On bipolar complex fuzzy sets and its application

Abstract

Many factors with the perspective of bipolarity in the traditional Chinese food system “Yin and Yang food system” manipulate with types of food simultaneously to have a balanced body. This research studies the multiple attributes decision making (MADM) problem that measuring the “bipolarity of periodic” variation in bipolar information with an illustration example in order to find an optimal nutrition program for a person X. To convey this type of data to a mathematical formula and vice versa without losing the full meaning of human knowledge, we use bipolar fuzzy set in a complex geometry by extending the range of bipolar fuzzy set to the realm of a complex number. This extension needs to be successful to study and introduce intensely a new mathematical structure called a bipolar complex fuzzy set (BCFS) with its properties. Ranges of values are extended to [0, 1] e^iα[0,1] and [-1, 0] e^iα[-1,0] for both positive and negative membership functions, respectively, as a replacement for [-1, 0] × [0, 1] , as in the bipolar fuzzy set. The main benefit of BCFS that the amplitude and phase terms of BCFSs can convey bipolar fuzzy information. Moreover, the formal definition of BCF distance measure and illustration application are introduced. Some basic mathematical operations on BCFS are also proposed and study its properties with arithmetical examples.

Keywords

Bipolar fuzzy set bipolar complex fuzzy distance bipolar fuzzy complement union and intersection fuzzy set complex fuzzy set

1 Introduction

One of the most important and valuable fields that essential in our daily life is decision making. Due to the current rising complications of the systems, decision-maker encounter a difficulties in creating suitable choice/s under changeable situations. Several uncertainty sets have been introduced for a smother dealing with decision-making problems. Such as fuzzy set theory [24], intuitionistic fuzzy set theory [23], bipolar fuzzy set [43], neutrosophic set [15, 16] and other researchers studied and introduced algorithms to solve MADM problems as Liu et al. [14], Si et al. [2], Zeng et al. [38], Ali et al. [27], (see [1 , 48]).

Fuzzy set (FS) was first introduced by Zadeh [24] whose membership degree lies in [0, 1]. FS has been used in decision-making prblems [2 , 44], improved to deal linguistic group decision making by proposing a novel model that depend on the employ of extended linguistic hierarchies [52] and contributed in developing the multiple criteria decision making (MCDM) problems depending on instable hesitant fuzzy linguistic term sets [42, 53]. A novel concept of intuitionistic fuzzy set (IFS) has been introduced with some restrictions by adding an additional membership degree to FS with the same range [0, 1]. However, I FS allows to get a wider range information that may represent a belongingness, non-belongingness and hesitancy degrees. Several applications were improved and manipulated the concept of IFS in decision making [16 , 47].

Bipolar fuzzy sets and relations (BFS) are introduced by [43]. BFS is a generalization of fuzzy sets [24] and IFS [23], whose membership degree range lies in [-1, 0] × [0, 1]. Bipolar equilibrium and non-equilibrium have been widely utilized in medical diagnosis, mental health, and bipolar disorder, decision making, optimization, classification, dynamic, relation and logic terms applications [57 –63]. Zhang has the biggest impact to erect the passage from static to dynamic, linear to non-linear and closed to open world of equilibria in dynamic neurobiological Modeling and bipolar disorders [62]. Zhang introduced the concepts of bipolar dynamic logic and fuzzy logic to show the unification of quantum fields with neural biology networks, bipolar disorder with equilibrium, and especial hypothesis with brain and behavior [58].

Bipolar fuzzy information is two sides in coordination and decision making. For instance, love and hate, male and female, competition and cooperation, feedback and feedforward. Therefore, many applications and studies on bipolar fuzzy set theory are rapidly and increasingly raised in real-life problems [25 , 43]. The coexistence, equilibrium, and harmony between two-sided in Yin and Yang food system are deemed a tone for physical health and mental for a person [43].

Generalizing uncertainty set from real-valued interval [0, 1] to the realm of complex numbers (unit disk) have been addressed [3 , 51]. Complex fuzzy set (CFS) is a fuzzy set described by membership function as [12]: $μ (x) : U \to {a | a \in C, | a | ⩽ 1} .$

The innovation of CFS appeared in further dimension membership. The CFS reduces to traditional fuzzy set without the phase membershipω (x) [12]. The idea of generalizing the range of fuzzy set to a wider range of complex fuzzy set lies in its ability to denote the semantics of uncertainty and periodicity information at the same time. They were identifying the phase of degrees to distinguish the different meanings in different phases or levels for similar data/degrees. The uncertainty and periodicity semantics can be denoted by amplitude r (x) and phase r (w) terms of the complex numbers r (x) e^iw(x). These terms have the ability to be fuzzy sets [9]. The significance of phase feature of complex-valued membership function separates between the work of [12] and [22]. The complex fuzzy membership grade can be represented in polar form and Cartesian form with two fuzzy components [9]. Tamir and Kandel [8] developed an axiomatic for propositional complex fuzzy logic, which settles the greatest restrictions of the complex fuzzy logic theory. Some restrictions and limitations in the concept of CFS and its resolution lead to develop an axiomatic for propositional complex fuzzy logic [8 –11].

The range of numerous uncertainty sets extended to the realm of the complex number through complex intuitionistic Atanassov’s fuzzy sets for instance (CIFS) [3, 5]. CIFS stretched the range of both membership and non-membership function to the complex plane. Alkouri and Saleh [5] studied the basic theoretical operations and relations between two CIFSs with suitable application in decision-making problems. Garg and Rani [21] generalized the concept of CIFSs to a higher stage and illustrated a good multi-criteria decision making in the complex realm. Also, Rani and Garg [13] introduced some distance measure of CIFS and their application in the decision-making process. In 2017, a Complex neutrosophic set [26] came as a generalization of the novel concept of neutrosophic set [15] which came to overcome the problem in handling imprecise, indeterminate, inconsistent, and incomplete information of periodic nature that has appeared in CFS and CIFS.

Many studies gave prospective tools such as bipolar fuzzy concept lattice and its properties in the complex vague plane which they characterize bipolar information, independently when confronting to vague set [54 –56]. Singh [54] mention that to give an accurate representation of bipolar information in the dynamic or complex data sets is computationally a hard mission. Singh, [36] proposed the bipolar complex fuzzy (BCFL) concept lattice with its application to resolve this issue. He studied the problem that calculating the periodic variation in bipolar information at the given phase of time and proposed three methods and illustrative examples for a suitable representation of bipolar complex data set. Singh represented the phase term in the complex fuzzy set as a real-valued component. Some questions are rised after Singh’s paper [36]: (1) what is the proper mathematical structure that has an ability to convey bipolarity of periodic variation in bipolar information (2) What are the benefits of manipulating the phase term as a bipolar fuzzy set to be represented in [-1, 0] × [0, 1] (3) How the optimal-desired solution can be defind for MADM problems that carry this type of information (4) What is a suitable application that may be represented by this information in real life.

This research aims to formalize a mathematical structure, its properties, operators and representing this structure in a suitable and effective application by using bipolar fuzzy set in the complex number realm. An interpretation of bipolar complex fuzzy positive and negative membership grade is obtained, in which the bipolar complex fuzzy membership grade can be represented with two bipolar fuzzy components. Problem that measuring the bipolarity of periodic variation in bipolar information at several types of phase, which it is conveys bipolar information as positive and negative values [-1, 0] × [0, 1], is also addressed differing from the work of Singh, P. k. [36], where he represented it in a real-valued (0, 2π] (see explicit example). We are presenting a general definition of the bipolar complex fuzzy set to represent bipolarity of uncertainty and bipolarity of periodicity semantics simultaneously by applying the range of bipolar fuzzy set in complex geometry and identifying the phase of degrees as a bipolar fuzzy set “with positive and negative pole” to distinguish the different meanings in different phases or levels for similar data/degrees. The bipolarity of uncertainty semantics can be represented in the form of amplitude terms as r⁺ (x) and r^- (x) and the bipolarity of periodicity semantics on the form of phase terms as w⁺ (x) and w^- (x) of complex numbers r⁺ (x) e^iαw⁺(x) and r^- (x) e^{iαw^-(x)} for positive and negative membership functions, respectively.

