop a more powerful technique which has the ability to capture the uncertainty in a more better way. We explain union, intersection, complement, Cartesian product and De-Morgan’s Laws of complex bipolar fuzzy sets. We prove some other properties and propositions of complex bipolar fuzzy sets. We explain complex bipolar fuzzy relations, fundamental operations on complex bipolar fuzzy matrices and some operators of complex bipolar fuzzy matrices. Furthermore, we also explain distance Measure on complex bipolar fuzzy sets and complex bipolar fuzzy aggregation. At the end we give a numerical example of distance measure between complex bipolar fuzzy sets.

2 Preliminaries

Here in this section we gathered some basic helping materiel.

Definition 2.1. [1] A function f is defined from a universe $U$ to a closed interval [0, 1] is called fuzzy set.

Definition 2.2. [5] Let B be a bipolar fuzzy set (BFS) over the universe $U$ , is of the form: $B = {〈 m, μ_{B}^{+} (m), ϑ_{B}^{-} (m) 〉 : m \in U},$ where the functions $μ_{B}^{+} (m) : U \to [0, 1]$ grade of positive membership and $ϑ_{B}^{-} (m) : U \to [- 1, 0]$ grade of negative membership of every element $m \in U,$ and for each element $m \in U,$ satisfying the condition, $- 1 \leq μ_{B}^{+} (m) + ϑ_{B}^{-} (m) \leq 1 .$

Definition 2.3. [16] Let C be a complex fuzzy set (CFS) is defined over the universe $U$ , is an object having the form $C = {(m, μ_{C} (m)) : m \in U},$ where μ_C (m) = r_C (m) · e^iw_C(m) where both amplitude and phase terms r_C (m), and w_C (m) are real valued, for every $m \in U,$ the amplitude term $r_{C} (m) : U \to [0, 1]$ and phase term w_C (m) lying in the interval [0, 2π] .

3 Complex bipolar fuzzy sets

In this part first we define S and T norms on bipolar fuzzy sets, by use of [9], after this we introduce the concept of CBFS with example. We also introduce some basic operations on it, such as union, intersection, complement and Cartesian product.

Definition 3.1. Let $S (S_{\overset{+}{μ}}, S_{\bar{ϑ}}) :$ [-1, 0] ² × [0, 1] ² → [-1, 0] × [0, 1] be a function, such that for all å, b̊ ∈ [0, 1] , and å^′, b̊^′ ∈ [-1, 0] , $S ((å, å^{'}), (b̊, {b̊}^{'})) = (S_{\overset{+}{μ}} (å, b̊), S_{\bar{ϑ}} (å^{'}, {b̊}^{'})),$ where $S_{\overset{+}{μ}} : [0, 1]^{2} \to [0, 1]$ and $S_{\bar{ϑ}} : [- 1, 0]^{2} \to [- 1, 0]$ . Then S is called an S-norm if the following axioms are satisfied:

Axiom 1. “Boundary condition”, $S_{\overset{+}{μ}} (å, 0) = å$ and $S_{\bar{ϑ}} (å^{'}, 0) = å^{'} .$

Axiom 2. “Commutative condition”, $S_{\overset{+}{μ}} (å, b̊) =$ $S_{\overset{+}{μ}} (b̊, å)$ and $S_{\bar{ϑ}} (å^{'}, {b̊}^{'}) = S_{\bar{ϑ}} ({b̊}^{'}, å^{'}) .$

Axiom 3. “Non-decreasing condition”, if å ≤ b̊, c̊ ≤ d̊, then $S_{\overset{+}{μ}} (å, c̊) \leq S_{\overset{+}{μ}} (b̊, d̊)$ and if å^′ ≤ b̊^′, c̊^′ ≤ d̊^′, then $S_{\bar{ϑ}} (å^{'}, {c̊}^{'}) \geq S_{\bar{ϑ}} ({b̊}^{'}, {d̊}^{'}) .$

Axiom 4. “Associative condition”, $S_{\overset{+}{μ}} (S_{\overset{+}{μ}} (å, b̊), c̊) =$ $S_{\overset{+}{μ}} (å, S_{\overset{+}{μ}} (b̊, c̊))$ and $S_{\bar{ϑ}} (S_{\bar{ϑ}} (å^{'}, {b̊}^{'}), {c̊}^{'}) = S_{\bar{ϑ}} (å^{'}, S_{\bar{ϑ}} ({b̊}^{'}, {c̊}^{'}) .$

Definition 3.2. Let $T (T_{\overset{+}{μ}}, T_{\bar{ϑ}}) :$ [-1, 0] ² × [0, 1] ² → [-1, 0] × [0, 1] be a function, such that for all å, b̊ ∈ [0, 1] , and å^′, b̊^′ ∈ [-1, 0] , $T ((å, å^{'}), (b̊, {b̊}^{'})) = (T_{\overset{+}{μ}} (å, b̊), T_{\bar{ϑ}} (å^{'}, {b̊}^{'})),$ where $T_{\overset{+}{μ}} : [0, 1]^{2} \to [0, 1]$ and $T_{\bar{ϑ}} : [- 1, 0]^{2} \to [- 1, 0]$ . Then T is called a T-norm if the following axioms are satisfied:

Axiom 1. “Boundary condition”,

$T_{\overset{+}{μ}} (å, 1) = T_{\overset{+}{μ}} (1, å) = å and T_{\bar{ϑ}} (å^{'}, 1) = T_{\bar{ϑ}} (1, å^{'}) = å^{'} .$

Axiom 2. “Commutative condition”, $T_{\overset{+}{μ}} (å, b̊) =$ $T_{\overset{+}{μ}} (b̊, å)$ and $T_{\bar{ϑ}} (å^{'}, {b̊}^{'}) = T_{\bar{ϑ}} ({b̊}^{'}, å^{'}) .$

Axiom 3. “Non-Decreasing condition”, if å ≤ b̊, c̊ ≤ d̊, then $T_{\overset{+}{μ}} (å, c̊) \leq$ $T_{\overset{+}{μ}} (b̊, d̊)$ and if å^′ ≤ b̊^′, c̊^′ ≤ d̊^′, then $T_{\bar{ϑ}} (å^{'}, {c̊}^{'}) \geq T_{\bar{ϑ}} ({b̊}^{'}, {d̊}^{'}) .$

Axiom 4. “Associative condition”, $\begin{matrix} T_{\overset{+}{μ}} (T_{\overset{+}{μ}} (å, b̊), c̊) & = & T_{\overset{+}{μ}} (å, T_{\overset{+}{μ}} (b̊, c̊)) \\ and T_{\bar{ϑ}} (T_{\bar{ϑ}} (å^{'}, {b̊}^{'}), {c̊}^{'}) & = & T_{\bar{ϑ}} (å^{'}, T_{\bar{ϑ}} ({b̊}^{'}, {c̊}^{'})) . \end{matrix}$

Following are some example of S-norm:

1. The Standard S-norm:

For any two BFS’s $B_{1} = {〈 m, μ_{B_{1}}^{+} (m), ϑ_{B_{1}}^{-} (m) 〉 : m \in U}$ and $B_{2} = {〈 m, μ_{B_{2}}^{+} (m), ϑ_{B_{2}}^{-} (m) 〉 : m \in U}$ in a universe of over $U,$ then B₁ ∪ B₂ is given by $B_{1} \cup B_{2} = {\begin{matrix} 〈 m, max (μ_{B_{1}}^{+} (m), μ_{B_{2}}^{+} (m)), \\ min (ϑ_{B_{1}}^{-} (m), ϑ_{B_{2}}^{-} (m)) 〉 : m \in U \end{matrix}} .$ This union is said to be the main BF union, and it is the smallest of the BFS, including the B₁ and B₂.

2. Yager S-norm: $S ((å, å^{'}), (b̊, {b̊}^{'})) = S (S_{w}^{\overset{+}{μ}} (å, b̊), S_{w}^{\bar{ϑ}} (å^{'}, {b̊}^{'}))$ where $S_{w}^{\overset{+}{μ}} (å, b̊) = m \dot{ı} n (1, (å^{w} + {b̊}^{w})^{1 / w}),$ and $\begin{matrix} S_{w}^{\bar{ϑ}} (å^{'}, {b̊}^{'}) \\ = & 1 - m \dot{ı} n (1, ((1 - å^{'})^{w} + (1 - {b̊}^{'})^{w})^{1 / w}), \end{matrix}$ with w ∈ (0, ∞) .

Following are some example of T-norm:

1. The Standard T-norm:

For any two BFS’s $B_{1} = {〈 m, μ_{B_{1}}^{+} (m), ϑ_{B_{1}}^{-} (m) 〉 : m \in U}$ and $B_{2} = {〈 m, μ_{B_{2}}^{+} (m), ϑ_{B_{2}}^{-} (m) 〉 : m \in U}$ in a universe of over $U,$ then B₁ ∩ B₂ is given by $B_{1} \cap B_{2} = {\begin{matrix} 〈 m, min (μ_{B_{1}}^{+} (m), μ_{B_{2}}^{+} (m)), \\ max (ϑ_{B_{1}}^{-} (m), ϑ_{B_{2}}^{-} (m)) 〉 : m \in U \end{matrix}} .$ This intersection is said to be the main BF intersection, and it is the smallest of the BFS, including the B₁ and B₂.

2. Yager T-norm: $T ((å, å^{'}), (b̊, {b̊}^{'})) = T (T_{w}^{\overset{+}{μ}} (å, b̊), T_{w}^{\bar{ϑ}} (å^{'}, {b̊}^{'}))$ where $T_{w}^{\overset{+}{μ}} (å, b̊) = 1 - m \dot{ı} n (1, ((1 - å)^{w} + (1 - b̊)^{w})^{1 / w}),$ and $T_{w}^{\bar{ϑ}} (å^{'}, {b̊}^{'}) = m \dot{ı} n (1, ((å^{'})^{w} + ({b̊}^{'})^{w})^{1 / w}),$ with w ∈ (0, ∞) .

Definition 3.3. A complex bipolar fuzzy set (CBFS) $C$ on a non-empty set over the universe $U,$ is defined as: $C = {\begin{matrix} 〈 m, (\overset{+}{μ_{C}} (m) = r_{C}^{+} (m) \cdot e^{{iw}_{C}^{+} (m)}), \\ (ϑ_{C}^{-} (m) = k_{C}^{-} (m) \cdot e^{i η_{C}^{-} (m)} 〉) : m \in U \end{matrix}},$ where $r_{C}^{+} (m), k_{C}^{-} (m)$ and $w_{C}^{+} (m), η_{C}^{-} (m)$ are real valued, where the amplitude terms $r_{C}^{+} (m) \in [0, 1],$ $k_{C}^{-} (m) \in [- 1, 0],$ for all $m \in U,$ and the phase terms $w_{C}^{+} (m) \in [0, 2 π],$ $η_{C}^{-} (m) \in [- 2 π, 0],$ satisfying the conditions $- 1 \leq r_{C}^{+} (m) + k_{C}^{-} (m) \leq 1$ and $- 2 π \leq w_{C}^{+} (m) + η_{C}^{-} (m) \leq 2 π .$

Example 3.4. Let $U = {m_{1,} m_{2}, m_{3}}$ be the universe set, and complex bipolar fuzzy set $C$ is given by $C = {\begin{matrix} 〈 m_{1}, 0.2 e^{0.5 π i}, - 0.3 e^{- 0.6 π i} 〉, \\ 〈 m_{2}, 0.4 e^{0.6 π i}, - 0.5 e^{- 1.3 π i} 〉, \\ 〈 m_{3}, 0.1 e^{0.6 π i}, - 0.3 e^{- 0.9 π i} 〉 \end{matrix}} .$

3.1 Some basic operations on complex bipolar fuzzy sets

In this section we discuss union, intersection, complement and cartesian product of complex bipolar fuzzy sets with examples.

Definition 3.5. Let $C_{1}$ and $C_{2}$ be two complex bipolar fuzzy sets, then we defined their union as follows: $C_{1} \cup C_{2} = {〈 m, μ_{C_{1} \cup C_{2}}^{+} (m), ϑ_{C_{1} \cup C_{2}}^{-} (m) 〉 : m \in U},$ where $μ_{C_{1} \cup C_{2}}^{+} (m) = [r_{C_{1}}^{+} (m) S_{\overset{+}{μ}} r_{C_{2}}^{+} (m)] \cdot e^{{iw}_{C_{1} \cup C_{2}}^{+} (m)}$ and $ϑ_{C_{1} \cup C_{2}}^{-} (m) = [k_{C_{1}}^{-} (m) S_{\bar{ϑ}} k_{C_{2}}^{-} (m)] \cdot e^{i η_{C_{1} \cup C_{2}}^{-} (m)} .$ The operations $S_{\overset{+}{μ}}$ and $S_{\bar{ϑ}}$ can be effectively denoted by Max and Min operators, respectively, or any S-norm. We call the explanation above with Max and Min operators for positive membership and negative membership amplitude functions, the basic complex bipolar fuzzy union.

