Bipolar Complex Fuzzy Soft Sets and Their Applications in Decision-Making

Abstract

This article introduces the notion of bipolar complex fuzzy soft set as a generalization of bipolar complex fuzzy set and soft set. Furthermore, this article contains elementary operations for bipolar complex fuzzy soft sets such as complement, union, intersection, extended intersection, and related properties. The OR and AND operations for bipolar complex fuzzy soft set are also initiated in this study. Moreover, this study contains the decision-making algorithm and real-life examples to display the success and usability of bipolar complex fuzzy soft sets. Finally, the comparative study of initiated notions with some prevailing ideas are also interpreted in this study.

1. Introduction

Complex issues in various fields such as medicine, economics, environmental sciences, engineering, and social sciences occur because of the use of classical mathematical modeling and of numerous kinds of ambiguities. As conventional mathematical techniques cannot succeed to resolve such types of complex issues, we employ certain mathematical modeling methods such as the notion of the fuzzy set (FS) [1], interval mathematics [2], rough set (RS) [3], and probability, which are famous and effective techniques for dealing with ambiguity and vagueness. However, these notions have their specific fundamental limitations and boundaries; one of their key drawbacks being the lack of parametrization tools. To overcome this, Molodtsov [4] invented the idea of the soft set (SS) which is an impressive mathematical tool to interpret existing ambiguities and vagueness. Recently, various researchers have employed the notion of SS. A new perspective regarding SSs was invented by Maji et al. [5]. Serval new operations for SSs were invented by Ali et al. [6]. Babitha et al. [7] devised SS relations and function. The SS theory (SST) and uni–int decision-making (DM) were explored by Cagman and Enginoglu [8]. A note on SST was given by Yang [9]. Herwan and Deris [10] interpreted a SS for association rules mining. Georgious and Megaritis [11] presented SST and topology. Yang and Guo [12] explored kernels and closures of SS relation and mapping. The similarity in SST was invented by Min [13]. Ma et al. [14] presented a survey of DM methods based on two classes of hybrid SS models. The SS applied to ideals in d-algebras was invented by Jun et al. [15]. Mahmood [16] conceived a new idea of bipolar SS (BSS).

FS was invented by Zadeh [1] in 1965, which is a handy technique to interpret circumstances in which the information is vague or ambiguous. FSs cope with such circumstances by providing a truth grade (TG) to which a particular element belongs to a fixed set. After the appearance of the idea of FS, a lot of scholars paid attention to improved FSs. Zimmermann [17] explored the applications of FS to mathematical programming. Adlassnig [18] utilized FS in medical diagnosis. The concept of fuzzy SS (FSS) was invented by Maji et al. [19]. Roy and Maji [20] employed FSS in DM problems. A novel algorithm of FSS based on DM was explored by Alcantud [21]. Jun et al. [22] applied the FSS to BCK/BCI-algebras. FS is a kind of vital mathematical form to signify a group of items whose boundary is imprecise. There are numerous generalizations of FS, such as intuitionistic FS [23], picture FS [24], spherical FS [25], etc. Bipolar FS (BFS) [26] is another generalization of FS that carries a positive TG (PTG), which lies in $\left[0, 1\right]$, and negative TG (NTG) which lies in $\left[-1, 0\right]$. Abdullah et al. [27] invented the concept of bipolar FSS (BFSS). Riaz and Tehrim [28] presented bipolar fuzzy (BF) soft mapping. The concept of fuzzy BSS was developed by Naz and Shabir [29]. Riaz and Tehrim [30] invented BF soft topology with DM.

The idea of complex FS (CFS), conceived by Ramot et al. [31], is the modification of FS theory whose range is expended from $\left[0, 1\right]$ to the unit disk in a complex plane. The TG of CFS carries an amplitude term and a phase term to cope with ambiguities and vagueness in two-dimensional planes. Later, Tamir et al. [32] invented a new interpretation of CFS. Ma et al. [33] invented a method for multiple periodic factor prediction problems employing CFS. The concept of complex FSS (CFSS) was interpreted by Thirunavukarasu et al. [34]. Selvachandran and Singh [35] invented interval-valued CFSS. Akram et al. [36] gave the idea of complex spherical FSSs. The concept of complex hesitant FS (CHFS) was interpreted by Mahmood et al. [37]. Chinram et al. [38] invented cosine similarity measures (SM) for CHFSs. The notion of complex dual hesitant FS (CDHFS) was established by ur Rehman et al. [39]. The complex fuzzy N-soft set (CFN-SS) was interpreted by Mahmood et al. [40]. When the decision analyst delivers the data in two dimensions along with positive and negative aspects, then the above prevailing concepts are not able to solve such kind of data. To overcome this, Mahmood and Ur Rehman [41] invented the concept of bipolar CFS (BCFS). BCFS is signified by PTG, which belongs to $\left[0, 1\right]+i \left[0, 1\right]$, and NTG which belongs to $\left[-1, 0\right]+i \left[-1, 0\right]$.