Undoubtedly, to solve a problem in decision making, we depend on the expert’s opinion. Experts will agree and disagree with an object affected by some factors. Depends on the expert’s opinion, each object and factor are represented by BFS. So, In BCFS, the object will be represented by amplitude terms and the factor will be represented by using the phase terms. Also, both object and factor will convey positive (agree) and negative (disagree) values of the expert’s opinion. As an example, Experts agreed 60% and 20% for the selection of food types and one of the factors “environment or daily activities or degree of spiritual awareness and transformation”, respectively, whereas 30% and 40% disagreed in food types and one of the factors “environment or daily activities or degree of spiritual awareness and transformation, respectively. To characterize this type of information, this research presents the properties of the bipolar complex fuzzy set. The phase term conveys the values of factor may affecting the amplitude term. Since this factor is represented by BFS, these values must have positive and negative values (See explicit example).

Explicit example. Relating and integral values of the Yin and Yang system are utilized to define the dynamic nature of the universe. The ideologies of Yin (negative pole) and Yang (positive pole) do not occur in this life without each other. Regarding the Chinese believed, all universe objects can be observed from the viewpoint of Yin and Yang elements [30]. Many factors with perspective of bipolarity have affected simultaneously with types of food (Yin and Yang) to have a balanced body similar to our daily activities (sitting still is Yin, exercise is Yang), our environment (a cold-weather climate and a sleepy country town is more Yin; a hotter climate and busy city is Yang), and our degree of spiritual awareness and transformation. Gabriel Cousens [30] presented that “the degree of spiritual awareness and transformation affects how much our mind is shifted by the yin and yang energy of foods in a somewhat different way than the other factors affecting yin and yang”. These types of information convey bipolarity of uncertainty (food types) and the bipolarity of periodicity (daily activities, environment, or degree of spiritual awareness and transformation). To represent this type of data we need a proper set that able to handle two variables at same time used to find the optimal body balance in the Yin and Yang food system. By incorporating the bipolarity of periodic factors information to BFS information, we are able to apply the BFS in complex geometry (called later BCFS).

We used the phase term in the polar form of a complex number to signify the bipolarity information with their bipolarity of periodicity in order to represent affected factors for each element (section 3). Balancing body by using Yin and Yang food system is presented in section 4 as an application in real daily life. Eventually, we present some basic mathematical operations and their properties under the concept of BCFS (see section 5).

2 Preliminaries

This section delivers a brief outline of the important literature. We recall the core operative definitions, theorems, and some results of complex fuzzy set and bipolar fuzzy set. Also, the most related background and importance to this paper are briefly summarized.

Definition 2.1 [24] A fuzzy set A in a universe of discourse X is characterized by a membership function μ_A (x) that takes values in the interval [0, 1] .

Zadeh [24] has defined the following basic operators: let A ={ (x, μ_A (x) : x ∈ X) } , B = { (x, μ_B (x) : x ∈ X) }. Then:

The complement of A is given as, A^c ={ (x, 1 - μ_A) : x ∈ X }

The union between two sets is given as A∪ B = { (x, μ_A ◊ μ_B) : x ∈ X }

The intersection between two sets is given as A∩ B = { (x, μ_A * μ_B : x ∈ X) }

Here, ◊ and * denote the s-norm and t-norm operators, respectively.

Let CFS (X) be a set of all complex fuzzy sets in X. Then we have

Definition 2.2 [12] A complex fuzzy set A, defined on a universe of discourse X, is characterized by a membership function μ_A (x), that assigns to any element x ∈ X a complex-valued grade of membership in A. By definition, the values of μ_A (x), may receive all lying within the unit circle in the complex plane, and are thus of the form μ_A (x) = r_A (x) e ^i ω_A (x), where $i = \sqrt{- 1}$ , each of r_A (x) and ω_A (x) are both real-valued, and r_A (x) ∈ [0, 1]. The CFS A may be represented as the set of ordered pairs A ={ (x, μ_A (x)) : x ∈ X }.

Definition 2.3 [12]. A complex fuzzy complement of A may be represented as follows: $\begin{matrix} \bar{A} & = {(x, μ_{\bar{A}} (x)) : x \in X} \\ = {(x, r_{\bar{A}} (x) \cdot e^{i ω_{\bar{A}} (x)}) : x \in X}, \end{matrix}$ where $r_{\bar{A}} (x) = 1 - r_{A} (x)$ and $ω_{\bar{A}} (x) = ω_{A} (x) = 2 π - ω_{A} (x) = π + ω_{A} (x)$ .

Definition 2.4. [19] Let A and B be two complex fuzzy sets on X, $μ_{A} (x) = r_{A} (x) \cdot e^{i {arg}_{A}^{(x)}}$ andμ_B (x) = r_B (x) · e^iarg_B(x) their membership functions, respectively. We say that A is greater than B, denoted by A ⊇ B or B ⊆ A, if for any x ∈ X, r_A (x) ⩽ r_B (x), and arg _A (x) ⩽ arg _B (x).

Definition 2.5 [12]. Let A and B be two complex fuzzy sets on X, with complex-valued membership function μ_A (x) and μ_B (x). The complex fuzzy union of A and B denoted A ∪ B, is specified by a function $\begin{matrix} u : {a | a \in C, | a | ⩽ 1} * {b | b \in C, | b | ⩽ 1} \\ \to {d | d \in C, | d | ⩽ 1} . \end{matrix}$

u assigns a complex value, u (μ_A (x) , μ_B (x)) = μ_A∪B (x) to all x in X.

The complex fuzzy union function u must satisfy at least the following axiomatic requirements, for any a, b, c, d∈ { x|x ∈ C, |x| ⩽ 1 }

Axiom 1 (boundary conditions), u (a, 0) = a.

Axiom 2 (monotonicity): |b| ⩽ |d| implies |u (a, b) | ⩽ |u (a, d) |.

Axiom 3 (commutativity): u (a, b) = u(b, a).

Axiom 4 (associativity): u (a, u(b, d)) = u(u(a, b), d).

Axiom 5 (continuity): u is a continuous function.

Axiom 6 (superidempotency):|u (a, a) | > |a|.

Axiom 7 (strict monotonicity):|a| ⩽ |c| and|d| ⇒ |u (a, b) | ⩽ |u (c, d) |.

Definition 2.5 [12]. let A and B be two complex fuzzy sets on X, with complex-valued membership function μ_A (x) and μ_B (x). The complex fuzzy union of A and B denoted A ∩ B, is specified by a function $\begin{matrix} i : {a | a \in C, | a | ⩽ 1} * {b | b \in C, | b | ⩽ 1} \\ \to {d | d \in C, | d | ⩽ 1} \end{matrix}$

i assigns a complex value, u (μ_A (x) , μ_B (x)) = μ_A∩B (x) to all x in X.

The complex fuzzy intersection function i must satisfy at least the following axiomatic requirements, for any a, b, c, d∈ { x|x ∈ C, |x| ⩽ 1 }

Axiom 1 (boundary conditions): |b| = 1, if |i (a, b) | = |a|.

Axiom 2 (monotonicity): |b| ⩽ |d|implies |i (a, b) | ⩽ |i (a, d) |.

Axiom 3 (commutativity): i(a, b) = i(b, a).

Axiom 4 (associativity): i(a, i(b, d)) = i(i(a, b), d).

Axiom 5 (continuity): i is a continuous function.

Axiom 6 (superidempotency):|i (a, a) | < |a|.

Axiom 7 (strict monotonicity):|a| ⩽ |c| and |b| ⩽ |d| ⇒ |i (a, b) | ⩽ |i (c, d) |.

Table 2.1 shows some examples of s-norms and t-norms under CFS.

Table 2.1
Some Examples of s-norms and t-norm

For any two CFSs

A ={ (x, r_A (x) e^iαω_A(x)) : x ∈ X } and

B ={ (x, r_B (x) e^iαω_B(x)) : x ∈ X } in a universe

of discourse U,

s-norm t-norm

•Basic s-norm: $\begin{matrix} A \cup B = \\ {〈 x, max (r_{A} (x), r_{B} (x)), \\ α max (ω_{A} (x), ω_{B} (x)) 〉 : x \in X} \end{matrix}$ •Basic t-norm: $\begin{matrix} A \cap B = \\ {〈 x, min (r_{A} (x), r_{B} (x)), \\ α min (ω_{A} (x), ω_{B} (x)) 〉 : x \in X} . \end{matrix}$

•Yager S-norm: $\begin{matrix} s (({ae}^{a^{'}}), ({be}^{b^{'}})) = \\ s (s_{ϖ}^{r} (a, b), s_{ϖ}^{ω} (a^{'}, b^{'})), \end{matrix}$ where $\begin{matrix} s_{ϖ}^{r} (a, b) = \\ min (1, {(a^{ϖ} + b^{ϖ})}^{1 / ϖ}) \end{matrix}$ and $\begin{matrix} s_{ϖ}^{ω} (a^{'}, b^{'}) = \\ min (1, {(a^{' ϖ} + b^{' ϖ})}^{1 / ϖ}) \end{matrix}$ with ϖ ∈ (0, ∞). •Yager T-norm: $\begin{matrix} t (({ae}^{a^{'}}), ({be}^{b^{'}})) = \\ t (t_{ϖ}^{r} (a, b), t_{ϖ}^{ω} (a^{'}, b^{'})), \end{matrix}$ where $\begin{matrix} t_{ϖ}^{r} (a, b) = \\ 1 - min (1, {((1 - a)^{ϖ} + (1 - b)^{ϖ})}^{1 / ϖ}), \end{matrix}$ and $\begin{matrix} t_{ϖ}^{ω} (a^{'}, b^{'}) = \\ min (1, {((a^{'})^{ϖ} + (b^{'})^{ϖ})}^{1 / ϖ}) \end{matrix}$ with ϖ ∈ (0, ∞).