Example 3.6. Let $U = {m_{1,} m_{2}, m_{3}}$ and let $C_{1} = {\begin{matrix} 〈 m_{1}, 0.2 e^{0.5 π i}, - 0.3 e^{- 0.6 π i} 〉, \\ 〈 m_{2}, 0.4 e^{0.6 π i}, - 0.5 e^{- 0.4 π i} 〉, \\ 〈 m_{3}, 0.1 e^{0.7 π i}, - 0.3 e^{- 0.8 π i} 〉 \end{matrix}}$ and $C_{2} = {\begin{matrix} 〈 m_{1}, 0.4 e^{0.6 π i}, - 0.5 e^{- 0.3 π i} 〉, \\ 〈 m_{2}, 0.5 e^{0.1 π i}, - 0.4 e^{- 0.2 π i} 〉, \\ 〈 m_{3}, 0.1 e^{0.9 π i}, - 0.6 e^{- 0.8 π i} 〉 \end{matrix}}$ be two complex bipolar fuzzy sets. Then its union is defined as: $C_{1} \cup C_{2} = {\begin{matrix} 〈 \begin{matrix} m_{1}, (0.2 \lor 0.4) e^{(0.5 \lor 0.6) π i}, \\ (- 0.3 \land - 0.5) e^{(- 0.6 \land - 0.3) π i} \end{matrix} 〉, \\ 〈 \begin{matrix} m_{2}, (0.4 \lor 0.5) e^{(0.6 \lor 0.1) π i}, \\ (- 0.5 \land - 0.4) e^{(- 0.4 \land - 0.2) π i} \end{matrix} 〉, \\ 〈 \begin{matrix} m_{3}, (0.1 \lor 0.1) e^{(0.7 \lor 0.9) π i}, \\ (- 0.3 \land - 0.6) e^{(- 0.8 \land - 0.8) π i} \end{matrix} 〉 \end{matrix}}$ $= {\begin{matrix} 〈 m_{1}, 0.4 e^{0.6 π i}, - 0.5 e^{- 0.6 π i} 〉, \\ 〈 m_{2}, 0.5 e^{0.6 π i}, - 0.5 e^{- 0.4 π i} 〉, \\ 〈 m_{3}, 0.1 e^{0.9 π i}, - 0.6 e^{- 0.8 π i} 〉 \end{matrix}} .$

Definition 3.7. Let $C_{1}$ and $C_{2}$ be two CBFSs on over the universe $U$ , with complex-valued positive membership function and negative membership function. The complex bipolar fuzzy union of $C_{1}$ and $C_{2}$ indicated by $C_{1} \cup C_{2},$ is specified by a function, $\begin{matrix} Ξ = & {(å, å^{'}) ∣ å, å^{'} \in ℂ : | å + å^{'} | \leq 1, | å |, | å^{'} | \leq 1} \\ \times {(b̊, {b̊}^{'}) ∣ b̊, {b̊}^{'} \in ℂ : | b̊ + {b̊}^{'} | \leq 1, | b̊ |, | {b̊}^{'} | \leq 1} \\ ⟼ & {(d, d^{'}) ∣ d̊, {d̊}^{'} \in ℂ : | d̊ + {d̊}^{'} | \leq 1, | d̊ |, | {d̊}^{'} | \leq 1}, \end{matrix}$

where å, b̊, d̊ and å^′, b̊^′d̊^′, are the complex positive membership and complex negative membership functions of $C_{1}$ , $C_{2}$ and $C_{1} \cup C_{2},$ respectively. Ξ assign a complex value, $\begin{matrix} Ξ ((å, å^{'}), (b̊, {b̊}^{'})) & = & (Ξ_{\overset{+}{μ}} (å, b̊), Ξ_{\bar{ϑ}} (å^{'}, {b̊}^{'})) \\ = & (μ_{C_{1} \cup C_{2}}^{+} (m), ϑ_{C_{1} \cup C_{2}}^{-} (m)) \\ = & (d̊, {d̊}^{'}), \end{matrix}$ to all m in $U .$

Remark 3.8. The complex bipolar fuzzy union function, Ξ should satisfy the following axiomatic conditions for each å, b̊, c̊, d̊, å^′, b̊^′, c̊^′ and ${d̊}^{'} \in {m ∣ m \in ℂ, | m | \leq 1} .$

Axiom 1 :“Boundary Condition”, $| Ξ_{\overset{+}{μ}} (å, 0) | = | å |$ and $| Ξ_{\bar{ϑ}} (å^{'}, 0) | = | å^{'} |$ .

Axiom 2 :“Commutative Condition”, $Ξ_{\overset{+}{μ}} (å, b̊) = Ξ_{\overset{+}{μ}} (b̊, å)$ and

$Ξ_{\bar{ϑ}} (å^{'}, {b̊}^{'}) = Ξ_{\bar{ϑ}} ({b̊}^{'}, å^{'})$ .

Axiom 3 :“Monotonic Condition” if |b̊| = |d̊| then $| Ξ_{\overset{+}{μ}} (å, b̊) | \leq | Ξ_{\overset{+}{μ}} (å, d̊) |$ and if |b̊^′| = |d̊^′| then $| Ξ_{\bar{ϑ}} (å^{'}, {b̊}^{'}) | \leq | Ξ_{\bar{ϑ}} (å^{'}, {d̊}^{'}) |$ .

Axiom 4 :“Associative Condition”, $Ξ_{\overset{+}{μ}} (Ξ_{\overset{+}{μ}} (å, b̊), c̊) = Ξ_{\overset{+}{μ}} (å, Ξ_{\overset{+}{μ}} (b̊, c̊))$ and

$Ξ_{\bar{ϑ}} (Ξ_{\bar{ϑ}} (å^{'}, {b̊}^{'}), {c̊}^{'}) = Ξ_{\bar{ϑ}} (å^{'}, Ξ_{\bar{ϑ}} ({b̊}^{'}, {c̊}^{'}))$ .

Remark 3.9. In a few cases it can be enviable that the following necessities are also satisfied:

Axiom 5 :“Continuity”, Ξ is a continuous function.

Axiom 6 :“Superidempotency”, $| Ξ_{\overset{+}{μ}} (å, å) | > | å |$ and $| Ξ_{\bar{ϑ}} (å^{'}, å^{'}) | < | å^{'} |$ .

Axiom 7 :“Strict Monotonicity”,|å| ≤ |c̊| and |b̊| ≤ |d̊|

$\Rightarrow | Ξ_{\overset{+}{μ}} (å, b̊) | \leq | Ξ_{\overset{+}{μ}} (c̊, d̊) |,$ also |å^′| ≥ |c̊^′| and $| {b̊}^{'} | \geq | {d̊}^{'} | \Rightarrow | Ξ_{\overset{+}{μ}} (å, b̊) | \leq | Ξ_{\overset{+}{μ}} (c̊, d̊) |$ .

The phase term of complex positive membership function belong to [-2π, 0] , and complex negative membership function belong to [0, 2π] , we define $w_{C_{1} \cup C_{2}}^{+} (m) a n d η_{C_{1} \cup C_{2}}^{-} (m)$ with some forms, that Ramot et al. [16] presented to compute $w_{C_{1} \cup C_{2}}^{+} (m)$ and $η_{C_{1} \cup C_{2}}^{-} (m)$ as follows:

(i) Sum: $w_{C_{1} \cup C_{2}}^{+} (m) = w_{C_{1}}^{+} (m) + w_{C_{2}}^{+} (m)$ and

(ii)Max: $w_{C_{1} \cup C_{2}}^{+} (m) =$ max $(w_{C_{1}}^{+} (m), w_{C_{2}}^{+} (m))$ and

(iii) Min: $w_{C_{1} \cup C_{2}}^{+} (m) =$ min $(w_{C_{1}}^{+} (m), w_{C_{2}}^{+} (m))$ and

(iϑ) ‘Winner Take All’: $w_{C_{1} \cup C_{2}}^{+} (m) = {\begin{matrix} w_{C_{1}}^{+} (m) r_{C_{1}}^{+} > r_{C_{2}}^{+} \\ w_{C_{2}}^{+} (m) r_{C_{1}}^{+} < r_{C_{2}}^{+} \end{matrix},$ and $η_{C_{1} \cup C_{2}}^{-} (m) = {\begin{matrix} η_{C_{1}}^{-} (m) k_{C_{1}}^{-} < k_{C_{2}}^{-} \\ η_{C_{2}}^{-} (m) k_{C_{1}}^{-} > k_{C_{2}}^{-} \end{matrix} .$ We conclude in this section that the conventional union functions given by (see Definition 3.1.) for bipolar fuzzy sets are relevant only to the amplitude terms of complex status of positive membership and negative membership functions (i.e. the properties for complex bipolar fuzzy union function that have to satisfy are the same as properties of function given by Mohammad Fathi, known as S-norm). Thus, we can use any S-norm such as intuitionistic algebraic sum, Yager S-norm and others that were introduced by Mohammad Fathi [9], to show a lot of applications (or examples) on amplitude phrase on complex bipolar fuzzy union. With the positive and negative membership functions of $C_{1} \cup C_{2}$ are known, respectively, by:

$μ_{C_{1} \cup C_{2}}^{+} (m) = [r_{C_{1}}^{+} (m) S_{\overset{+}{μ}} r_{C_{2}}^{+} (m)] \cdot e^{{iw}_{C_{1} \cup C_{2}}^{+} (m)}$

and $ϑ_{C_{1} \cup C_{2}}^{-} (m) = [k_{C_{1}}^{-} (m) S_{\bar{ϑ}} k_{C_{2}}^{-} (m)] \cdot e^{i η_{C_{1} \cup C_{2}}^{-} (m)} .$

Definition 3.10. Let $C_{1}$ and $C_{2}$ be two complex bipolar fuzzy sets, then their intersection as follows: $C_{1} \cap C_{2} = {〈 m, μ_{C_{1} \cap C_{2}}^{+} (m), ϑ_{C_{1} \cap C_{2}}^{-} (m) 〉 : m \in U},$ where $μ_{C_{1} \cap C_{2}}^{+} (m) = [r_{C_{1}}^{+} (m) T_{\overset{+}{μ}} r_{C_{2}}^{+} (m)] \cdot e^{{iw}_{C_{1} \cap C_{2}}^{+} (m)}$ and $ϑ_{C_{1} \cap C_{2}}^{-} (m) = [k_{C_{1}}^{-} (m) T_{\bar{ϑ}} k_{C_{2}}^{-} (m)] \cdot e^{i η_{C_{1} \cap C_{2}}^{-} (m)} .$ The operators $T_{\overset{+}{μ}}$ and $T_{\bar{ϑ}}$ can be effectively denoted by Min and Max operators, respectively or any T-norm operators on the amplitude components. We called the explanation above with Min and Max operators for positive membership and negative membership amplitude functions, the basic complex bipolar fuzzy intersection.