The concept of the bipolar complex fuzzy soft set (BCFSS) was interpreted by Alqaraleh et al. [42] by modifying the theory of BFSS. For this modification, they considered the BCFS described by Alkouri et al. [43]. However, Mahmood and ur Rehman [41] argued that this idea by Alkouri et al. [43] contradicts the basic definition of the polar form of the complex number as Alkouri et al. [43] took the value of amplitude term of the NTG from a negative interval, i.e., $\left[-1, 0\right]$ which is not possible, and propounded a novel definition of BCFS which is in the cartesian form instead of polar form and more accurate. Then here, a question arises: how can the utilization of such an idea, which contradicts the basic definition of a complex number, be correct? That is why, in this article, we interpret a novel concept named BCFSS, which is based on the correct notion of BCFS presented in [41]. We are also motivated by the need for BCFSS to cope with complicated and vagueness information which involved the positive and negative aspects, fuzzy information, and the parameters in a single set. The concept is inspired by the bipolarity of parameters and then the complex fuzziness of the information comes into play. We combine three notions, i.e., parameterization, complex fuzziness, and bipolarity, which simplify the modeling of a problem where these three factors are included. The BCFSS is the fusion of BCFS and SS and modified prevailing theories such as BCFS, BFSS, BFS, FS, SS, etc. The invented notion is a substantial technique to handle clumsy and problematic real-life matters and plays a significant role in the DM process, which we will demonstrate in this study with the assistance of two genuine life applications.

The rest of the article is constructed as follows: In Section 2, we review some prevailing notions including SS, BFS, CFS, and BCFS with their few properties. Section 3 contains three subsections; in Section 3.1, we introduce the idea of BCFSS, null BCFSS, absolute BCFSS; in Section 3.2, we establish a few elementary operations for BCFSSs such as complement, union, γ intersection, extended intersection, and related properties; and in Section 3.3, we describe OR and AND operations for BCFSSs. In Section 4, we present a DM procedure for solving information in the setting of BCFSSs and apply it to real-life DM problems. In Section 5, we offer a comparative study and evaluate the advantages of the invented concepts to display the benefits and accomplishments of the established BCFSSs. In Section 6 of the article, the conclusion of this study is provided.

2. Preliminaries

[4] This section contains the definitions of some prevailing notions including SS, BFS, CFS, and BCFS.

Definition 1.

Suppose $\stackrel{=}{T}$ is a fixed set, $\stackrel{=}{\mathfrak{P}}$ are parameters set, $\stackrel{=}{\wp }\subset \stackrel{=}{\mathfrak{P}}$, then the pair $(\stackrel{=}{\phi }, \stackrel{=}{\wp })$ is called SS, where $\stackrel{=}{\phi }:\stackrel{=}{\wp }\to \mathcal{P}\left(\stackrel{=}{T}\right)$, $\mathcal{P}\left(\stackrel{=}{T}\right)$ is the power set of $\stackrel{=}{T}$.

Definition 2.

[6] For two SSs $(\stackrel{=}{{\phi }_1}, \stackrel{=}{{\wp }_1})$ and $(\stackrel{=}{{\phi }_2}, \stackrel{=}{{\wp }_2})$ with $\stackrel{=}{{\wp }_1}{{\displaystyle \cap }}^{}\stackrel{=}{{\wp }_2}\ne \varnothing$, their restricted union and intersection are signified and invented as

$$(\stackrel{=}{{\phi }_1},\stackrel{=}{{\wp }_1}){{{\displaystyle \cup }}^{}}_R(\stackrel{=}{{\phi }_2},\stackrel{=}{{\wp }_2})=(\stackrel{=}{{\alpha }_R},\stackrel{=}{{\wp }_1}{{\displaystyle \cap }}^{}\stackrel{=}{{\wp }_2})$$

where $\stackrel{=}{{\alpha }_R}(\stackrel{=}{\mathfrak{p}})=\stackrel{=}{{\phi }_1}(\stackrel{=}{\mathfrak{p}}){{\displaystyle \cup }}^{}\stackrel{=}{{\phi }_2}(\stackrel{=}{\mathfrak{p}})$ $\forall \stackrel{=}{\mathfrak{p}}$ $\in \stackrel{=}{{\wp }_1}{{\displaystyle \cap }}^{}\stackrel{=}{{\wp }_2}$.

$$(\stackrel{=}{{\phi }_1},\stackrel{=}{{\wp }_1}){{{\displaystyle \cap }}^{}}_R(\stackrel{=}{{\phi }_2},\stackrel{=}{{\wp }_2})=(\stackrel{=}{{\beta }_R},\stackrel{=}{{\wp }_1}{{\displaystyle \cap }}^{}\stackrel{=}{{\wp }_2})$$

where $\stackrel{=}{{\beta }_R}(\stackrel{=}{\mathfrak{p}})=\stackrel{=}{{\phi }_1}(\stackrel{=}{\mathfrak{p}}){{\displaystyle \cap }}^{}\stackrel{=}{{\phi }_2}(\stackrel{=}{\mathfrak{p}})\forall \stackrel{=}{\mathfrak{p}}\in \stackrel{=}{{\wp }_1}{{\displaystyle \cap }}^{}\stackrel{=}{{\wp }_2}$.