For any two CFSs
•Basic s-norm: $\begin{matrix} A \cup B = \\ {〈 x, max (r_{A} (x), r_{B} (x)), \\ α max (ω_{A} (x), ω_{B} (x)) 〉 : x \in X} \end{matrix}$	•Basic t-norm: $\begin{matrix} A \cap B = \\ {〈 x, min (r_{A} (x), r_{B} (x)), \\ α min (ω_{A} (x), ω_{B} (x)) 〉 : x \in X} . \end{matrix}$
•Yager S-norm: $\begin{matrix} s (({ae}^{a^{'}}), ({be}^{b^{'}})) = \\ s (s_{ϖ}^{r} (a, b), s_{ϖ}^{ω} (a^{'}, b^{'})), \end{matrix}$ where $\begin{matrix} s_{ϖ}^{r} (a, b) = \\ min (1, {(a^{ϖ} + b^{ϖ})}^{1 / ϖ}) \end{matrix}$ and $\begin{matrix} s_{ϖ}^{ω} (a^{'}, b^{'}) = \\ min (1, {(a^{' ϖ} + b^{' ϖ})}^{1 / ϖ}) \end{matrix}$ with ϖ ∈ (0, ∞).	•Yager T-norm: $\begin{matrix} t (({ae}^{a^{'}}), ({be}^{b^{'}})) = \\ t (t_{ϖ}^{r} (a, b), t_{ϖ}^{ω} (a^{'}, b^{'})), \end{matrix}$ where $\begin{matrix} t_{ϖ}^{r} (a, b) = \\ 1 - min (1, {((1 - a)^{ϖ} + (1 - b)^{ϖ})}^{1 / ϖ}), \end{matrix}$ and $\begin{matrix} t_{ϖ}^{ω} (a^{'}, b^{'}) = \\ min (1, {((a^{'})^{ϖ} + (b^{'})^{ϖ})}^{1 / ϖ}) \end{matrix}$ with ϖ ∈ (0, ∞).

Definition 2.6.. A distance of complex fuzzy sets is a function ρ : CFS (X) × CFS (X) → [0, 1] such that for any A, B, C ∈ CFS (X)

(D1) ρ (A, B) ⩾0, ρ (A, B) =0 iff A = B,

(D2) ρ (A, B) = ρ (B, A) ,

(D3) ρ (A, B) ⩽ ρ (A, C) + ρ (C, B) .

In the following, Zhang et al. [20] introduced a distance function d, defined as follows: $\begin{matrix} d (A, B) = max (sup_{x \in U} | r_{A} (x) - r_{B} (x) |, . \\ \frac{1}{2 π} sup_{x \in U} | {arg}_{A} (x) - {arg}_{B} (x) |) \end{matrix}$

Definition 2.7. [49]. Let d be a mapping d : IFS (X) × IFS (X) → [0, 1] if d(A, B) satisfies the following properties:

(DP1) 0 ⩽ d (A, B) ⩽1.

(DP2) d (A, B) =0 if and only if A = B.

(DP3) d(A, B) = d(B, A).

(DP4) If A ⊂ B ⊂ C and A, B, C ∈ IFS (U) then d (A, C) > d (A, B) , d (A, C) > d (B, C),

i.e. d(A, B) is a distance measure between IFS A and B.

Song et al. [49] introduced the function d : IFS (X) × IFS (X) → [0, 1] between two IFSs A and B. defined as follows: $d (A, B) = \frac{1}{\sum_{i = 1}^{n} w_{i}} [\sum_{i = 1}^{n} w_{i} [(α \cdot | μ_{A} (x_{i}) - μ_{B} (x_{i}) | +$ $β \cdot | γ_{A} (x_{i}) - γ_{B} (x_{i}) | +$ $σ \cdot max (| μ_{A} (x_{i}) - μ_{B} (x_{i}) |, | γ_{A} (x_{i}) - γ_{B} (x_{i}) |)]],$ where α, β and σ ∈[0, 1], α + β + σ = 1 and w_i ∈ [0, 1], i∈ { 1, 2, . . . , m }.

Definition 2.8. [43]. Let X be the universe of discourse. The canonical representation of bipolar fuzzy sets A in X has the following shape: $A = {(x, μ_{A}^{P} (x), μ_{A}^{N} (x)) : x \in X},$ where $\begin{matrix} μ_{A}^{P} (x) : X \to [0, 1] \\ μ_{A}^{N} (x) : X \to [- 1, 0] \end{matrix}$

The positive membership function $μ_{A}^{P} (x)$ denotes the satisfaction degree of an element x to the property corresponding to a bipolar fuzzy set A, and The negative membership function $μ_{A}^{N} (x)$ denotes the satisfaction degree of an element x to some implicit counter property to A.

Suppose A and B are two bipolar fuzzy sets expressed as $A = {(x, μ_{A}^{P} (x), μ_{A}^{N} (x)) : x \in X}$ , and $B = {(x, μ_{B}^{P} (x), μ_{B}^{N} (x)) : x \in X}$ , the set operations of the bipolar fuzzy set are defined as follows:

$\begin{matrix} 1) A \cup B = {(x, μ_{A \cup B}^{P} (x), μ_{A \cup B}^{N} (x)) : x \in X} \\ μ_{A \cup B}^{P} (x) = max {μ_{A}^{P} (x), μ_{B}^{P} (x)} \\ μ_{A \cup B}^{N} (x) = min {μ_{A}^{N} (x), μ_{B}^{N} (x)} . \end{matrix}$

$\begin{matrix} 2) A \cap B = {(x, μ_{A \cap B}^{P} (x), μ_{A \cap B}^{N} (x)) : x \in X} \\ μ_{A \cap B}^{P} (x) = min {μ_{A}^{P} (x), μ_{B}^{P} (x)} \\ μ_{A \cap B}^{N} (x) = max {μ_{A}^{N} (x), μ_{B}^{N} (x)} . \end{matrix}$

$\begin{matrix} 3) A^{c} = {(x, μ_{A^{c}}^{P} (x), μ_{A^{c}}^{N} (x)) : x \in X} \\ μ_{A^{c}}^{P} (x) = 1 - μ_{A}^{P} (x) \\ μ_{A^{c}}^{N} (x) = - 1 - μ_{A}^{N} (x) . \end{matrix}$

3 Bipolar complex fuzzy set and its illustration

The phase term of complex numbers is considered as an effective strategy to introduce the concept of BCFS. The phase term is employed to extend the range-valued membership functions. This extension has used to represent the problems with uncertainty and periodicity simultaneously. Complex numbers are used in the description of the facts of our lives and they were used almost in all fields of physics (e.g in electricity, thermodynamics, and theory of relativity) [33]. For instance, the signals that change periodically are expressed in terms of a sine or cosine or a group of them together (i.e. in complex numbers, e^iθ = cos θ + i sin θ). However, adding a polarity perspective, we may represent signals as BCFS for identifying a particular signal of interest out of a large number of signals received by a TV, radio or digital receiver. Another application of complex numbers is a power factor correction, which uses lots of complex numbers and phase vectors [33]. We may use BCFS by adding a polarity perspective to represent the power factors to find out the optimal device to reduce power bills and save our money by using a proper method.

We extend the range of amplitude and phase terms of BFS to convey BCFS information, not only bipolar fuzzy information as presented by Zhang [43]. The new concept of BCFS can be employed to represent the uncertainty and periodicity problems of BF information in complex geometry simultaneously, Figure 3.1. We are using a polar form (r · e^iθ) to show the advantage of the nature of periodic that appears in the complex-valued membership functions. See Table 3.1.