Example 3.11. Let $U = {m_{1,} m_{2}, m_{3}}$ and let $C_{1} = {\begin{matrix} 〈 m_{1}, 0.3 e^{0.4 π i}, - 0.2 e^{- 0.1 π i} 〉, \\ 〈 m_{2}, 0.6 e^{0.3 π i}, - 0.7 e^{- 0.2 π i} 〉, \\ 〈 m_{3}, 0.9 e^{0.3 π i}, - 0.1 e^{- 0.4 π i} 〉 \end{matrix}}$ and $C_{2} = {\begin{matrix} 〈 m_{1}, 0.9 e^{0.5 π i}, - 0.1 e^{- 0.8 π i} 〉, \\ 〈 m_{2}, 0.3 e^{0.7 π i}, - 0.3 e^{- 0.5 π i} 〉, \\ 〈 m_{3}, 0.6 e^{0.8 π i}, - 0.7 e^{- 0.1 π i} 〉 \end{matrix}}$ be two complex bipolar fuzzy sets. Then its intersection is defined as: $C_{1} \cap C_{2} = {\begin{matrix} 〈 \begin{matrix} m_{1}, (0.3 \land 0.9) e^{(0.4 \land 0.5) π i}, \\ (- 0.2 \lor - 0.1) e^{(- 0.1 \lor - 0.8) π i} \end{matrix} 〉, \\ 〈 \begin{matrix} m_{2}, (0.6 \land 0.3) e^{(0.3 \land 0.7) π i}, \\ (- 0.7 \lor - 0.3) e^{(- 0.2 \lor - 0.5) π i} \end{matrix} 〉, \\ 〈 \begin{matrix} m_{3}, (0.9 \land 0.6) e^{(0.3 \land 0.8) π i}, \\ (- 0.1 \lor - 0.7) e^{(- 0.4 \lor - 0.1) π i} \end{matrix} 〉 \end{matrix}}$

$= {\begin{matrix} 〈 m_{1}, 0.3 e^{0.4 π i}, - 0.1 e^{- 0.1 π i} 〉, \\ 〈 m_{2}, 0.3 e^{0.3 π i}, - 0.3 e^{- 0.2 π i} 〉, \\ 〈 m_{3}, 0.6 e^{0.3 π i}, - 0.1 e^{- 0.1 π i} 〉 \end{matrix}} .$

Definition 3.12. Let $C_{1}$ and $C_{2}$ be two CBFSs on over the universe $U$ , with complex-valued positive membership function and negative membership function. The complex bipolar fuzzy intersection of $C_{1}$ and $C_{2}$ , indicated by $C_{1} \cap C_{2}$ , is specified by a function, $\begin{matrix} Θ = & {(å, å^{'}) ∣ å, å^{'} \in ℂ : | å + å^{'} | \leq 1, | å |, | å^{'} | \leq 1} \\ \times {(b̊, {b̊}^{'}) ∣ b̊, {b̊}^{'} \in ℂ : | b̊ + {b̊}^{'} | \leq 1, | b̊ |, | {b̊}^{'} | \leq 1} \\ ⟼ & {(d̊, {d̊}^{'}) ∣ d̊, {d̊}^{'} \in ℂ : | d̊ + {d̊}^{'} | \leq 1, | d̊ |, | {d̊}^{'} | \leq 1}, \end{matrix}$

where å, b̊, d̊ and å^′, b̊^′, d̊^′ are the complex positive membership and complex negative membership functions of $C_{1}$ , $C_{2}$ and $C_{1} \cap C_{2}$ respectively. Θ assign a complex value, $\begin{matrix} Θ ((å, å^{'}), (b̊, {b̊}^{'})) & = & (Θ_{\overset{+}{μ}} (å, b̊), Θ_{\bar{ϑ}} (å^{'}, {b̊}^{'})) \\ = & (μ_{C_{1} \cap C_{2}}^{+} (m), ϑ_{C_{1} \cap C_{2}}^{-} (m)) \\ = & (d̊, {d̊}^{'}), \end{matrix}$ to all m in $U .$

Remark 3.13. The complex bipolar fuzzy intersection function, Θ should satisfy the following axiomatic conditions, for each å, b̊, c̊, d̊, å^′, b̊^′, c̊^′ and ${d̊}^{'} \in {m ∣ m \in ℂ, | m | \leq 1} .$

Axiom 1 :“Boundary Condition”, if |b̊|=1, $| Θ_{\overset{+}{μ}} (å, b̊) | = | å | and if | {b̊}^{'} | = 0, | Θ_{\bar{ϑ}} (å^{'}, {b̊}^{'}) | = | å^{'} | .$

Axiom 2 :“Commutative Condition”, $Θ_{\overset{+}{μ}} (å, b̊) = Θ_{\overset{+}{μ}} (b̊, å)$ and $Θ_{\bar{ϑ}} (å^{'}, {b̊}^{'}) = Θ_{\bar{ϑ}} ({b̊}^{'}, å^{'})$ .

Axiom 3 :“Monotonic Condition”, if |b̊| ≤ |d̊|, then $| Θ_{\overset{+}{μ}} (å, b̊) | \leq | Θ_{\overset{+}{μ}} (å, d̊) |$ and if |b̊^′| ≤ |d̊^′| then $| Θ_{\bar{ϑ}} (å^{'}, {b̊}^{'}) | \leq | Θ_{\bar{ϑ}} (å^{'}, {d̊}^{'}) |$ .

Axiom 4 :“Associative Condition”, $Θ_{\overset{+}{μ}} (Θ_{\overset{+}{μ}} (å, b̊), c̊) = Θ_{\overset{+}{μ}} (å, Θ_{\overset{+}{μ}} (b̊, c̊))$ and

$Θ_{\bar{ϑ}} (Θ_{\bar{ϑ}} (å^{'}, {b̊}^{'}), {c̊}^{'}) = Θ_{\bar{ϑ}} (å^{'}, Θ_{\bar{ϑ}} ({b̊}^{'}, {c̊}^{'}))$ .

Remark 3.4. In a few cases it can be enviable that the following necessities are also satisfied:

Axiom 5 :“Continuity”, Θ is a continuous function.

Axiom 6 :“Superidempotency”, $| Θ_{\overset{+}{μ}} (å, å) | < | å |$ and $| Θ_{\bar{ϑ}} (å^{'}, å^{'}) | > | å^{'} |$ .

Axiom 7: $“ StrictMonotonicity ”, | å | \leq | c̊ | and | b̊ | \leq | d̊ | \Rightarrow | Θ_{\overset{+}{μ}} (å, b̊) | \leq | Θ_{\overset{+}{μ}} (c̊, d̊) |, also | å^{'} | \geq | {c̊}^{'} |$ and

$| {b̊}^{'} | \geq | {d̊}^{'} | \Rightarrow | Θ_{\bar{ϑ}} (å^{'}, {b̊}^{'}) | \geq | Θ_{\bar{ϑ}} ({c̊}^{'}, {d̊}^{'}) |$ .

Definition 3.15. Let $C = {〈 m, r_{C}^{+} (m) \cdot e^{{iw}_{C}^{+} (m)}, k_{C}^{-} (m) \cdot e^{i η_{C}^{-} (m)} 〉 : m \in U}$ be a complex bipolar fuzzy set. The complement of $C$ is defined as; $\begin{matrix} c (C) & = {〈 m, r_{\overline{C}}^{+} (m) \cdot e^{{iw}_{\overline{C}}^{+} (m)}, k_{\overline{C}}^{-} (m) \cdot e^{i η_{\overline{C}}^{-} (m)} 〉} \\ = {〈 \begin{matrix} m, (1 - r_{C}^{+} (m)) \cdot e^{(2 π - w_{C}^{+} (m)) i}, \\ (- 1 - k_{C}^{-} (m)) \cdot e^{(- 2 π - η_{C}^{-} (m)) i} \end{matrix} 〉} . \end{matrix}$

Example 3.16. Let $U = {m_{1,} m_{2}, m_{3}}$ and let $C = {\begin{matrix} 〈 m_{1}, 0.8 e^{0.7 π i}, - 0.6 e^{- 0.5 π i} 〉, \\ 〈 m_{2}, 0.3 e^{0.4 π i}, - 0.5 e^{- 0.2 π i} 〉, \\ 〈 m_{3}, 0.5 e^{0.3 π i}, - 0.1 e^{- 0.4 π i} 〉 \end{matrix}}$ be a complex bipolar fuzzy set. Then its complement is defined as: $\begin{matrix} c (C) & = & {\begin{matrix} 〈 \begin{matrix} m_{1}, (1 - 0.8) e^{(2 π - 0.7 π) i}, \\ (- 1 - (- 0.6)) e^{(- 2 π - (- 0.5 π)) i} \end{matrix} 〉, \\ 〈 \begin{matrix} m_{2}, (1 - 0.3) e^{(2 π - 0.4 π) i}, \\ (- 1 - (- 0.5)) e^{(- 2 π - (- 0.2 π)) i} \end{matrix} 〉, \\ 〈 \begin{matrix} m_{3}, (1 - 0.5) e^{(2 π - 0.3 π) i}, \\ (- 1 - (- 0.1)) e^{(- 2 π - (- 0.4 π)) i} \end{matrix} 〉 \end{matrix}} \\ = & {\begin{matrix} 〈 m_{1}, 0.2 e^{1.3 π i}, - 0.4 e^{- 1.5 π i} 〉, \\ 〈 m_{2}, 0.7 e^{1.6 π i}, - 0.5 e^{- 1.8 π i} 〉, \\ 〈 m_{3}, 0.5 e^{1.7 π i}, - 0.9 e^{- 1.6 π i} 〉 \end{matrix}} . \end{matrix}$

Definition 3.17. Let $C_{1}, C_{2}, . . ., C_{n}$ be complex bipolar fuzzy sets are non-empty in $U_{1}, U_{2}, . . ., U_{n}$ , respectively. The cartesian product $C_{1} \times C_{2} \times . . . \times C_{n}$ is a complex bipolar fuzzy set is defined by,

$C_{1} \times C_{2} \times . . . \times C_{n}$ $= {\begin{matrix} 〈 \begin{matrix} (m_{1}, m_{2}, . . ., m_{n}), μ_{C_{1} \times C_{2} \times . . . \times C_{n}}^{+} (m_{1}, m_{2}, . . ., m_{n}), \\ ϑ_{C_{1} \times C_{2} \times . . . \times C_{n}}^{-} (m_{1}, m_{2}, . . ., m_{n}) \end{matrix} 〉 \\ : m_{1} \times m_{2} \times . . . \times m_{n} \in U_{1} \times U_{2} \times . . . \times U_{n} \end{matrix}}$

where $μ_{C_{1} \times C_{2} \times . . . \times C_{n}}^{+} (m_{1}, m_{2}, . . ., m_{n})$ $= [min (r_{C_{1}}^{+} (m_{1}), . ., r_{C_{n}}^{+} (m_{n})) \cdot e^{i min (w_{C_{1}}^{+} (m_{1}), . . ., w_{C_{n}}^{+} (m_{n}))}]$ and $ϑ_{C_{1} \times C_{2} \times . . . \times C_{n}}^{-} (m_{1}, m_{2}, . . ., m_{n})$ $= [max (k_{C_{1}}^{-} (m_{1}), . ., k_{C_{n}}^{-} (m_{n})) \cdot e^{i max (η_{C_{1}}^{-} (m_{1}), . . ., η_{C_{n}}^{-} (m_{n}))}] .$

Remark 3.18. So in case of two complex bipolar fuzzy sets $C_{1}$ and $C_{2}$ we have $\begin{matrix} C_{1} \times C_{2} \\ = {〈 \begin{matrix} (m, n), μ_{C_{1} \times C_{2}}^{+} (m, n), \\ ϑ_{C_{1} \times C_{2}}^{-} (m, n) \end{matrix} 〉 : (m, n) \in M \times N} \end{matrix}$ where $\begin{matrix} μ_{C_{1} \times C_{2}}^{+} (m, n) \\ = & min (r_{C_{1}}^{+} (m), r_{C_{2}}^{+} (n)) \cdot e^{i min (w_{C_{1}}^{+} (m), w_{C_{2}}^{+} (n))} \end{matrix}$ and $\begin{matrix} ϑ_{C_{1} \times C_{2}}^{-} (m, n) \\ = & max (k_{C_{1}}^{-} (m), k_{C_{2}}^{-} (n)) \cdot e^{i max (η_{C_{1}}^{-} (m), η_{C_{2}}^{-} (n))} . \end{matrix}$

4 Properties of complex bipolar fuzzy sets

Here we begin with some basic Algebraic properties of CBFS’s. Also we define some propositions on CBFS’s, which illustrate the relationship between the set theoretic operations in above section.