Fig. 3.1

(a) Phase terms in BCFS, (b) Phase term in BFS.

Table 3.1

A comparative study among different uncertainty sets (FS, IFS, BFS, CFS, CIFS, and BCFL) with BCFS

	FS	IFS	BFS	CFS	CIFS	BCFL	BCFS
Domain	Universe of discourse	Universe of discourse	Universe of discourse	Universe of discourse	Universe of discourse	Universe of discourse	Universe of discourse
Co-domain	Single-valued in [0, 1]	Two valued in [0, 1]	values in [-1, 0] × [0, 1]	One value in [0, 1] e^iα[0, 1]	Two values in [0, 1] e^iα[0, 1]	values in [-1, 0] × [0, 1] e^i(0, 2π]	Values in [0, 1] e^iα[0,1] and [-1, 0] e^iα[-1,0]
Uncertainty measurement	Yes	Yes	Yes	Yes	Yes	yes	Yes
Fluctuation measurement	Yes	Yes	Yes	Yes	Yes	yes	Yes
Unit disk	No	No	No	Yes	Yes	yes	Yes
Periodic measurement	No	No	No	Yes	Yes	yes	Yes
Positive Amplitude term	Single valued in [0, 1]	Two valued in [0, 1]	Single valued in [0, 1]	Single valued in [0, 1]	Two values in [0, 1]	Single valued in [0, 1]	Single valued in [0, 1]
Negative amplitude term	No	NO	Single valued in [-1, 0]	No	No	Single valued in [-1, 0]	Single valued in [-1, 0]
Positive phase term	No	No	No	Single valued in [0, 1]	Two values in [0, 1]	Real valued in (0, 2π]	Single valued in [0, 1]
Negative phase term	No	No	No	No	No	Real valued in (0, 2π]	Single valued in [-1, 0]
Ability to convey bipolar information	No	No	Yes, for amplitude term.	No	No	Yes, for amplitude term.	Yes, for both amplitude and phase terms.

Definition 3.1 A Bipolar complex fuzzy set A, is characterized by positive $μ_{A}^{+} (x)$ and negative $μ_{A}^{-} (x)$ membership functions, that assign to any element x on a universe of discourse X. A complex-valued grade of both positive and negative memberships in A. By definition, the values of $μ_{A}^{+} (x)$ and $μ_{A}^{-} (x)$ may receive all values lies within the unit disks in complex plane and are of the form $μ_{A}^{+} (x) = r_{A}^{+} (x) \cdot e^{i α ω_{A}^{+} (x)}$ for positive membership function in A and $μ_{A}^{-} (x) = r_{A}^{-} (x) \cdot e^{i α ω_{A}^{-} (x)}$ for negative membership function in A, where $i = \sqrt{- 1}$ , each of $r_{A}^{+} (x)$ and $ω_{A}^{+} (x)$ belong to [0, 1] and $r_{A}^{-} (x) and$ $ω_{A}^{+} (x)$ are belong to [-1, 0], We represent the CBFS A as $A = {〈 x, μ_{A}^{+} (x), μ_{A}^{-} (x) 〉 : x \in X}$ , where $μ_{A}^{+} (x) : U \to [0, 1] e^{i α [0, 1]} and$ $μ_{A}^{-} (x) : U \to [- 1, 0] e^{i α [- 1, 0]},$

The scaling factor α, α ∈ (0, 2π], is used to confine the interpretation of the phases within the unit disk and the interval (0, 2π], and the unit disk. Therefore, the phase terms may denote the BF information, and phase terms values in this demonstration case belong to [0, 1] and satisfied BFS constraint. So, α in this research will always be considered equal to 2π.

Remark: In a BCFS $\begin{matrix} A = {x, μ_{A}^{+} (x) = r^{+} (x) e^{i α ω^{+} (x)}, \\ μ_{A}^{-} (x) = r^{-} (x) e^{i α ω^{-} (x)} 〉 : x \in X} . \end{matrix}$

(1) If both r⁺ (x) and r^- (x) equal to 0, then it is the circumstances that x is observed as neutral having no positive or negative satisfaction for A with affected by any bipolar phase value ω⁺ (x) and ω^- (x).

(2) If r⁺ (x) ≠0 and r^- (x) =0, then it is the circumstances that x is observed as having only positive satisfaction for A, with respect to affected positive phase value ω⁺ (x).

(3) If r⁺ (x) =0 and r^- (x) ≠0, then it is the circumstances that x does not satisfy the property of A, but somewhat satisfies the counter property of A, with respect to affected negative phase value ω^- (x).

(4) If r⁺ (x) ≠0 and r^- (x) ≠0, then the membership function of the property overlaps that of its counter property over some portion of the domain, with respect to affected positive and negative phase values ω⁺ (x) and ω^- (x).

In contrast to the bipolar complex fuzzy lattice presented by Singh [36], the phase term takes into account both positive and negative membership functions. BCFS generates two additional phase terms allowing wider range of values of irrelevant to the corresponding property, satisfies the property and the implicit counter-property to represent the uncertainty and periodicity semantics simultaneously. In other words, the phase term of complex numbers represents the bipolarity of periodicity semantic, and the amplitude term of complex numbers represents the bipolarity of uncertainty semantic. Undoubtedly, the main indication of the present concepts is the nature of periodic in the complex numbers.

Theorem 3.1. Some bipolar fuzzy sets can be represented as a bipolar complex fuzzy set.

Proof. Let us assume that the ordinary BFS (A) is characterized by $η_{A}^{+} (x)$ , positive membership function, and $η_{A}^{-} (x)$ , negative membership function. The transformation of A to CBFS is done by setting $r_{A}^{+} (x) = η_{A}^{+} (x)$ , $r_{A}^{-} (x) = η_{A}^{-} (x)$ and for all x, the phase terms equal to zero.

We conclude that without the phase term, BCFS is reduced to conventional BFS. Therefore, BCFS theory is an extension of the innovative BFS by asserting that, at least in some cases, a second dimension is required to be added to the expression of positive and negative membership functions. However, the second dimension does not change the basic concept of BFS.

Definition 3.2. If A and B are BCFSs in a universe of discourse X, where $A = {〈 x, r_{A}^{+} (x) \cdot e^{i α ω_{A}^{+} (x)}, r_{A}^{-} (x) \cdot e^{i α ω_{A}^{-} (x)} 〉 : x \in X}$ and $B = {〈 x, r_{B}^{+} (x) \cdot e^{i α ω_{B}^{+} (x)}, r_{B}^{+} (x) \cdot e^{i α ω_{B}^{-} (x)} 〉 : x \in X}$ , then

A ⊂ B if and only if $r_{A}^{+} (x) < r_{B}^{+} (x) and r_{A}^{-} (x) > r_{B}^{-} (x)$ , for amplitude terms and the phase terms $ω_{A}^{+} (x) < ω_{B}^{+} (x)$ and $ω_{A}^{-} (x) > ω_{B}^{-} (x)$ , for all x ∈ X.

A = B if and only if $r_{A}^{+} (x) = r_{B}^{+} (x) and r_{A}^{-} (x) = r_{B}^{-} (x)$ , for amplitude terms and the phase terms $ω_{A}^{+} (x) = ω_{B}^{+} (x)$ and $ω_{A}^{-} (x) = ω_{B}^{-} (x)$ , for all x ∈ X.

Let BCFS (X) be the set of all bipolar complex fuzzy sets on X.

An application of BCFS is given to demonstrate the way of employing both components of positive and negative poles to convey bipolar fuzzy information in the form of amplitude and phase terms. Section 4. We used one BCFS to predict/select the optimal nutrition program by using Yin and Yang food system. Instead of using two BFSs to represent the optimal food types and optimal activities or environment or degree of spiritual awareness and transformation, we use the benefit of using BCFS appears in representing information of two dimensions for one object (person) simultaneously to get the optimal nutrition program. We can save expert’s time by creating one set (BCFS) instead of two sets (BFSs) and dropping the number of mathematical calculations and examination of numerous operators in joining two BFSs to announce the best solution as well.

The relation between the amplitude terms and phase terms clearly appears/emerges in Application 4.1. Each object (person) has several phases/levels of evolution of which the relation can represent the interpretation of the BCF proposition very clearly. When the expert/committee represents the information mathematically for selecting the wanted nutrition program, they can directly identifying the nutrition program evolve as a phase term. Further advantage of BCFS is representing the information involving human decision created on the positive and negative side (bipolar judgment thinking). For example male and female, sunrise and sunset, common and conflict interest, hostility and friendship [43].