Definition 4.1. Let

$C_{1} = {〈 m, r_{C_{1}}^{+} (m) \cdot e^{{iw}_{C_{1}}^{+} (m)}, k_{C_{1}}^{-} (m) ϑ e^{i η_{C_{1}}^{-} (m)} 〉 : m \in U}$

and

$C_{2} = {〈 m, r_{C_{2}}^{+} (m) \cdot e^{{iw}_{C_{2}}^{+} (m)}, k_{C_{2}}^{-} (m) \cdot e^{i η_{C_{2}}^{-} (m)} 〉 : m \in U}$

be two CBFSs in $U$ . Then the following operations on complex bipolar fuzzy sets is define as;

1. $C_{1} \oplus C_{2} = 〈 \begin{matrix} r_{C_{1}}^{+} (m) \cdot e^{{iw}_{C_{1}}^{+} (m)} + r_{C_{2}}^{+} (m) \cdot e^{{iw}_{C_{2}}^{+} (m)} \\ - (r_{C_{1}}^{+} (m) \cdot r_{C_{2}}^{+} (m)) \cdot e^{{iw}_{C_{1}}^{+} (m) \cdot w_{C_{2}}^{+} (m)}, \\ - | k_{C_{1}}^{-} (m) \cdot e^{i η_{C_{1}}^{-} (m)} | \cdot | k_{C_{2}}^{-} (m) \cdot e^{i η_{C_{2}}^{-} (m)} | \end{matrix} 〉$

2. $C_{1} \otimes C_{2} = 〈 \begin{matrix} (r_{C_{1}}^{+} (m) \cdot r_{C_{2}}^{+} (m)) \cdot e^{{iw}_{C_{1}}^{+} (m) \cdot w_{C_{2}}^{+} (m)}, \\ k_{C_{1}}^{-} (m) \cdot e^{i η_{C_{1}}^{-} (m)} \\ + k_{C_{2}}^{-} (m) \cdot e^{i η_{C_{2}}^{-} (m)} \\ - (k_{C_{1}}^{-} (m) \cdot k_{C_{2}}^{-} (m)) \cdot e^{i η_{C_{1}}^{-} (m) \cdot η_{C_{2}}^{-} (m)} \end{matrix} 〉$

3. $C_{1} - C_{2} = 〈 \begin{matrix} \min (\begin{matrix} r_{C_{1}}^{+} (m) \cdot e^{{iw}_{C_{1}}^{+} (m)}, \\ r_{C_{2}}^{+} (m) \cdot e^{{iw}_{C_{2}}^{+} (m)} \end{matrix}), \\ \max (k_{C_{1}}^{-} (m) \cdot e^{i η_{C_{1}}^{-} (m)}, \\ k_{C_{2}}^{-} (m) \cdot e^{i η_{C_{2}}^{-} (m)}) \end{matrix} 〉 .$

Definition 4.2. Let

$C_{1} = {\begin{matrix} 〈 m, r_{C_{1}}^{+} (m) \cdot e^{{iw}_{C_{1}}^{+} (m)}, \\ k_{C_{1}}^{-} (m) \cdot e^{i η_{C_{1}}^{-} (m)} 〉 : m \in U \end{matrix}}$ and

$C_{2} = {\begin{matrix} 〈 m, r_{C_{2}}^{+} (m) \cdot e^{{iw}_{C_{2}}^{+} (m)}, \\ k_{C_{2}}^{-} (m) \cdot e^{i η_{C_{2}}^{-} (m)} 〉 : m \in U \end{matrix}}$ be two CBFSs in $U$ , λ > 0. Then,

1. $C_{1} \oplus C_{2} = 〈 r_{C_{1}}^{+} (m) \cdot e^{i w_{C_{1}}^{+} (m)} + r_{C_{2}}^{+} (m) \cdot e^{i w_{C_{2}}^{+} (m)} - (r_{C_{1}}^{+} (m) \cdot r_{C_{2}}^{+} (m)) \cdot e^{i w_{C_{1}}^{+} (m) \cdot w_{C_{2}}^{+} (m)}, - | k_{C_{1}}^{-} (m) \cdot e^{i η_{C_{1}}^{-} (m)} | \cdot | k_{C_{2}}^{-} (m) \cdot e^{i η_{C_{2}}^{-} (m)} | 〉$

2. $C_{1} \otimes C_{2} = 〈 \begin{matrix} (r_{C_{1}}^{+} (m) \cdot r_{C_{2}}^{+} (m)) \cdot e^{{iw}_{C_{1}}^{+} (m) \cdot w_{C_{2}}^{+} (m)}, k_{C_{1}}^{-} (m) \cdot e^{i η_{C_{1}}^{-} (m)} \\ + k_{C_{2}}^{-} (m) \cdot e^{i η_{C_{2}}^{-} (m)} - (k_{C_{1}}^{-} (m) \cdot k_{C_{2}}^{-} (m)) \cdot e^{i η_{C_{1}}^{-} (m) \cdot η_{C_{2}}^{-} (m)} \end{matrix} 〉$

3. $λ C_{2} = 〈 1 - (1 - r^{+} (m) \cdot e^{{iw}^{+} (m)})^{λ}, - | k^{-} (m) \cdot e^{i η^{-} (m)} |^{λ} 〉$

4. $C_{2} = 〈 (r^{+} (m) \cdot e^{{iw}^{+} (m)})^{λ}, - 1 + | - 1 + k^{-} (m) \cdot e^{i η^{-} (m)} |^{λ} 〉 .$

Proposition 4.3. Let $C_{1}$ , $C_{2}$ and $C_{3}$ be any three CBFSs over $U$ . Then the following holds:

(i) $C_{1} \cup C_{1} = C_{1},$

(ii) $C_{1} \cap C_{1} = C_{1},$

(iii) $C_{1} \cup C_{2} = C_{2} \cup C_{1},$

(iv) $C_{1} \cap C_{2} = C_{2} \cap C_{1},$

(v) $C_{1} \cup (C_{2} \cup C_{3}) = (C_{1} \cup C_{2}) \cup C_{3},$

(vi) $C_{1} \cap (C_{2} \cap C_{3}) = (C_{1} \cap C_{2}) \cap C_{3},$

(vii) $C_{1} \cup (C_{2} \cap C_{3}) = (C_{1} \cup C_{2}) \cap (C_{1} \cup C_{3}),$

(viii) $C_{1} \cap (C_{2} \cup C_{3}) = (C_{1} \cap C_{2}) \cup (C_{1} \cap C_{3}) .$

Proposition 4.4. Let $C_{1}$ and $C_{2}$ be CBFSs over universe of discourse $U$ . Then

(i) $(C_{1})^{c} = C_{1},$

(ii) $(C_{1} \cup C_{2})^{c} = C_{1}^{c} \cap C_{2}^{c},$

(iii) $(C_{1} \cap C_{2})^{c} = C_{1}^{c} \cup C_{2}^{c} .$

► Let us see the following example to explain the Demorgan’s Law.

Example 4.5. Let $U = {m_{1,} m_{2}}$ and let $C_{1} = {\begin{matrix} 〈 m_{1}, 0.8 e^{0.3 π i}, - 0.5 e^{- 0.2 π i} 〉, \\ 〈 m_{2}, 0.5 e^{0.4 π i}, - 0.3 e^{- 0.7 π i} 〉 \end{matrix}}$ and $C_{2} = {\begin{matrix} 〈 m_{1}, 0.7 e^{0.6 π i}, - 0.4 e^{- 0.1 π i} 〉, \\ 〈 m_{2}, 0.2 e^{0.3 π i}, - 0.5 e^{- 0.6 π i} 〉 \end{matrix}}$ be two complex bipolar fuzzy sets. Consider $\begin{matrix} C_{1} \cap C_{2} & = {\begin{matrix} 〈 \begin{matrix} m_{1}, (0.8 \land 0.7) e^{(0.3 \land 0.6) π i}, \\ (- 0.5 \lor - 0.4) e^{(- 0.2 \lor - 0.1) π i} \end{matrix} 〉, \\ 〈 \begin{matrix} m_{2}, (0.5 \land 0.2) e^{(0.4 \land 0.3) π i}, \\ (- 0.3 \lor - 0.5) e^{(- 0.7 \lor - 0.6) π i} \end{matrix} 〉 \end{matrix}} \\ = {\begin{matrix} 〈 m_{1}, 0.7 e^{0.3 π i}, - 0.4 e^{- 0.1 π i} 〉, \\ 〈 m_{2}, 0.2 e^{0.3 π i}, - 0.3 e^{- 0.6 π i} 〉 \end{matrix}} \end{matrix}$

$\begin{matrix} {(C_{1} \cap C_{2})}^{c} & = {\begin{matrix} 〈 \begin{matrix} m_{1}, (1 - 0.7) e^{(2 π - 0.3 π) i}, \\ (- 1 - (- 0.4)) e^{(- 2 π - (- 0.1 π)) i} \end{matrix} 〉, \\ 〈 \begin{matrix} m_{2}, (1 - 0.2) e^{(2 π - 0.3 π) i}, \\ (- 1 - (- 0.3)) e^{(- 2 π - (- 0.6 π)) i} \end{matrix} 〉 \end{matrix}} \\ = {\begin{matrix} 〈 m_{1}, 0.3 e^{1.7 π i}, - 0.6 e^{- 1.9 π i} 〉, \\ 〈 m_{2}, 0.8 e^{1.7 π i}, - 0.7 e^{- 1.4 π i} 〉 \end{matrix}} ⟶ (1) \end{matrix}$

Now $\begin{matrix} C_{1}^{c} & = & {\begin{matrix} 〈 \begin{matrix} m_{1}, (1 - 0.8) e^{(2 π - 0.3 π) i}, \\ (- 1 - (- 0.5)) e^{(- 2 π - (- 0.2 π)) i} \end{matrix} 〉, \\ 〈 \begin{matrix} m_{2}, (1 - 0.5) e^{(2 π - 0.4 π) i}, \\ (- 1 - (- 0.3)) e^{(- 2 π - (- 0.7 π)) i} \end{matrix} 〉 \end{matrix}} \\ = & {\begin{matrix} 〈 m_{1}, 0.2 e^{1.7 π i}, - 0.5 e^{- 1.8 π i} 〉, \\ 〈 m_{2}, 0.5 e^{1.6 π i}, - 0.7 e^{- 1.3 π i} 〉 \end{matrix}} \end{matrix}$ $\begin{matrix} C_{2}^{c} & = & {\begin{matrix} 〈 \begin{matrix} m_{1}, (1 - 0.7) e^{(2 π - 0.6 π) i}, \\ (- 1 - (- 0.4)) e^{(- 2 π - (- 0.1 π)) i} \end{matrix} 〉, \\ 〈 \begin{matrix} m_{2}, (1 - 0.2) e^{(2 π - 0.3 π) i}, \\ (- 1 - (- 0.5)) e^{(- 2 π - (- 0.6 π)) i} \end{matrix} 〉 \end{matrix}} \\ = & {\begin{matrix} 〈 m_{1}, 0.3 e^{1.4 π i}, - 0.6 e^{- 1.9 π i} 〉, \\ 〈 m_{2}, 0.8 e^{1.7 π i}, - 0.5 e^{- 1.4 π i} 〉 \end{matrix}} \end{matrix}$ $\begin{matrix} C_{1}^{c} \cup C_{2}^{c} & = {\begin{matrix} 〈 \begin{matrix} m_{1}, (0.2 \lor 0.3) e^{(1.7 \lor 1.4) π i}, \\ (- 0.5 \land - 0.6) e^{(- 1.8 \land - 1.9) π i} \end{matrix} 〉, \\ 〈 \begin{matrix} m_{2}, (0.5 \lor 0.8) e^{(1.6 \lor 1.7) π i}, \\ (- 0.7 \land - 0.5) e^{(- 1.3 \land - 1.4) π i} \end{matrix} 〉 \end{matrix}} \\ = {\begin{matrix} 〈 m_{1}, 0.3 e^{1.7 π i}, - 0.6 e^{- 1.9 π i} 〉, \\ 〈 m_{2}, 0.8 e^{1.7 π i}, - 0.7 e^{- 1.4 π i} 〉 \end{matrix}} ⟶ (2) \end{matrix}$

From (1) and (2), we get that ${(C_{1} \cap C_{2})}^{c} = C_{1}^{c} \cup C_{2}^{c} .$ In similar way we can prove that $(C_{1} \cup C_{2})^{c} = C_{1}^{c} \cap C_{2}^{c} .$

5 Complex bipolar fuzzy matrices

In this section we define complex bipolar fuzzy relations and complex bipolar fuzzy matrices with examples and basic operations.

Definition 5.1. Let $C_{1}$ and $C_{2}$ be two complex bipolar fuzzy sets, then we define the complex bipolar fuzzy relation as, $R (C_{1}, C_{2}) = {\begin{matrix} ((m, n), r_{R}^{+} (m, n) \cdot e^{{iw}_{R}^{+} (m, n)}, \\ k_{R}^{-} (m, n) \cdot e^{i η_{R}^{-} (m, n)}) / (m, n) \in C_{1} \times C_{2}, \\ r_{R}^{+} (m, n) \in [0, 1], w_{R}^{+} (m, n) \in [0, 2 π], \\ k_{R}^{-} (m, n) \in [- 1, 0], η_{R}^{-} (m, n) \in [- 2 π, 0] \end{matrix}} .$

Example 5.2. Consider the complex bipolar fuzzy relation which consists of finite number of ordered pairs,

$R (C_{1}, C_{2}) = {\begin{matrix} ((m_{1}, n_{1}), 0.2 e^{0.3 π i}, - 0.3 e^{- 0.1 π i}), \\ ((m_{1}, n_{2}), 0.4 e^{0.5 π i}, - 0.5 e^{- 0.3 π i}), \\ ((m_{1}, n_{3}), 0.5 e^{0.1 i}, - 0.7 e^{- 0.4 π i}), \\ ((m_{2}, n_{1}), 0.1 e^{0.4 π i}, - 0.3 e^{- 0.5 π i}), \\ ((m_{2}, n_{2}), 0.4 e^{0.5 π i}, - 0.6 e^{- 0.6 π i}), \\ ((m_{2}, n_{3}), 0.7 e^{0.4 π i}, - 0.5 e^{- 0.2 π i}), \\ ((m_{3}, n_{1}), 0.8 e^{0.1 π i}, - 0.7 e^{- 0.9 π i}), \\ ((m_{3}, n_{2}), 0.5 e^{0.4 π i}, - 0.6 e^{- 0.1 π i}), \\ ((m_{3}, n_{3}), 0.1 e^{0.5 π i}, - 0.4 e^{- 0.9 π i}) \end{matrix}} .$

Example 5.3. complex bipolar fuzzy matrix is defined as follows.

If $a_{ij} = (r_{R}^{+} (m, n) . e^{{iw}_{R}^{+} (m, n)}, k_{R}^{-} (m, n) . e^{i η_{R}^{-} (m, n)})),$ we define the matrix $[a_{ij}]_{\tilde{m} \times \tilde{n}} = [\begin{matrix} a_{11} & a_{12} & . & . & . & a_{1 \tilde{n}} \\ a_{21} & a_{22} & . & . & . & a_{2 \tilde{n}} \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ a_{\tilde{m} 1} & a_{\tilde{m} 2} & . & . & . & a_{\tilde{m} \tilde{n}} \end{matrix}]$ which is an $\tilde{m} \times \tilde{n}$ complex bipolar fuzzy matrix (CBFM) of the complex bipolar fuzzy set $C_{1}$ and $C_{2}$ over $U .$ Thus we can say that any complex bipolar set is uniquely characterized by the complex bipolar fuzzy matrix (CBFM) and conversely. The set of all $\tilde{m} \times \tilde{n}$ complex bipolar fuzzy matrices indicated by ${CBFM}_{\tilde{m} \times \tilde{n}} .$

5.1 Operations on complex bipolar fuzzy matrices

Here we give some operations of complex bipolar fuzzy matrices.

Definition 5.4. Let $C_{1} = [〈 \begin{matrix} r_{ij C_{1}}^{+} (m, n) \cdot e^{{iw}_{ij C_{1}}^{+} (m, n)}, \\ k_{ij C_{1}}^{-} (m, n) \cdot e^{i η_{ij C_{1}}^{-} (m, n)} \end{matrix} 〉]$ and $C_{2} = [〈 \begin{matrix} r_{ij C_{2}}^{+} (m, n) \cdot e^{{iw}_{ij C_{2}}^{+} (m, n)}, \\ k_{ij C_{2}}^{-} (m, n) \cdot e^{i η_{ij C_{2}}^{-} (m, n)} \end{matrix} 〉],$ be two complex bipolar fuzzy matrices, which is the subsets of universal set $U .$ Then the basic operation are defined is follows as,

1. Addition: $\begin{matrix} C_{1} + C_{2} & = [〈 \begin{matrix} max {r_{ij C_{1}}^{+} (m, n) \cdot e^{{iw}_{ij C_{1}}^{+} (m, n)}, \\ r_{ij C_{2}}^{+} (m, n) \cdot e^{{iw}_{ij C_{2}}^{+} (m, n)}}, \\ min {k_{ij C_{1}}^{-} (m, n) \cdot e^{i η_{ij C_{1}}^{-} (m, n)}, \\ k_{ij C_{2}}^{-} (m, n) \cdot e^{i η_{ij C_{2}}^{-} (m, n)}} \end{matrix} 〉] \\ = [〈 \begin{matrix} (r_{ij C_{1}}^{+} \lor r_{ij C_{2}}^{+}) (m, n) \cdot e^{i (w_{ij C_{1}}^{+} \lor w_{ij C_{2}}^{+}) (m, n)}, \\ (k_{ij C_{1}}^{-} \land k_{ij C_{2}}^{-}) (m, n) \cdot e^{i (η_{ij C_{1}}^{-} \land η_{ij C_{2}}^{-}) (m, n)} \end{matrix} 〉] . \end{matrix}$

2. Subtraction: $C_{1} - C_{2} = [〈 \begin{matrix} (r_{ij C_{1}}^{+} - r_{ij C_{2}}^{+}) (m, n) \cdot e^{i (w_{ij C_{1}}^{+} - w_{ij C_{2}}^{+}) (m, n)}, \\ (k_{ij C_{1}}^{-} - k_{ij C_{2}}^{-}) (m, n) \cdot e^{i (η_{ij C_{1}}^{-} - η_{ij C_{2}}^{-}) (m, n)} \end{matrix} 〉]$

where $r_{i j C_{1}}^{+} - r_{i j C_{2}}^{+} = {\begin{matrix} r_{i j C_{1}}^{+} r_{ij C_{1}}^{+} \geq r_{ij C_{2}}^{+} \\ 0 elsewhere \end{matrix}$ and $k_{i j C_{1}}^{-} - k_{i j C_{2}}^{-} = {\begin{matrix} k_{i j C_{1}}^{-} k_{ij C_{1}}^{-} < k_{ij C_{2}}^{-} \\ 0 otherwise \end{matrix} .$

3. Componentwise Multiplication: $\begin{matrix} C_{1} \circ C_{2} & = [\begin{matrix} min {r_{ij C_{1}}^{+} (m, n) \cdot e^{{iw}_{ij C_{1}}^{+} (m, n)}, \\ r_{ij C_{2}}^{+} (m, n) \cdot e^{{iw}_{ij C_{2}}^{+} (m, n)}}, \\ max {k_{ij C_{1}}^{-} (m, n) \cdot e^{i η_{ij C_{1}}^{-} (m, n)}, \\ k_{ij C_{2}}^{-} (m, n) \cdot e^{i η_{ij C_{2}}^{-} (m, n)}} \end{matrix}] \\ = [〈 \begin{matrix} (r_{ij C_{1}}^{+} \land r_{ij C_{2}}^{+}) (m, n) \cdot e^{i (w_{ij C_{1}}^{+} \land w_{ij C_{2}}^{+}) (m, n)}, \\ (k_{ij C_{1}}^{-} \lor k_{ij C_{2}}^{-}) (m, n) \cdot e^{i (η_{ij C_{1}}^{-} \lor η_{ij C_{2}}^{-}) (m, n)} \end{matrix} 〉] . \end{matrix}$

4. Product: Let $C_{1}$ and $C_{2}$ be two CBFMs of order $\tilde{m} \times \tilde{n}$ and $\tilde{n} \times \tilde{t}$ . Then the matrix product $C_{1} C_{2}$ is follows; $C_{1} C_{2} = [〈 \begin{matrix} \sum_{p} min {r_{ip C_{1}}^{+} (m, n) \cdot e^{{iw}_{ip C_{1}}^{+} (m, n)}, \\ r_{pj C_{2}}^{+} (m, n) \cdot e^{{iw}_{pj C_{2}}^{+} (m, n)}}, \\ \prod_{p} max {k_{ip C_{1}}^{-} (m, n) \cdot e^{i η_{ip C_{1}}^{-} (m, n)}, \\ k_{pj C_{2}}^{-} (m, n) \cdot e^{i η_{pj C_{2}}^{-} (m, n)}} \end{matrix} 〉] \in F_{\tilde{m} \times \tilde{t}} .$

5. Scalar Multiplication: Let κ be a scalar, where 0 ≤ κ ≤ 1, $κ C_{1} = [〈 \begin{matrix} κ (r_{ij C_{1}}^{+} (m, n) \cdot e^{{iw}_{ij C_{1}}^{+} (m, n)}), \\ κ (k_{ij C_{1}}^{-} (m, n) \cdot e^{i η_{ij C_{1}}^{-} (m, n)}) \end{matrix} 〉] for all i and j .$

6. Union: $\begin{matrix} C_{1} \cup C_{2} & = [〈 \begin{matrix} max {r_{ij C_{1}}^{+} (m, n) \cdot e^{{iw}_{ij C_{1}}^{+} (m, n)}, \\ r_{ij C_{2}}^{+} (m, n) \cdot e^{{iw}_{ij C_{2}}^{+} (m, n)}}, \\ min {k_{ij C_{1}}^{-} (m, n) \cdot e^{i η_{ij C_{1}}^{-} (m, n)}, \\ k_{ij C_{2}}^{-} (m, n) \cdot e^{i η_{ij C_{2}}^{-} (m, n)}} \end{matrix} 〉] \\ = [〈 \begin{matrix} (r_{ij C_{1}}^{+} \lor r_{ij C_{2}}^{+}) (m, n) \cdot e^{i (w_{ij C_{1}}^{+} \lor w_{ij C_{2}}^{+}) (m, n)}, \\ (k_{ij C_{1}}^{-} \land k_{ij C_{2}}^{-}) (m, n) \cdot e^{i (η_{ij C_{1}}^{-} \land η_{ij C_{2}}^{-}) (m, n)} \end{matrix} 〉] \\ for all i and j . \end{matrix}$

7. Intersection: $\begin{matrix} C_{1} \cap C_{2} = & [〈 \begin{matrix} min {r_{ij C_{1}}^{+} (m, n) \cdot e^{{iw}_{ij C_{1}}^{+} (m, n)}, \\ r_{ij C_{2}}^{+} (m, n) \cdot e^{{iw}_{ij C_{2}}^{+} (m, n)}}, \\ max {k_{ij C_{1}}^{-} (m, n) \cdot e^{i η_{ij C_{1}}^{-} (m, n)}, \\ k_{ij C_{2}}^{-} (m, n) \cdot e^{i η_{ij C_{2}}^{-} (m, n)}} \end{matrix} 〉] \\ = & [〈 \begin{matrix} (r_{ij C_{1}}^{+} \land r_{ij C_{2}}^{+}) (m, n) \cdot e^{i (w_{ij C_{1}}^{+} \land w_{ij C_{2}}^{+}) (m, n)}, \\ (k_{ij C_{1}}^{-} \lor k_{ij C_{2}}^{-}) (m, n) \cdot e^{i (η_{ij C_{1}}^{-} \lor η_{ij C_{2}}^{-}) (m, n)} \end{matrix} 〉] \\ for all i and j . \end{matrix}$

8. Complement: The complement of CBFM $C_{1}$ is indicated by $C_{1}$ ⁰ is defined as, $\begin{matrix} C_{1}^{0} = & [〈 (1 - r_{ij C_{1}}^{+} (m, n)) \cdot e^{i (2 π - w_{ij C_{1}}^{+} (m, n))}, . . \\ . . (- 1 - k_{ij C_{1}}^{-} (m, n)) \cdot e^{i (- 2 π - η_{ij C_{1}}^{-} (m, n))} 〉] \end{matrix}$

for all i and j .

9. Zero Matrix: Let $C_{1} = [〈 r_{ij C_{1}}^{+} (m, n) \cdot e^{{iw}_{ij C_{1}}^{+} (m, n)}, k_{ij C_{1}}^{-} (m, n) \cdot e^{i η_{ij C_{1}}^{-} (m, n)} 〉]$ . Then $C_{1}$ is called a Zero CBFM indicated by $\tilde{0}$ = [〈0, 0〉] , if $r_{ij C_{1}}^{+} (m, n) = 0$ and $k_{ij C_{1}}^{-} (m, n) = 0$ for all i and j .

10. Identity Matrix: A complex Bipolar fuzzy Matrix of order $\tilde{m} \times \tilde{n}$ is called complex Bipolar fuzzy Identity Matrix if $r_{ij C_{1}}^{+} (m, n) = 1 + 0 i$ and $k_{ij C_{1}}^{-} (m, n) = - 1 + 0 i$ for all i and j .It is indicated by $\tilde{I} .$

11. Square Matrix: Let $C_{1}$ $= [〈 r_{ij C_{1}}^{+} (m, n) \cdot e^{{iw}_{ij C_{1}}^{+} (m, n)}, k_{ij C_{1}}^{-} (m, n) \cdot e^{i η_{ij C_{1}}^{-} (m, n)} 〉]$ be CBFM of order $\tilde{m} \times \tilde{n} .$ Then $C_{1}$ is said to be complex Bipolar fuzzy Square Matrix if $\tilde{m} = \tilde{n}$ for all i and j.