Figure 3.1. The characteristic and possible position of phase terms, i.e. how the phase terms have the ability to convey and represent a positive and negative membership value in [-1, 0] × [0, 1](bipolar fuzzy information) in complex form. Color gradation has used from the lighter to the darker when the positive phase values changing inside the unit disk and from the darker to lighter when the negative phase term values changing inside the unit disk. Also, the length of the line represents the amplitude values for both positive and negative poles in [-1, 0] × [0, 1] as in the conventional BFS.

4 Distance measure and application on bipolar complex fuzzy sets

The idea of distance is used as a difference between sets or a measure of similarity. The distance measure is considered as the main term because the ‘spatial distribution’ of sets has fixed with respect to the selected reference set [50]. Therefore, in momentous real-life application, such as reasoning and multi-attributes decision making are using distance measure term [35 , 50]. All distance measures among uncertainty sets treating with real-valued parameters. However fuzzy set and bipolar fuzzy set only represented the uncertainty semantic in term of the distance function. Complex-valued parameters in BCFS are used in the distance measures form and have uncertainty and periodicity semantics, where the uncertainty semantics is equivalent to the one in classical BFS. Here we present a distance measure that may involve both positive and negative uncertainty and periodic semantics in one formula to simplify and shorting BCFS expression. We demonstrate MADM problems in BCF Fields with bipolarity of periodic factors. See application 4.1.

We generalize the distance measure in Song et al. [49] to be imposed on BCFSs for implementing a model selects the optimal product based on two dimensions characterized by multiple attributes. The bipolarity of the periodic dimension represents the pivotal influence for considering an appropriate decision.

Definition 4.1 A distance function of bipolar complex fuzzy sets is given as follows

d : BCFS (X) × BCFS (X) → [0, 1], such that for any A, B and C ∈BCFS (X) , satisfies the following constrictions:

(D1) 0 ⩽ d (A, B) ⩽1.

(D2) d (A, B) =0 if and only if A = B.

(D3) d(A, B) = d(B, A).

(D4) If A ⊂ B ⊂ C and A, B, C ∈ CBFS (X) then d (A, C) > d (A, B), d (A, C) > d (B, C).

We present d : BCFS (X) × BCFS (X) → [0, 1] as a function between two sets of BCFSs, say A and B. formed as follows:

$\begin{matrix} d (A, B) = \frac{1}{2 \sum_{i = 1}^{n} w_{i}} [\sum_{i = 1}^{n} w_{i} [(α_{1} \cdot | r_{A}^{+} (x_{i}) - r_{B}^{+} (x_{i}) | + \\ β_{1} \cdot | r_{A}^{-} (x_{i}) - r_{B}^{-} (x_{i}) | + \end{matrix}$ $σ_{1} \cdot max (| r_{A}^{+} (x_{i}) - r_{B}^{+} (x_{i}) |, | r_{A}^{-} (x_{i}) - r_{B}^{-} (x_{i}) |) +$ $\frac{1}{2 π} (α_{2} . | ω_{A}^{+} (x_{i}) - ω_{B}^{+} (x_{i}) | + β_{2} | ω_{A}^{-} (x_{i}) - ω_{B}^{-} (x_{i}) | +$ $σ_{2} \cdot max (| ω_{A}^{+} (x_{i}) - ω_{B}^{+} (x_{i}) |, | ω_{A}^{-} (x_{i}) - ω_{B}^{-} (x_{i}) |))]] .$ (4.1)

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

Theorem 4.1 The function d (A, B) presented in Equation (4.1) is a distance measure of bipolar complex fuzzy set between two sets A and B under CBFSs in the universe of discourse X.

Proof. (1) By using BCFS definition, we have for all x_i in X, where i = 1, 2, . . . , n, each of $r_{A}^{+} (x_{i}), r_{B}^{+} (x_{i})$ , $r_{A}^{-} (x_{i}) and r_{B}^{-} (x_{i})$ lies in [0, 1] and [–1.0], respectively. Also, by letting α = 2π, each $ω_{A}^{+} (x_{i}), ω_{B}^{+} (x_{i})$ , $ω_{A}^{-} (x_{i}) and ω_{B}^{-} (x_{i})$ lies in [0, 1] and [-1, 0]. So it is easy to conclude that each of the following lies in [0, 1]: $| r_{A}^{+} (x_{i}) - r_{B}^{+} (x_{i}) |,$ and $| r_{A}^{-} (x_{i}) - r_{B}^{-} (x_{i}) | .$ and, $| ω_{A}^{+} (x_{i}) - ω_{B}^{+} (x_{i}) |,$ and $| ω_{A}^{-} (x_{i}) - ω_{B}^{-} (x_{i}) |$ lie in the interval [0, 1]. Thus, we have

$\begin{matrix} d (A, B) \\ ⩽ \frac{1}{2 \sum_{i = 1}^{n} w_{i}} [\sum_{i = 1}^{n} w_{i} [(α_{1} \cdot 1 + β_{1} \cdot 1 + σ_{1} \cdot max (1, 1)) + \\ (α_{2} \cdot 1 + β_{2} \cdot 1 + σ_{2} \cdot max (1, 1)]] \end{matrix}$ $= \frac{1}{2 \sum_{i = 1}^{n} w_{i}} [\sum_{i = 1}^{n} w_{i} [(α_{1} + β_{1} + σ_{1}) + (α_{2} + β_{2} + σ_{2})]] .$

Since α₁ + β₁ + σ₁ = 1 and α₂ + β₂ + σ₂ = 1, we have $0 ⩽ d (A, B) ⩽ \frac{1}{2 \sum_{i = 1}^{n} w_{i}} [\sum_{i = 1}^{n} 2 w_{i}] = 1 .$

(2) By Definition 3.2, it is easy to see that d (A, B) satisfies.

(3) By using Definition 3.2, we have $1 ⩾ r_{A}^{+} (x_{i}) ⩾ r_{B}^{+} (x_{i}) ⩾ r_{c}^{+} (x) ⩾ 0 and$ $2 π ⩾ ω_{A}^{+} (x_{i}) ⩾ ω_{B}^{+} (x_{i}) ⩾ ω_{C}^{+} (x_{i}) ⩾ 0 .$ $0 ⩾ r_{A}^{-} (x_{i}) ⩾ r_{B}^{-} (x_{i}) ⩾ r_{C}^{-} (x) ⩾ - 1 and$ $0 ⩾ ω_{A}^{-} (x_{i}) ⩾ ω_{B}^{-} (x_{i}) ⩾ ω_{C}^{-} (x_{i}) ⩾ - 2 π .$

Then, $\begin{matrix} | r_{A}^{+} (x_{i}) - r_{B}^{+} (x_{i}) | ⩽ | r_{A}^{+} (x_{i}) - r_{C}^{+} (x) | and \\ | ω_{A}^{+} (x_{i}) - ω_{B}^{+} (x_{i}) | ⩽ | ω_{A}^{+} (x_{i}) - ω_{C}^{+} (x_{i}) |, \\ | r_{A}^{-} (x_{i}) - r_{B}^{-} (x_{i}) | ⩾ | r_{A}^{-} (x_{i}) - r_{C}^{-} (x) | and \\ | ω_{A}^{-} (x_{i}) - ω_{B}^{-} (x_{i}) | ⩾ | ω_{A}^{-} (x_{i}) - ω_{C}^{-} (x_{i}) | . \end{matrix}$

So, $d (A, B) ⩽ d (A, C) .$

Similarly, we show d (B, C) ⩽ d (A, C).

From (1), (2), and (3), we displayed that d (A, B) is a distance measure between two sets of BCFS.

Application 4.1 is presented below by generalizing a selection model in Song et al. [49] to be covered by BCFS. Our Application aims to catch the wanted or optimal nutrition Program (NP). for a person X to choose an NP from five NPs (food and daily activities are two-dimensional problems).

Application 4.1. Assume that a nutrition company would like to construct a nutrition program (A₁) based on several expert’s opinions. Experts agreed 60% and 20% for the selection of food types and daily activity, respectively, whereas 30% and 40% disagreed in food types and daily activity, respectively. To characterize this type of information, this research presents the properties of the bipolar complex fuzzy set A₁ = (0.6e^i2π(0.2), - 0.3e^i2π(-0.4)). So, suppose the nutrition company constructs five nutrition programs based on expert’s opinions regarding four attributes (B₁: Healthy food, B₂: Purchasing cost, B₃: Applicability or Durability, B₄: Availability of NP elements) for each program. And, a person X would like to choose the optimal nutrition program for him/her, we propose the following table to find out the optimal program by using the distance measure of BCFS. Let the ideal nutrition program for person X is given by (0.7e^i2π(0.8), - 0.2e^i2π(-0.4)) regarding his ability and vision to have the daily type of foods and daily activities.

To continue our application, we proposed the following algorithm to find the optimal NP for person X.

Step 1: we recall an Ideal NP for a person X. to get an optimal NP from A_j, j = 1, 2, 3, 4, 5.

Step 2: Let d (A_j, A_I.N.P.) be a distance measure between bipolar complex fuzzy sets nutrition program A_j, and Ideal NP for person X. So, the proper model can be presented as follows: $\begin{matrix} M = {A_{j} | d (A_{j}, A_{I . N . P .}) = min (\sum_{i = 1}^{n} d (A_{1, i}, A_{I . N . P .}), \dots, \\ \sum_{i = 1}^{n} d (A_{k, i}, A_{I . N . P .})), j = 1, \dots, 5 and i = 1, \dots, 4} \end{matrix}$

Step 3: Suppose the nutrition company gives the weight for each NP and experts as follows:

Let w₁ = 0.3, w₂ = 0.2, w₃ = 0.6, w₄ = 0.8, be the weight for Attribute. Let α₁ = 0.1, β₁ = 0.4, and σ₁ = 0.5 be the weight of amplitude term, while α₂ = 0.3, β₂ = 0.3, and σ₂ = 0.4 being the weight of phase term.

Step 4: To discover the nearest NP to the ideal NP we plug the data in Table 4.1 into formula (4.1) to calculate the values between each NP (A_j, j = 1, 2, 3, 4, 5) and the Ideal NP for a person X. $\begin{matrix} d (A_{1}, A_{I . N . P .}) = \sum_{i = 1}^{4} d (A_{1, i}, A_{I . N . P .}) \\ = \frac{1}{2 \cdot \sum_{i = 1}^{4} w_{i}} [\sum_{i = 1}^{4} w_{i} [(α_{1} \cdot | r_{A_{1}}^{+} (B_{i}) - r_{A_{I . N . P}}^{+} (B_{i}) | \\ + β_{1} \cdot | r_{A}^{-} (B_{i}) - r_{A_{I . N . P}}^{-} (B_{i}) | + \\ σ_{1} \cdot max (| r_{A_{1}}^{+} (B_{i}) - r_{A_{I . N . P}}^{+} (B_{i}) |, | r_{A_{1}}^{-} (B_{i}) - r_{A_{I . N . P}}^{-} (B_{i}) |) + \\ \frac{1}{2 π} (α_{2} \cdot | ω_{A_{1}}^{+} (B_{i}) - ω_{A_{I . N . P}}^{+} (B_{i}) | + β_{2} \cdot | ω_{A_{1}}^{-} (B_{i}) - ω_{A_{I . N . P}}^{-} (B_{i}) | + \\ σ_{2} \cdot max (| ω_{A_{1}}^{+} (B_{i}) - ω_{A_{I . N . P}}^{+} (B_{i}) |, | ω_{A_{1}}^{-} (B_{i}) - ω_{A_{I . N . P}}^{-} (B_{i}) |)]] . \end{matrix}$

Table 4.1

NP Data for Four Attribute

	B ₁	B ₂	B ₃	B ₄
A ₁	$\begin{matrix} (0.7 e^{i 2 π (0.2)}, - 0.4 e i 2 π (- 0.5)) \end{matrix}$	$\begin{matrix} (0.6 e^{i 2 π (0.4)}, \\ - 0.2 e^{i 2 π (- 0.5)}) \end{matrix}$	$\begin{matrix} (0.5 e^{i 2 π (0.5)}, \\ - 0.8 e^{i 2 π (- 0.3)}) \end{matrix}$	$\begin{matrix} (0.4 e^{i 2 π (0.3)}, \\ - 0.4 e^{i 2 π (- 0.6)}) \end{matrix}$
A ₂	$\begin{matrix} (0.4 e^{i 2 π (0.4)}, \\ - 0.4 e^{i 2 π (- 0.5)}) \end{matrix}$	$\begin{matrix} (0.3 e^{i 2 π (0.5)}, \\ - 0.5 e^{i 2 π (- 0.1)}) \end{matrix}$	$\begin{matrix} (0.7 e^{i 2 π (0.4)}, \\ - 0.2 e^{i 2 π (- 0.4)}) \end{matrix}$	$\begin{matrix} (0.9 e^{i 2 π (0.1)}, \\ - 1 e^{i 2 π (- 0.5)}) \end{matrix}$
A ₃	$\begin{matrix} (0.7 e^{i 2 π (0.3)}, \\ - 0.5 e^{i 2 π (- 0.2)}) \end{matrix}$	$\begin{matrix} (0.4 e^{i 2 π (0.3)}, \\ - 0.7 e^{i 2 π (- 0.9)}) \end{matrix}$	$\begin{matrix} (0.5 e^{i 2 π (0.3)}, \\ - 0.5 e^{i 2 π (- 0.5)}) \end{matrix}$	$\begin{matrix} (0.5 e^{i 2 π (0.7)}, \\ - 0.4 e^{i 2 π (- 0.5)}) \end{matrix}$
A ₄	$\begin{matrix} (0.5 e^{i 2 π (0.8)}, \\ - 0.3 e^{i 2 π (- 0.4)}) \end{matrix}$	$\begin{matrix} (0.3 e^{i 2 π (0.1)}, \\ - 0.7 e^{i 2 π (- 0.3)}) \end{matrix}$	$\begin{matrix} (0.8 e^{i 2 π (0.7)}, \\ - 0.3 e^{i 2 π (- 0.4)}) \end{matrix}$	$\begin{matrix} (0.1 e^{i 2 π (0.7)}, \\ - 0.7 e^{i 2 π (- 0.2)}) \end{matrix}$
A ₅	$\begin{matrix} (0.3 e^{i 2 π (0.2)}, \\ - 0.7 e^{i 2 π (- 0.6)}) \end{matrix}$	$\begin{matrix} (0.6 e^{i 2 π (0.2)}, \\ - 0.4 e^{i 2 π (- 0.5)}) \end{matrix}$	$\begin{matrix} (0.6 e^{i 2 π (0.7)}, \\ - 0.3 e^{i 2 π (- 0.8)}) \end{matrix}$	$\begin{matrix} (0.3 e^{i 2 π (0.6)}, \\ - 0.7 e^{i 2 π (- 0.3)}) \end{matrix}$

For the (j = 1), the distance is calculated as follows:

$\begin{matrix} d (A_{1}, A_{I . N . P .}) = \frac{1}{2 \cdot (1.9)} ([0.3 [(0.1 \cdot | 0.7 - 0.7 | + \\ 0.4 \cdot | - 0.4 - (- 0.2) | + 0.5 \cdot max (0, 0.2) + \frac{1}{2 π} (0.3 \cdot \\ | 0.2 (2 π) - 0.8 (2 π) | + 0.3 \cdot | - 0.5 (2 π) - (- 0.4) (2 π) | + \\ 0.4 \cdot max (0.6 (2 π), 0.1 (2 π))]] + [0.2 [(0.1 \cdot | 0.6 - 0.7 | \\ + 0.4 \cdot | - 0.2 - (- 0.2) | + 0.5 \cdot max (0.1, 0) + \frac{1}{2 π} \\ (0.3 \cdot | 0.8 (2 π) - 0.4 (2 π) | + 0.3 \cdot | - 0.5 (2 π) - (- 0.4) (2 π) | + \\ 0.4 \cdot max (0.3 (2 π), 0.1 (2 π))]] + [0.6 [(0.1 \cdot | 0.5 - 0.7 | + \\ 0.4 \cdot | - 0.8 - (- 0.2) | + 0.5 \cdot max (0.2, 0.6) + \frac{1}{2 π} (0.3 \cdot \\ | 0.8 (2 π) - 0.5 (2 π) | + 0.3 \cdot | - 0.4 (2 π) - (- 0.3) (2 π) | + \\ 0.4 \cdot max (0.3 (2 π), 0.1 (2 π))]] + \end{matrix}$

$\begin{matrix} [0.8 [(0.1 \cdot | 0.4 - 0.7 | + 0.4 \cdot | - 0.4 - (- 0.2) | + \\ 0.5 \cdot max (0.3, 0.2) + \frac{1}{2 π} (0.3 \cdot | 0.8 (2 π) - 0.3 (2 π) | + 0.3 \cdot \\ | - 0.4 (2 π) - (- 0.6) (2 π) | + 0.4 \cdot max (0.5 (2 π), 0.2 (2 π))]]) \\ = \frac{1}{2 \cdot (1.9)} (0.3 (1.35) + 0.2 (0.3) + 0.6 (0.8) + 0.8 (0.67)) \\ = \frac{1}{2 \cdot (1.9)} (1.481) = 0.3897368421 . \end{matrix}$

Similarly, we compute the distance values for d (A₂, A_I.N.P.) = 0.4060526316 ., d (A₃, A_I.N.P.) =3736842105 ., and d (A₄, A_I.N.P.) =0.2463157895 ., and d (A₅, A_I.N.P.) =0.3036842105 ..