12. Max-min Product Matrix: Let $C_{1}$ and $C_{2}$ be two CBFMs of order $\tilde{m} \times \tilde{n}$ and $\tilde{n} \times \tilde{p}$ respectively. Then the Max-Min Product of $C_{1}$ and $C_{2}$ is indicated by $C_{1} ★ C_{2}$ is defined as,

$\begin{matrix} C_{1} ★ C_{2} = & [\begin{matrix} max - min (r_{ij C_{1}}^{+} (m, n) \cdot e^{{iw}_{ij C_{1}}^{+} (m, n)}, \\ r_{ij C_{2}}^{+} (m, n) \cdot e^{{iw}_{ij C_{2}}^{+} (m, n)}), \\ min - max (k_{ij C_{1}}^{-} (m, n) \cdot e^{i η_{ij C_{1}}^{-} (m, n)}, \\ k_{ij C_{2}}^{-} (m, n) \cdot e^{i η_{ij C_{2}}^{-} (m, n)}) \end{matrix}] \\ for all i and j . \end{matrix}$

13. Equal Matrix: Let

$\begin{matrix} C_{1} & = [〈 \begin{matrix} r_{ij C_{1}}^{+} (m, n) \cdot e^{{iw}_{ij C_{1}}^{+} (m, n)}, \\ k_{ij C_{1}}^{-} (m, n) \cdot e^{i η_{ij C_{1}}^{-} (m, n)} \end{matrix} 〉] \in {CBFM}_{\tilde{m} \times \tilde{n}}, \\ C_{2} & = [〈 \begin{matrix} r_{ij C_{2}}^{+} (m, n) \cdot e^{{iw}_{ij C_{2}}^{+} (m, n)}, \\ k_{ij C_{2}}^{-} (m, n) \cdot e^{i η_{ij C_{2}}^{-} (m, n)} \end{matrix} 〉] \in {CBFM}_{\tilde{m} \times \tilde{n}} . \end{matrix}$

Then $C_{1}$ is equal to $C_{2}$ indicated by $C_{1} = C_{2}$ if $r_{ij C_{1}}^{+} (m, n) = r_{ij C_{2}}^{+} (m, n),$ $w_{ij C_{1}}^{+} (m, n) = w_{ij C_{2}}^{+} (m, n)$ and $k_{ij C_{1}}^{-} (m, n) = k_{ij C_{2}}^{-} (m, n),$ $η_{ij C_{1}}^{-} (m, n) = η_{ij C_{2}}^{-} (m, n)$ for all i and j .

14. Super Matrix: Let $\begin{matrix} C_{1} & = [〈 \begin{matrix} r_{ij C_{1}}^{+} (m, n) \cdot e^{{iw}_{ij C_{1}}^{+} (m, n)}, \\ k_{ij C_{1}}^{-} (m, n) \cdot e^{i η_{ij C_{1}}^{-} (m, n)} \end{matrix} 〉] \in {CBFM}_{\tilde{m} \times \tilde{n}}, \\ C_{2} & = [〈 \begin{matrix} r_{ij C_{2}}^{+} (m, n) \cdot e^{{iw}_{ij C_{2}}^{+} (m, n)}, \\ k_{ij C_{2}}^{-} (m, n) \cdot e^{i η_{ij C_{2}}^{-} (m, n)} \end{matrix} 〉] \in {CBFM}_{\tilde{m} \times \tilde{n}} . \end{matrix}$ Then $C_{1}$ is called complex bipolar fuzzy super matrix of $C_{2}$ indicated by $C_{1}$ $\supseteq C_{2}$ if $r_{ij C_{1}}^{+} (m, n) \geq r_{ij C_{2}}^{+} (m, n),$ $w_{ij C_{1}}^{+} (m, n) \geq w_{ij C_{2}}^{+} (m, n)$ and $k_{ij C_{1}}^{-} (m, n) \leq k_{ij C_{2}}^{-} (m, n),$ $η_{ij C_{1}}^{-} (m, n) \leq η_{ij C_{2}}^{-} (m, n)$ for all i and j .

15. Row Matrix: Let

16. Column Matrix: Let

17. Diagonal Matrix: Let

18. Transpose of Matrix: Let

$C_{1} = [〈 \begin{matrix} r_{ij C_{1}}^{+} (m, n) \cdot e^{{iw}_{ij C_{1}}^{+} (m, n)}, \\ k_{ij C_{1}}^{-} (m, n) \cdot e^{i η_{ij C_{1}}^{-} (m, n)} \end{matrix} 〉] \in$ ${CBFM}_{\tilde{m} \times \tilde{n}} .$ Then transpose of matrix $C_{1}$ is defined as, $C_{1}^{T} = [〈 \begin{matrix} r_{ji C_{1}}^{+} (m, n) \cdot e^{{iw}_{ji C_{1}}^{+} (m, n)}, \\ k_{ji C_{1}}^{-} (m, n) \cdot e^{i η_{ji C_{1}}^{-} (m, n)} \end{matrix} 〉] \in {CBFM}_{\tilde{n} \times \tilde{m}} .$

Definition 5.5. Let $C_{1} = [〈 \begin{matrix} r_{ij C_{1}}^{+} (m, n) \cdot e^{{iw}_{ij C_{1}}^{+} (m, n)}, \\ k_{ij C_{1}}^{-} (m, n) \cdot e^{i η_{ij C_{1}}^{-} (m, n)} \end{matrix} 〉]$ and $C_{2} = [〈 \begin{matrix} r_{ij C_{2}}^{+} (m, n) \cdot e^{{iw}_{ij C_{2}}^{+} (m, n)}, \\ k_{ij C_{2}}^{-} (m, n) \cdot e^{i η_{ij C_{2}}^{-} (m, n)} \end{matrix} 〉]$ be two CBFMs of order $\tilde{m} \times \tilde{n} .$ Then CBFM $C_{3} = [〈 \begin{matrix} r_{ij C_{3}}^{+} (m, n) \cdot e^{{iw}_{ij C_{3}}^{+} (m, n)}, \\ k_{ij C_{3}}^{-} (m, n) \cdot e^{i η_{ij C_{3}}^{-} (m, n)} \end{matrix} 〉]$ is said to be

(1) “Probabilistic Sum”(“†”) operation of $C_{1}$ and $C_{2},$ and indicated by $C_{3} =$ $C_{1} † C_{2},$ if $r_{ij C_{3}}^{+} \cdot e^{{iw}_{ij C_{3}}^{+}} = r_{ij C_{1}}^{+} \cdot e^{{iw}_{ij C_{1}}^{+}} + r_{ij C_{2}}^{+} \cdot e^{{iw}_{ij C_{2}}^{+}} - (r_{ij C_{1}}^{+} \cdot r_{ij C_{2}}^{+}) \cdot e^{i (w_{ij C_{1}}^{+} \cdot w_{ij C_{2}}^{+})}$ and $k_{ij C_{3}}^{-} \cdot e^{i η_{ij C_{3}}^{-} (m, n)} = (k_{ij C_{1}}^{-} \cdot k_{ij C_{2}}^{-}) \cdot e^{i (η_{ij C_{1}}^{-} \cdot η_{ij C_{2}}^{-})}$ all i and j .

(2) “Product”(“∗”) operation of $C_{1}$ and $C_{2},$ indicated by $C_{3} =$ $C_{1} * C_{2},$ if $r_{ij C_{3}}^{+} \cdot e^{{iw}_{ij C_{3}}^{+}} = (r_{ij C_{1}}^{+} \cdot r_{ij C_{2}}^{+}) \cdot e^{i (w_{ij C_{1}}^{+} \cdot w_{ij C_{2}}^{+})}$ and $k_{ij C_{3}}^{-} \cdot e^{i η_{ij C_{3}}^{-}} = k_{ij C_{1}}^{-} \cdot e^{i η_{ij C_{1}}^{-}} + k_{ij C_{2}}^{-} \cdot e^{i η_{ij C_{2}}^{-}} - (k_{ij C_{1}}^{-} \cdot k_{ij C_{2}}^{-}) \cdot e^{i (η_{ij C_{1}}^{-} \cdot η_{ij C_{2}}^{-})}$ all i and j .

(3) “Arithmetic Mean”(“ψ”) operation of $C_{1}$ and $C_{2},$ and indicated by $C_{3} =$ $C_{1} ψ C_{2},$ if $r_{ij C_{3}}^{+} \cdot e^{{iw}_{ij C_{3}}^{+}} = (\frac{r_{ij C_{1}}^{+} + r_{ij C_{2}}^{+}}{2}) \cdot e^{i (\frac{w_{ij C_{1}}^{+} + w_{ij C_{2}}^{+}}{2})}$ and $k_{ij C_{3}}^{-} \cdot e^{i η_{ij C_{3}}^{-}} = (\frac{k_{ij C_{1}}^{-} + k_{ij C_{2}}^{-}}{2}) \cdot e^{i (\frac{η_{ij C_{1}}^{-} + η_{ij C_{2}}^{-}}{2})}$ all i and j .

(4) “Weighted Arithmetic Mean”(“ψ^ω”) operation of $C_{1}$ and $C_{2},$ and indicated by $C_{3} =$ $C_{1} ψ^{ω} C_{2},$ if $r_{ij C_{3}}^{+} \cdot e^{{iw}_{ij C_{3}}^{+}} = (\frac{ω_{1} r_{ij C_{1}}^{+} + ω_{2} r_{ij C_{2}}^{+}}{ω_{1} + ω_{2}}) \cdot e^{i (\frac{ω_{1} w_{ij C_{1}}^{+} + ω_{2} w_{ij C_{2}}^{+}}{ω_{1} + ω_{2}})}$ and $k_{ij C_{3}}^{+} \cdot e^{i η_{ij C_{3}}^{-}} = (\frac{ω_{1} k_{ij C_{1}}^{-} + ω_{2} k_{ij C_{2}}^{-}}{ω_{1} + ω_{2}}) \cdot e^{i (\frac{ω_{1} η_{ij C_{1}}^{-} + ω_{2} η_{ij C_{2}}^{-}}{ω_{1} + ω_{2}})}$ all i and j . ω₁ > 0, ω₂ > 0 .

(5) “Geometric Mean”(“ρ”) operation of $C_{1}$ and $C_{2},$ and indicated by $C_{3} =$ $C_{1} ρ C_{2},$ if $r_{ij C_{3}}^{+} \cdot e^{{iw}_{ij C_{3}}^{+}} = \sqrt{(r_{ij C_{1}}^{+} \cdot r_{ij C_{2}}^{+})} \cdot e^{\sqrt{(w_{ij C_{1}}^{+} \cdot w_{ij C_{2}}^{+}) i}}$ and $k_{ij C_{3}}^{-} \cdot e^{i η_{ij C_{3}}^{-}} = \sqrt{(k_{ij C_{1}}^{-} \cdot k_{ij C_{2}}^{-})} \cdot e^{\sqrt{(η_{ij C_{1}}^{-} \cdot η_{ij C_{2}}^{-}) i}}$ all i and j .

(6) “Weighted Geometric Mean”(“ρ^ω”) operation of $C_{1}$ and $C_{2},$ and indicated by $C_{3} =$ $C_{1} ρ^{ω} C_{2},$ if $r_{ij C_{3}}^{+} \cdot e^{{iw}_{ij C_{3}}^{+}} = ((r_{ij C_{1}}^{+})^{ω_{1}} \cdot (r_{ij C_{2}}^{+})^{ω_{2}})^{\frac{1}{ω_{1} + ω_{2}}} \cdot e^{i ((w_{ij C_{1}}^{+})^{ω_{1}} \cdot (w_{ij C_{2}}^{+})^{ω_{2}})^{\frac{1}{ω_{1} + ω_{2}}}}$ and $k_{i j C_{3}}^{-} \cdot e^{i}^{η_{i j C_{3}}^{-}} = (\frac{ω_{1} k_{i j C_{1}}^{-} + ω_{2} k_{i j C_{2}}^{-}}{ω_{1} + ω_{2}}) \cdot e^{i (\frac{ω_{1} η_{i j C_{1}}^{-} + ω_{2} η_{i j C_{2}}^{-}}{ω_{1} + ω_{2}})}$ all i and j. ω₁ > 0,ω₂ > 0.

(7) “Harmonic Mean”(“φ”) operation of $C_{1}$ and $C_{2},$ and indicated by $C_{3} =$ $C_{1} φ C_{2},$ if $r_{ij C_{3}}^{+} \cdot e^{{iw}_{ij C_{3}}^{+}} = (2 . \frac{r_{ij C_{1}}^{+} . r_{ij C_{2}}^{+}}{r_{ij C_{1}}^{+} + r_{ij C_{2}}^{+}}) \cdot e^{i (2 . \frac{w_{ij C_{1}}^{+} . w_{ij C_{2}}^{+}}{w_{ij C_{1}}^{+} + w_{ij C_{2}}^{+}})}$ and $k_{ij C_{3}}^{-} \cdot e^{i η_{ij C_{3}}^{-}} = (2 . \frac{k_{ij C_{1}}^{-} . k_{ij C_{2}}^{-}}{k_{ij C_{1}}^{-} + k_{ij C_{2}}^{-}}) \cdot e^{i (2 . \frac{η_{ij C_{1}}^{-} . η_{ij C_{2}}^{-}}{η_{ij C_{1}}^{-} + η_{ij C_{2}}^{-}})}$ all i and j .