Step 5: give the decision by applying the model (4.2). Therefore, d (A₄, A_I.N.P.) =0.2463157895 is the smallest value, so A₄ is the desired nutrition program for a person X.

Note 4.1. If we changed some weights of attributes then we get different consequences. A company needs to fix suitable weights for attributes, in order to get the right consequences.

If we recognize identical food types of NP but with reformed daily exercise for the first NP, then definitely the expert will decide a new value for the first NP. Because every exercise has its affected values to the balanced body. Even though the changes are a slight improvement, people’s acceptance and appreciation may change to choose a new NP

5 Basic operations on bipolar complex fuzzy sets

We show in this section that the extension of BFS to BCFS highlights a necessity to define a suitable formula for some basic operations on the BCFSs, for instance, complement, union, and intersection. Each bipolar complex fuzzy complement, union, and intersection are defined below.

Zhang (1994) introduced the complement of BFS A. Ramot et al. [12] obtained several possible techniques for computing the membership phase of complex fuzzy complement, $ω_{\bar{A}} (x)$ . For example, $ω_{\bar{A}} (x)$ is defined as $ω_{\bar{A}} (x) = ω_{A} (x)$ , also $ω_{\bar{A}} (x)$ may be defined by the relation $ω_{\bar{A}} (x) = 2 π - ω_{A} (x) = - ω_{A} (x)$ , where Zhang et al. [20] used this relation to define the complement for the phase component, also the rotation of ω_A (x) by π radians may be a respectable manner for computing the complement of the phase term as $ω_{\bar{A}} (x) = ω_{A} (x) + π$ .

In this research, we combine Zhang’s definition [43] for complement and Ramot et al. [12] suggestions to define the complement of a BCFS. Thus, we present the complement of a bipolar complex fuzzy set as follows.

Definition 5.1. Let A be BCFS, where A is given as: $\begin{matrix} A = {〈 x, μ_{A}^{+} (x) = r_{A}^{+} (x) \cdot e^{i α (ω_{A}^{+} (x))}, \\ μ_{A}^{-} (x) = r_{A}^{-} (x) \cdot e^{i α (ω_{A}^{-} (x))} 〉 : x \in X} \end{matrix}$

Thus, the complement of A, c (A) , or $\bar{A}$ defined as $c (A) = \bar{A} = {〈 x, μ_{\bar{A}}^{+} (x), μ_{\bar{A}}^{-} (x) 〉 : x \in U},$ where $μ_{\bar{A}}^{+} (x) = [1 - r_{\bar{A}}^{+} (x)] \cdot e^{i α (1 - ω_{\bar{A}}^{+} (x))}$ $μ_{\bar{A}}^{-} (x) = [- 1 - r_{\bar{A}}^{-} (x)] \cdot e^{i α (- 1 - ω_{\bar{A}}^{-} (x))} .$

Example 5.1. Let A be a BCFS. Determine the complement of A, c (A), if

$\begin{matrix} A = {〈 x_{1}, 0.6 e^{i 1.2 π}, - 0.2 e^{- i 0.5 π} 〉, 〈 x_{2}, 0.7 e^{i 0.7 π}, - 0.4 e^{- i π} 〉, \\ 〈 x_{3}, 0.2 e^{i π}, - 0.6 e^{- i 0.3 π} 〉, 〈 x_{4}, 0.7 e^{i 0.9 π}, - 0.6 e^{- i 0.2 π} 〉} . \end{matrix}$

Solution. By applying Definition 5.1 on the BCFS A, we have

$\begin{matrix} \bar{A} = {〈 x_{1}, 0.4 e^{i 0.8 π}, - 0.8 e^{- i 1.5 π} 〉, 〈 x_{2}, 0.3 e^{i 1.3 π}, - 0.6 e^{- i π} 〉, \\ 〈 x_{3}, 0.8 e^{i π}, - 0.4 e^{- i 1.7 π} 〉, 〈 x_{4}, 0.3 e^{i 1.1 π}, - 0.4 e^{- i 1.8 π} 〉} . \end{matrix}$

Ramot et al. [12] commented that “since, r_A (x) and r_B (x) are belong to [0, 1], operators such as Max and Min can be applied to them”. Thus μ⁺ (x) and μ^- (x) are belong to [0, 1] and [–1, 0] in CBFS. We call the interpretation above to define the basic bipolar complex fuzzy union and intersection as follows.

Definition 5.2. Let A and B be two BCFSs on the universe of discourse X, with positive and negative complex-valued membership functions such as: $A = {〈 x, μ_{A}^{+} (x), μ_{A}^{-} (x) 〉 : x \in X} and$ $B = {〈 x, μ_{B}^{+} (x), μ_{B}^{-} (x) 〉 : x \in X} .$

The bipolar complex fuzzy union of A and B, denoted by A ∪ B, is specified $A \cup B = {〈 x, μ_{A \cup B}^{+} (x), μ_{A \cup B}^{-} (x) 〉 : x \in U}$ where, $μ_{A \cup B}^{+} (x) = max [r_{A}^{+} (x), r_{B}^{+} (x)] \cdot e^{i α max [ω_{A}^{+} (x), ω_{B}^{+} (x)]}$ and $μ_{A \cup B}^{-} (x) = min [r_{A}^{-} (x), r_{B}^{-} (x)] \cdot e^{i α min [ω_{A}^{-} (x), ω_{B}^{-} (x)]}$ for all x in U.

Example 5.2. Let A and B be two bipolar complex fuzzy sets in the same common universe X ={ x₁, x₂, x₃, x₄ }, suppose that $\begin{matrix} A = {〈 x_{1}, 0.2 e^{i π}, - 0.8 e^{- i 0.6 π} 〉, 〈 x_{2}, 0.7 e^{i 0.7 π}, - 0.3 e^{- i π} 〉, \\ 〈 x_{3}, 0.1 e^{i 1.5 π}, - 0.6 e^{- i 0.3 π} 〉, 〈 x_{4}, 0.6 e^{i 0.2 π}, - 0.3 e^{- i π} 〉}, and \end{matrix}$ $\begin{matrix} B = {〈 x_{1}, 0.1 e^{i 0.3 π}, - 0.7 e^{- i 1.6 π} 〉, 〈 x_{2}, 0.3 e^{i 0.7 π}, - 0.6 e^{- i 1.3 π} 〉, \\ 〈 x_{3}, 0.1 e^{i 0.4 π}, - 0.6 e^{- i 0.7 π} 〉, 〈 x_{4}, 0.1 e^{i 0.7 π}, - 0.8 e^{- i 0.3 π} 〉} . \end{matrix} .$

For the bipolar complex fuzzy union, we have $\begin{matrix} A \cup B = {〈 x_{1}, max (0.2, 0.1) e^{i max (π, 0.2 π)}, min (- 0.8, - 0.7) e^{i min (- 0.6 π, - 1.6 π)} 〉, \\ 〈 x_{2}, max (0.7, 0.3) e^{i max (0.7 π, 0.7 π)}, min (- 0.3, - 0.6) e^{i min (- π, - 1.3 π)} 〉, \\ 〈 x_{3}, max (0.1, 0.1) e^{i max (1.5 π, 0.4 π)}, min (- 0.6, - 0.6) e^{i min (- 0.3 π, - 0.7 π)} 〉, \\ 〈 x_{4}, max (0.6, 0.1) e^{i max (0.2 π, 0.7 π)}, min (- 0.3, - 0.8) e^{i min (- π, - 0.3 π)} 〉 : x \in X} \end{matrix}$ $\begin{matrix} = {〈 x_{1}, 0.2 e^{i π}, - 0.8 e^{- i 1.6 π} 〉, 〈 x_{2}, 0.7 e^{i 0.7 π}, - 0.6 e^{- i 1.3 π} 〉, \\ 〈 x_{3}, 0.1 e^{i 1.5 π}, - 0.6 e^{- i 0.7 π} 〉, 〈 x_{4}, 0.6 e^{i 0.7 π}, - 0.8 e^{- i π} 〉 : x \in X} . \end{matrix}$