(8) “Harmonic Mean”(“φ^ω”) operation of $C_{1}$ and $C_{2},$ and indicated by $C_{3} =$ $C_{1} φ^{ω} C_{2},$ if $r_{ij C_{3}}^{+} \cdot e^{{iw}_{ij C_{3}}^{+}} = (\frac{ω_{1} + ω_{2}}{\frac{ω_{1}}{r_{ij C_{1}}^{+}} + \frac{ω_{2}}{r_{ij C_{2}}^{+}}}) \cdot e^{i (\frac{ω_{1} + ω_{2}}{\frac{ω_{1}}{w_{ij C_{1}}^{+}} + \frac{ω_{2}}{w_{ij C_{2}}^{+}}})}$ and $k_{ij C_{3}}^{+} \cdot e^{i η_{ij C_{3}}^{-}} = (\frac{ω_{1} + ω_{2}}{\frac{ω_{1}}{k_{ij C_{1}}^{+}} + \frac{ω_{2}}{k_{ij C_{2}}^{-}}}) \cdot e^{i (\frac{ω_{1} + ω_{2}}{\frac{ω_{1}}{η_{ij C_{1}}^{-}} + \frac{ω_{2}}{η_{ij C_{2}}^{-}}})}$ all i and j . ω₁ > 0, ω₂ > 0 .

6 Distance measure and aggregation operators

In this part we define distance measure and aggregation operators on complex bipolar fuzzy sets.

Definition 6.1. If $C_{1}$ and $C_{2}$ are CBFSs in a universe of discourse $U$ , where $C_{1} = {〈 m, r_{C_{1}}^{+} (m) e^{{iw}_{C_{1}}^{+} (m)}, k_{C_{1}}^{-} (m) e^{i η_{C_{1}}^{-} (m)} 〉}$ and $C_{2} = {〈 m, r_{C_{2}}^{+} (m) e^{{iw}_{C_{2}}^{+} (m)}, k_{C_{2}}^{-} (m) e^{i η_{C_{2}}^{-} (m)} 〉},$ then

$(1) C_{1} \subset C_{2}$ iff amplitude terms $r_{C_{1}}^{+} (m) < r_{C_{2}}^{+} (m)$ and $k_{C_{1}}^{-} (m) > k_{C_{2}}^{-} (m)$ and the phase terms $w_{C_{1}}^{+} (m) < w_{C_{2}}^{+} (m)$ and $η_{C_{1}}^{-} (m) > η_{C_{2}}^{-} (m),$ for all $m \in U .$

$(2) C_{1} = C_{2}$ iff amplitude terms $r_{C_{1}}^{+} (m) = r_{C_{2}}^{+} (m)$ and $k_{C_{1}}^{-} (m) = k_{C_{2}}^{-} (m)$ and the phase terms $w_{C_{1}}^{+} (m) = w_{C_{2}}^{+} (m)$ and $η_{C_{1}}^{-} (m) = η_{C_{2}}^{-} (m),$ for all $m \in U .$

Definition 6.2. Let CBFS( $U$ ) be the set of all complex bipolar fuzzy sets on $U .$ A distance of complex bipolar fuzzy sets is a function,

∂ :CBFS $(U) \times$ CBFS $(U) \to [[0, 1]$ for each $C_{1}, C_{2}$ and $C_{3} \in$ CBFS, the following properties are satisfying:

(∂₁) $0 \leq \partial (C_{1}, C_{2}) \leq 1 .$

(∂₂) $\partial (C_{1}, C_{2}) = 0$ iff $C_{1} = C_{2} .$

$(\partial_{3}) \partial (C_{1}, C_{2}) = \partial (C_{2}, C_{1}) .$

(∂₄) Let $C_{1}, C_{2}, C_{3} \in$ CBFS $(U),$ and if $C_{1} \subset C_{2} \subset C_{3}$ then $\partial (C_{1}, C_{2}) < \partial (C_{1}, C_{3}), \partial (C_{2}, C_{3}) < \partial (C_{1}, C_{3}) .$

Definition 6.3. We explain a function ∂ :CBFS $(U) \times$ CBFS $(U) ⟶ [[0, 1]$ between CBFSs $C_{1}$ and $C_{2},$ is defined as: $\begin{matrix} \partial (C_{1}, C_{2}) = \frac{1}{2 \sum_{i = 1}^{n} ω_{i}} \times [\sum_{i = 1}^{n} ω_{i} [\begin{matrix} (α_{1} \cdot | r_{C_{1}}^{+} (m_{i}) - r_{C_{2}}^{+} (m_{i}) | + \\ β_{1} \cdot | k_{C_{1}}^{-} (m_{i}) - k_{C_{2}}^{-} (m_{i}) | + σ_{1} \\ \cdot max (| r_{C_{1}}^{+} (m_{i}) - r_{C_{2}}^{+} (m_{i}) |, \\ | k_{C_{1}}^{-} (m_{i}) - k_{C_{2}}^{-} (m_{i}) |)) + \\ \frac{1}{2 π} (α_{2} \cdot | w_{C_{1}}^{+} (m_{i}) - w_{C_{2}}^{+} (m_{i}) | + β_{2} \\ \cdot | η_{C_{1}}^{-} (m_{i}) - η_{C_{2}}^{-} (m_{i}) | + σ_{2} \cdot \\ max (| w_{C_{1}}^{+} (m_{i}) - w_{C_{2}}^{+} (m_{i}) |, \\ | η_{C_{1}}^{-} (m_{i}) - η_{C_{2}}^{-} (m_{i}) |)) \end{matrix}]] \end{matrix}$

where α₁, β₁, σ₁, α₂, β₂, σ₂ ∈ [0, 1] , α₁ + β₁ + σ₁ = 1 and α₂ + β₂ + σ₂ = 1 . w_i ∈ [0, 1] , i ∈ {1, 2, . . . m} .

Theorem 6.4. The distance measure of complex bipolar fuzzy set of a function $\partial (C_{1}, C_{2})$ defined in Definition 6, between two CBFSs $C_{1}$ and $C_{2}$ in $U$ .

Proof. (a) By definition of CBFS, we have all m_i in $U$ , where i= 1, 2, . . . . n, are each $r_{C_{1}}^{+} (m_{i})$ , $r_{C_{2}}^{+} (m_{i}) \in [0, 1]$ , and $k_{C_{1}}^{-} (m_{i}),$ $k_{C_{2}}^{-} (m_{i})$ ∈ [-1, 0] . Also each of $w_{C_{1}}^{+} (m_{i})$ , $w_{C_{2}}^{+} (m_{i})$ belong to [0, 2π] and $η_{C_{1}}^{-} (m_{i}),$ $η_{C_{2}}^{-} (m_{i})$ lie in the interval [-2π, 0], so it is easy to conclude that each of $| r_{C_{1}}^{+} (m_{i}) - r_{C_{2}}^{+} (m_{i}) |$ and $| k_{C_{1}}^{-} (m_{i}) - k_{C_{2}}^{-} (m_{i}) |$ lie between 0 and 1..

Also $| w_{C_{1}}^{+} (m_{i}) - w_{C_{2}}^{+} (m_{i}) |$ and $| η_{C_{1}}^{-} (m_{i}) - η_{C_{2}}^{-} (m_{i}) |$ lie between 0 and 2π .

Thus we have

$\partial (C_{1}, C_{2}) \leq \frac{1}{2 \sum_{i = 1}^{n} ω_{i}} \times [\begin{matrix} \sum_{i = 1}^{n} ω_{i} [(α_{1} \cdot 1 + β_{1} \cdot 1 + σ_{1} \cdot max (1, 1)) + \frac{1}{2 π} (α_{2} \cdot 2 π \\ + β_{2} \cdot 2 π + σ_{2} \cdot max (2 π, 2 π))] \end{matrix}]$ $= \frac{1}{2 \sum_{i = 1}^{n} ω_{i}} \times [\sum_{i = 1}^{n} ω_{i} [(α_{1} + β_{1} + σ_{1}) + (α_{2} + β_{2} + σ_{2})]] .$

Since α₁ + β₁ + σ₁ = 1 and α₂ + β₂ + σ₂ = 1, we have

$0 \leq \partial (C_{1}, C_{2}) \leq \frac{1}{2 \sum_{i = 1}^{n} ω_{i}} \times [\sum_{i = 1}^{n} 2 ω_{i}] = 1 .$

(b) By Definition 6.1, it is simple to see that $\partial (C_{1}, C_{2})$ satisfies the 2nd and 3rd conditions of Definition 6.2.

$- 1 \leq k_{C_{3}}^{-} (m_{i}) \leq k_{C_{2}}^{-} (m_{i}) \leq k_{C_{1}}^{-} (m_{i}) \leq 0,$ $- 2 π \leq η_{C_{3}}^{-} (m_{i}) \leq η_{C_{2}}^{-} (m_{i}) \leq η_{C_{1}}^{-} (m_{i}) \leq 0 .$ Then, we can conclude that $\begin{matrix} | r_{C_{1}}^{+} (m_{i}) - r_{C_{2}}^{+} (m_{i}) | & \leq & | r_{C_{1}}^{+} (m_{i}) - r_{C_{3}}^{+} (m_{i}) |, \\ | w_{C_{1}}^{+} (m_{i}) - w_{C_{2}}^{+} (m_{i}) | & \leq & | w_{C_{1}}^{+} (m_{i}) - w_{C_{3}}^{+} (m_{i}) |, \end{matrix}$ $\begin{matrix} | k_{C_{1}}^{-} (m_{i}) - k_{C_{2}}^{-} (m_{i}) | & \geq & | k_{C_{1}}^{-} (m_{i}) - k_{C_{3}}^{-} (m_{i}) |, \\ | η_{C_{1}}^{-} (m_{i}) - η_{C_{2}}^{-} (m_{i}) | & \geq & | η_{C_{1}}^{-} (m_{i}) - η_{C_{3}}^{-} (m_{i}) | . \end{matrix}$ Thus, $\partial (C_{1}, C_{3}) \geq \partial (C_{1}, C_{2}) .$ Similarly, we can obtain $\partial (C_{1}, C_{3}) \geq \partial (C_{2}, C_{3}) .$

Definition 6.5. Let $C_{1}, C_{2}, . . . . . . ., C_{n}$ are complex bipolar fuzzy sets defined over the universe $U,$ then the vector aggregation on $C_{1}, C_{2}, . . . . . . ., C_{n}$ by a function is follows;

$v : {(å, å^{'}) | å, å^{'} \in ℂ, | å |, | å^{'} | \leq 1}^{n} ⟼ {(b̊, {b̊}^{'}) | b̊, {b̊}^{'} \in ℂ, | b̊ |, | {b̊}^{'} | \leq 1} .$

This function produce an aggregate fuzzy set $C$ operating on positive memberships grade and negative memberships grade of $C_{i}$ for each $m \in U :$ $\begin{matrix} (μ_{C}^{+} (m), \bar{ϑ_{C}} (m)) \\ = v ((μ_{C_{1}}^{+} (m), \bar{ϑ_{C_{1}}} (m)) \cdot (μ_{C_{2}}^{+} (m), \bar{ϑ_{C_{2}}} (m)) \\ . . . . . . . . (μ_{C_{n}}^{+} (m), \bar{ϑ_{C_{n}}} (m))) \\ = \sum_{i = 1}^{n} ω_{i} (μ_{C}^{+} (m), \bar{ϑ_{C}} (m)) \end{matrix}$ where $μ_{C}^{+} (m) = r_{C}^{+} (m) \cdot e^{{iw}_{C}^{+} (m)}$ and $\bar{ϑ_{C}} (m) = k_{C}^{-} (m) \cdot e^{i η_{C}^{-} (m)}$ $ω_{i} \in {(å, å^{'}) | å, å^{'} \in ℂ, | å |, | å^{'} | \leq 1}, \sum_{i = 1}^{n} | ω_{i} | = 1 .$ Usually ω_i is selected to be $\frac{1}{n}$ since no method for choosing complex ω_i has been developed(2003) yielding: $(μ_{C}^{+} (m), \bar{ϑ_{C}} (m)) = \frac{1}{n} \sum_{i = 1}^{n} (μ_{C}^{+} (m), \bar{ϑ_{C}} (m)) .$ The meaning of vector aggregations is that all $C_{i}$ are complex valued.

7 Application

As an application of the presented theory we in this section quoted a real life example. We us the following algorithm to solve the real life problem.

Algorithm:

Input:

Step-1. First develop the complex bipolar fuzzy matrix by gathering data with the help of experts.

Step-2. Give weights to the opinion of experts, for amplitude and phase term.

Step-3. Calculate the complex bipolar distance measure using the Definition 6.2 and Definition 6.3.

Step-4. Finally rank all the complex bipolar distance measure. S

Output: Smallest the value of complex bipolar distance measure, greater the chances of having the optimal choice.

Example 7.1. Assume that the company A settles on a choice to purchase bus from a busmaker B. The busmaker B offers company A several data buses to three exemplars of each exemplar for different periods. Thus the company A got three exemplars (Bus₁, Bus₂, and Bus₃) on the same time for selection. The investigators’ group of the company chose that five qualities ought to be considered. They are Q1: constancy, Q2: most elevated shipment, Q3: buying value, Q4: greatest quickness, and Q5: dependability. But these qualities will be influenced and modified if the making date is diverse for the coordinating exemplar of buses. That judgment is made by the group to build on its experience and information. In this way, we can get "yes,""no," "I don’t identify," or "I am not certain" In response to this decision, the preferred buses. Accordingly, as noted above, the best technique is, this kind of data to CBFS (i.e. are data on individual choice, which once happened in a while), where in each of the exemplar of a new bus, prospects and customers have divergent thinking. To be precise, we assume that the investigators’ group of the company has proposed a perfect bus, before accepting the particular data from busmaker B. The reason for the group is to pick a fitting bus recognized by busmaker B that is the most plausible to be the perfect bus. A while later, every investigator in the choice group gives each bus’s quality a rang -1 or 1 they show what the bus corresponds to the standard or not, and gives a score of 0, for example when you are not sure of the date bus. For example, assuming that the panel found that no less than 60% consider that the bus is the perfect exemplar of appropriate quality primary; and not more than 10% of the investigators imagine a perfect exemplar of the poor quality of the bus, where this method is used to detect the amplitude terms of both the positive and negative membership work individually in CBFS. The phase terms that present making date for first nature of perfect bus can be given as below: if the group of experts felt that no less than 70% of them assume that the perfect making date of bus is fitting at the principal quality; and not over 16% of them trust that the perfect making date of bus is of low standard. Thus, the perfect bus’s first quality can be given as (0.6e^i2π(0.7), - 0.1e^-2π(0.16)i. In this manner, all data is made in good CBFS, where both amplitude and phase terms of bipolar can show data on vulnerability (see Table 1). Let ∂(Bus_j, P.B.) be a separation calculate between complex bipolar fuzzy sets Bus_j and P.B. Along these lines, the exemplar can be presented as takes after: $M = {\begin{matrix} {Bus}_{j} | \partial ({Bus}_{j}, P . B .) = min (\sum_{i = 1}^{n} \partial ({Bus}_{1, i}, {P . B}_{i} .), . . ., \sum_{i = 1}^{n} \partial ({Bus}_{1, i}, {P . B}_{i} .)), \\ j = 1, . . ., k, i = 1, . . ., n \end{matrix}} .$

Table 1
Bus data and the perfect bus.

(Q_1,Bus₁) (0.5e^i2π(0.4), - 0.2e^-2π(0.3)i)

(Q_1,Bus₂) (0.7e^i2π(0.3), - 0.3e^-2π(0.4)i)

(Q_1,Bus₃) (0.6e^i2π(0.1), - 0.4e^-2π(0.7)i)

(Q_1,Perfect Bus) (0.8e^i2π(0.3), - 0.5e^-2π(0.1)i)

(Q_2,Bus₁) (0.7e^i2π(0.3), - 0.5e^-2π(0.6)i)

(Q_2,Bus₂) (0.8e^i2π(0.1), - 0.9e^-2π(0.2)i)

(Q_2,Bus₃) (0.3e^i2π(0.5), - 0.8e^-2π(0.2)i)

(Q_2,Perfect Bus) (0.4e^i2π(0.1), - 0.3e^-2π(0.4)i)

(Q_3,Bus₁) (0.3e^i2π(0.4), - 0.4e^-2π(0.5)i)

(Q_3,Bus₂) (0.2e^i2π(0.3), - 0.5e^-2π(0.6)i)

(Q_3,Bus₃) (0.9e^i2π(0.2), - 0.1e^-2π(0.4)i)

(Q_3,Perfect Bus) (0.2e^i2π(0.5), - 0.9e^-2π(0.3)i)

(Q_4,Bus₁) (0.7e^i2π(0.6), - 0.5e^-2π(0.4)i)

(Q_4,Bus₂) (0.5e^i2π(0.6), - 0.8e^-2π(0.1)i)

(Q_4,Bus₃) (0.6e^i2π(0.9), - 0.2e^-2π(0.4)i)

(Q_4,Perfect Bus) (0.8e^i2π(0.3), - 0.5e^-2π(0.7)i)

(Q_5,Bus₁) (0.8e^i2π(0.5), - 0.4e^-2π(0.6)i)

(Q_5,Bus₂) (0.6e^i2π(0.3), - 0.8e^-2π(0.3)i)

(Q_5,Bus₃) (0.2e^i2π(0.5), - 0.9e^-2π(0.3)i)

(Q_5,Perfect Bus) (0.3e^i2π(0.4), - 0.2e^-2π(0.7)i)

Step-1.

Step-2. Assume the investigators’ group gives the weight for each quality as below. Let ω₁ = 0.1, ω₂ = 0.7, ω₃ = 0.6, ω₄ = 0.3, ω₅ = 0.3, ,and show the weight for each quality. Let α₁ = 0.1, β₁ = 0.6 and σ₁ = 0.3 weight for amplitude term, and α₂ = 0.3, β₂ = 0.3 and σ₂ = 0.4 weight for phase term.

Step-3. To recognize the perfect bus data in Table 1, we need to exchange information distance formula (Definitions 6, 6) to calculate the values between each bus (Bus_j, j = 1, 2, 3) and the perfect bus (P,B.): $\begin{matrix} \partial ({Bus}_{1}, P . B .) & = & \sum_{i = 1}^{5} \partial ({Bus}_{1, i}, P . B_{i} .) \\ = & \frac{1}{2 \cdot (2.9)} \times [\sum_{i = 1}^{5} ω_{i} [\begin{matrix} (α_{1} \cdot | r_{C_{1}}^{+} (m_{i}) - r_{C_{2}}^{+} (m_{i}) | + β_{1} \cdot | k_{C_{1}}^{-} (m_{i}) - k_{C_{2}}^{-} (m_{i}) | \\ + σ_{1} \cdot max (| r_{C_{1}}^{+} (m_{i}) - r_{C_{2}}^{+} (m_{i}) |, | k_{C_{1}}^{-} (m_{i}) - k_{C_{2}}^{-} (m_{i}) |)) \\ + \frac{1}{2 π} (α_{2} \cdot | w_{C_{1}}^{+} (m_{i}) - w_{C_{2}}^{+} (m_{i}) | + β_{2} \cdot | η_{C_{1}}^{-} (m_{i}) - η_{C_{2}}^{-} (m_{i}) | \\ + σ_{2} \cdot max (| w_{C_{1}}^{+} (m_{i}) - w_{C_{2}}^{+} (m_{i}) |, | η_{C_{1}}^{-} (m_{i}) - η_{C_{2}}^{-} (m_{i}) |)) \end{matrix}]] . \end{matrix}$ For the first quality Q₁, (i = 1), the distance is given as takes after: $\begin{matrix} \partial ({Bus}_{1, 1}, P . B_{1} .) \\ = & \frac{1}{2 \cdot (2.9)} [0.1 [\begin{matrix} (0.1 \cdot | 0.5 - 0.8 | + 0.6 \cdot | - 0.2 - (- 0.5) | \\ + 0.3 \cdot max (| 0.5 - 0.8 |, | - 0.2 - (- 0.5) |)) \\ + \frac{1}{2 π} (0.3 \cdot | 0.4 (2 π) - 0.3 (2 π) | + 0.3 \cdot | 0.3 (- 2 π) - 0.1 (- 2 π) | \\ + 0.4 \cdot max (| 0.4 (2 π) - 0.3 (2 π) |, | 0.3 (- 2 π) - 0.1 (- 2 π) |)) \end{matrix}]] \end{matrix}$ $= \frac{1}{2 \cdot (2.9)} [0.074] .$ Similarly, we compute the distances for Bus₁ for each quality, ∂ (Bus_1,2, P. $B_{2}) = \frac{1}{2 \cdot (2.9)},$ ∂ (Bus_1,3, P. $B_{3}) = \frac{1}{2 \cdot (2.9)} [0.378],$ ∂ (Bus_1,4, P. $B_{4}) = \frac{1}{2 \cdot (2.9)} [0.102], \partial ($ Bus_1,5, P. $B_{5}) = \frac{1}{2 \cdot (2.9)} [0.126] .$

Step-4. Then, ∂ (Bus₁, P.B. $) = \frac{1}{2 \cdot (2.9)} \times [0.074 + 0.308 + 0.378 + 0.102 + 0.126] = 0.1703 \Rightarrow \partial ($ Bus₁, P.B.) =0.1703 . Similarly, we compute the distance values for ∂ (Bus₂, P.B.) =0.2464, and ∂ (Bus₃, P.B.) =0.3043 . Therefore, ∂ (Bus₁, P.B.) =0.1703 is the smallest value, so Bus₁ is the desired bus.

8 Comparison analysis

The idea of complex bipolar fuzzy sets generalize the idea of bipolar fuzzy sets and as well as the idea of complex fuzzy sets. If we restrict the newly defined complex bipolar fuzzy sets to the real parts only by assuming the imaginary parts equal to zero we get bipolar fuzzy sets defined by Zhang et al. [5, 6]. On the other hand if we consider only the positive membership in complex bipolar fuzzy sets we get complex fuzzy sets [16].

9 Conclusions

Here we initiated in this paper the idea of complex bipolar fuzzy sets. We defined different operations and properties regarding complex bipolar fuzzy sets. Moreover, we elaborated the complex bipolar fuzzy relations, basic operations on complex bipolar fuzzy matrices and some operators of complex bipolar fuzzy matrices. We explained distance measure on complex bipolar fuzzy sets and complex bipolar fuzzy aggregation. Lastly we proved a numerical example of distance measure between complex bipolar fuzzy sets. In future we are focusing at complex bipolar fuzzy groups, complex bipolar fuzzy graphs, complex bipolar decision making problems and complex bipolar neutrosophic sets.

Footnotes

Acknowledgment

The authors extend their appreciation to the Deanship of Scientific Research at Majmaah University for funding this work under project number (RGP-2019-4).

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(Q_1,Bus₁)	(0.5e^i2π(0.4), - 0.2e^-2π(0.3)i)
(Q_1,Bus₂)	(0.7e^i2π(0.3), - 0.3e^-2π(0.4)i)
(Q_1,Bus₃)	(0.6e^i2π(0.1), - 0.4e^-2π(0.7)i)
(Q_1,Perfect Bus)	(0.8e^i2π(0.3), - 0.5e^-2π(0.1)i)
(Q_2,Bus₁)	(0.7e^i2π(0.3), - 0.5e^-2π(0.6)i)
(Q_2,Bus₂)	(0.8e^i2π(0.1), - 0.9e^-2π(0.2)i)
(Q_2,Bus₃)	(0.3e^i2π(0.5), - 0.8e^-2π(0.2)i)
(Q_2,Perfect Bus)	(0.4e^i2π(0.1), - 0.3e^-2π(0.4)i)
(Q_3,Bus₁)	(0.3e^i2π(0.4), - 0.4e^-2π(0.5)i)
(Q_3,Bus₂)	(0.2e^i2π(0.3), - 0.5e^-2π(0.6)i)
(Q_3,Bus₃)	(0.9e^i2π(0.2), - 0.1e^-2π(0.4)i)
(Q_3,Perfect Bus)	(0.2e^i2π(0.5), - 0.9e^-2π(0.3)i)
(Q_4,Bus₁)	(0.7e^i2π(0.6), - 0.5e^-2π(0.4)i)
(Q_4,Bus₂)	(0.5e^i2π(0.6), - 0.8e^-2π(0.1)i)
(Q_4,Bus₃)	(0.6e^i2π(0.9), - 0.2e^-2π(0.4)i)
(Q_4,Perfect Bus)	(0.8e^i2π(0.3), - 0.5e^-2π(0.7)i)
(Q_5,Bus₁)	(0.8e^i2π(0.5), - 0.4e^-2π(0.6)i)
(Q_5,Bus₂)	(0.6e^i2π(0.3), - 0.8e^-2π(0.3)i)
(Q_5,Bus₃)	(0.2e^i2π(0.5), - 0.9e^-2π(0.3)i)
(Q_5,Perfect Bus)	(0.3e^i2π(0.4), - 0.2e^-2π(0.7)i)