Definition 5.3. Let A and B be two BCFSs on the universe of discourse X, with positive and negative complex-valued membership functions such as: $A = {〈 x, μ_{A}^{+} (x), μ_{A}^{-} (x) 〉 : x \in X} and$ $B = {〈 x, μ_{B}^{+} (x), μ_{B}^{-} (x) 〉 : x \in X} .$

The bipolar complex fuzzy intersection of A and B, denoted by A ∩ B, is specified by $A \cap B = {〈 x, μ_{A \cap B}^{+} (x), μ_{A \cap B}^{-} (x) 〉 : x \in U}$

where, $μ_{A \cap B}^{+} (x) = min [r_{A}^{+} (x), r_{B}^{+} (x)] \cdot e^{i α min [ω_{A}^{+} (x), ω_{B}^{+} (x)]}$ and

$μ_{A \cap B}^{-} (x) = max [r_{A}^{-} (x), r_{B}^{-} (x)] \cdot e^{i α max [ω_{A}^{-} (x), ω_{B}^{-} (x)]}$ for all x in X.

Example 5.3. Let A and B be two bipolar complex fuzzy sets in the same common universe X = { x₁, x₂, x₃, x₄ } . Suppose that

$\begin{matrix} A = {〈 x_{1}, 0.2 e^{i π}, - 0.8 e^{- i 0.6 π} 〉, 〈 x_{2}, 0.4 e^{i 0.5 π}, - 0.5 e^{- i 0.3 π} 〉, \\ 〈 x_{3}, 0.3 e^{i 1.7 π}, - 0.5 e^{- i 0.4 π} 〉, 〈 x_{4}, 0.5 e^{i 0.3 π}, - 0.4 e^{- i 1.1 π} 〉}, and \end{matrix}$ $\begin{matrix} B = {〈 x_{1}, 0.1 e^{i 0.3 π}, - 0.6 e^{- i 1.6 π} 〉, 〈 x_{2}, 0.6 e^{i 1.7 π}, - 0.3 e^{- i π} 〉, \\ 〈 x_{3}, 0.3 e^{i 0.6 π}, - 0.9 e^{- i 0.9 π} 〉, 〈 x_{4}, 0.3 e^{i 1.7 π}, - 0.9 e^{- i 1.3 π} 〉} . \end{matrix}$ $\begin{matrix} A \cap B = {〈 x_{1}, min (0.2, 0.1) e^{i min (π, 0.3 π)}, max (- 0.8, - 0.6) e^{i max (- 0.6 π, - 1.6 π)} 〉, \\ 〈 x_{2}, min (0.4, 0.6) e^{i min (0.5 π, 1.7 π)}, max (- 0.5, - 0.3) e^{i max (- 0.3 π, - π)} 〉, \\ 〈 x_{3}, min (0.3, 0.3) e^{i min (1.7 π, 0.6 π)}, max (- 0.5, - 0.9) e^{i max (- 0.4 π, - 0.9 π)} 〉, \\ 〈 x_{4}, min (0.5, 0.3) e^{i min (0.3 π, 1.7 π)}, max (- 0.4, - 0.9) e^{i max (- 1.1 π, - 1.3 π)} 〉 : x \in X} \end{matrix}$ $\begin{matrix} = {〈 x_{1}, 0.1 e^{i 0.3 π}, - 0.6 e^{- i 0.6 π} 〉, 〈 x_{2}, 0.4 e^{i 0.6 π}, - 0.3 e^{- i 0.3 π} 〉, \\ 〈 x_{3}, 0.3 e^{i 0.6 π}, - 0.5 e^{- i 0.4 π} 〉, 〈 x_{4}, 0.3 e^{i 0.3 π}, - 0.4 e^{- i 1.1 π} 〉 : x \in X} . \end{matrix}$

Here, we introduce some properties and De Morgan’s laws theorem under bipolar complex fuzzy theoretic operations, which are a complement, union, and intersection.

Proposition 5.1 Let A, B, and R be any three BCFSs. Then the following are true statements:

(A^c) ^c = A,

A ∪ A = A,

A ∩ A = A,

A ∪ B = B ∪ A,

A ∩ B = B ∩ A

A ∪ (B ∩ R) = (A ∪ B) ∩ (A ∪ R) ,

A ∩ (B ∪ R) = (A ∩ B) ∪ (A ∩ R) ,

A ∪ (B ∪ R) = (A ∪ B) ∪ R,

A ∩ (B ∩ R) = (A ∩ B) ∩ R .

Proof. Trivial.

Theorem 5.1 (De Morgan’s law) Let A and B be CBFSs. Then $i . {(A \cup B)}^{c} = A^{c} \cap B^{c},$ $ii . {(A \cap B)}^{c} = A^{c} \cup B^{c} .$

Proof. Trivial.

6 Conclusion

bipolar complex fuzzy set is presented in more detail comparing with Singh’s approach by employing bipolar fuzzy set to the phase term in the complex numbers. The advantages of the bipolar complex fuzzy set may be concluded by its ability to represent problems. Not only the positive features of the problems but also the negative features are affected by a factor that also conveys bipolar fuzzy information simultaneously. BCFS can be applicable in several decision-making problems to select the optimal solution as BCFS uses two variables to represent the information instead of one variable. Yin and Yang food system of BCFS in real-life application was illustrated to select the optimal nutrition program for a person X by using distance measures under BCFS. No bipolar complex interaction is defined in this paper as a limitation. However, we may have a new extension of bipolar fuzzy set theory by defining bipolar complex interaction due to background independence and definable causality, since Yin-Yang bipolarity is space-time transcendent and geometrically background-independent [64]. Further limitations of BCFS may appeared in representing a membership value for a collection of approximate descriptions of an object. This limitation can be covered by incorporating BCFS and the concept of the soft set as a future research. BCFS has no ability to represent the truth, intermediate, and falsity information as in complex neutrosophic set. So the future combination may be considered to introduce the concept of complex neutrosophic bipolar fuzzy set.

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For any two CFSs
A ={ (x, r_A (x) e^iαω_A(x)) : x ∈ X } and
B ={ (x, r_B (x) e^iαω_B(x)) : x ∈ X } in a universe
of discourse U,
s-norm	t-norm
•Basic s-norm: $\begin{matrix} A \cup B = \\ {〈 x, max (r_{A} (x), r_{B} (x)), \\ α max (ω_{A} (x), ω_{B} (x)) 〉 : x \in X} \end{matrix}$	•Basic t-norm: $\begin{matrix} A \cap B = \\ {〈 x, min (r_{A} (x), r_{B} (x)), \\ α min (ω_{A} (x), ω_{B} (x)) 〉 : x \in X} . \end{matrix}$
•Yager S-norm: $\begin{matrix} s (({ae}^{a^{'}}), ({be}^{b^{'}})) = \\ s (s_{ϖ}^{r} (a, b), s_{ϖ}^{ω} (a^{'}, b^{'})), \end{matrix}$ where $\begin{matrix} s_{ϖ}^{r} (a, b) = \\ min (1, {(a^{ϖ} + b^{ϖ})}^{1 / ϖ}) \end{matrix}$ and $\begin{matrix} s_{ϖ}^{ω} (a^{'}, b^{'}) = \\ min (1, {(a^{' ϖ} + b^{' ϖ})}^{1 / ϖ}) \end{matrix}$ with ϖ ∈ (0, ∞).	•Yager T-norm: $\begin{matrix} t (({ae}^{a^{'}}), ({be}^{b^{'}})) = \\ t (t_{ϖ}^{r} (a, b), t_{ϖ}^{ω} (a^{'}, b^{'})), \end{matrix}$ where $\begin{matrix} t_{ϖ}^{r} (a, b) = \\ 1 - min (1, {((1 - a)^{ϖ} + (1 - b)^{ϖ})}^{1 / ϖ}), \end{matrix}$ and $\begin{matrix} t_{ϖ}^{ω} (a^{'}, b^{'}) = \\ min (1, {((a^{'})^{ϖ} + (b^{'})^{ϖ})}^{1 / ϖ}) \end{matrix}$ with ϖ ∈ (0, ∞).