Definition 3.

[6] For two SSs $(\stackrel{=}{{\phi }_1}, \stackrel{=}{{\wp }_1})$ and $(\stackrel{=}{{\phi }_2}, \stackrel{=}{{\wp }_2})$ with $\stackrel{=}{{\wp }_1}{{\displaystyle \cap }}^{}\stackrel{=}{{\wp }_2}\ne \varnothing$, their extended intersection is signified and invented as

$$(\stackrel{=}{{\phi }_1},\stackrel{=}{{\wp }_1}){{{\displaystyle \cap }}^{}}_{ℰ}(\stackrel{=}{{\phi }_2},\stackrel{=}{{\wp }_2})=(\stackrel{=}{{\beta }_{ℰ}},\stackrel{=}{{\wp }_1}{{\displaystyle \cap }}^{}\stackrel{=}{{\wp }_2})$$

where $\stackrel{=}{{\beta }_{ℰ}}(\stackrel{=}{\mathfrak{p}})=\{\begin{array}{cc}\stackrel{=}{{\phi }_1}(\stackrel{=}{\mathfrak{p}})& i\mathfrak{f}\stackrel{=}{\mathfrak{p}}\in \stackrel{=}{{\wp }_1}\backslash \stackrel{=}{{\wp }_2}\\ \stackrel{=}{{\phi }_2}(\stackrel{=}{\mathfrak{p}})& i\mathfrak{f}\stackrel{=}{\mathfrak{p}}\in \stackrel{=}{{\wp }_2}\backslash \stackrel{=}{{\wp }_1}\\ \stackrel{=}{{\phi }_1}(\stackrel{=}{\mathfrak{p}}){{\displaystyle \cap }}^{}\stackrel{=}{{\phi }_2}(\stackrel{=}{\mathfrak{p}})& \stackrel{=}{\mathfrak{p}}\in \stackrel{=}{{\wp }_1}{{\displaystyle \cup }}^{}\stackrel{=}{{\wp }_2}\end{array}$         $\forall \stackrel{=}{\mathfrak{p}}$ $\in \stackrel{=}{{\wp }_1}{{\displaystyle \cup }}^{}\stackrel{=}{{\wp }_2}$.

Definition 4.

[26] A BFS $\stackrel{=}{\mathfrak{M}}$ over $\stackrel{=}{T}$ is of the structure

$$\stackrel{=}{\mathfrak{M}}=\left\{\left(\stackrel{=}{\tau }, {\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{\mathfrak{M}}}^{+}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{\mathfrak{M}}}^{-}(\stackrel{=}{\tau })\right) | \stackrel{=}{\tau }\in \stackrel{=}{T}\right\}$$

where ${\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{\mathfrak{M}}}^{+}:\stackrel{=}{T}\to \left[0, 1\right]$, ${\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{\mathfrak{M}}}^{-}:\stackrel{=}{T}\to \left[-1, 0\right]$ are the PTG and NTG.

Definition 5.

[26] For two BFSs, $\stackrel{=}{{\mathfrak{M}}_1}=\left\{\left(\stackrel{=}{\tau }, {\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{+}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{-}(\stackrel{=}{\tau })\right) | \stackrel{=}{\tau }\in \stackrel{=}{T}\right\}$ and $\stackrel{=}{{\mathfrak{M}}_2}=\left\{\left(\stackrel{=}{\tau }, {\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_2}}^{+}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_2}}^{-}(\stackrel{=}{\tau })\right) | \stackrel{=}{\tau }\in \stackrel{=}{T}\right\}$, we have

1. 

${\stackrel{=}{{\mathfrak{M}}_1}}^c=c(\stackrel{=}{{\mathfrak{M}}_1})=\left\{\stackrel{=}{\tau }, \left( 1-{\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{+}(\stackrel{=}{\tau }),-1-{\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{-}(\stackrel{=}{\tau })\right) | \stackrel{=}{\tau }\in \stackrel{=}{T}\right\}$;

2. 

$\stackrel{=}{{\mathfrak{M}}_1}{{\displaystyle \cup }}^{}\stackrel{=}{{\mathfrak{M}}_2}=\left\{\stackrel{=}{\tau }, \left(\mathit{max}\left({\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{+}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_2}}^{+}(\stackrel{=}{\tau })\right), \mathit{min}\left({\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{-}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_2}}^{-}(\stackrel{=}{\tau })\right)\right) |\stackrel{=}{\tau }\in \stackrel{=}{T}\right\}$;

3. 

$\stackrel{=}{{\mathfrak{M}}_1}{{\displaystyle \cap }}^{}\stackrel{=}{{\mathfrak{M}}_2}=\left\{\stackrel{=}{\tau }, \left(\mathit{min}\left({\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{+}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_2}}^{+}(\stackrel{=}{\tau })\right), \mathit{max}\left({\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{-}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_2}}^{-}(\stackrel{=}{\tau })\right)\right) |\stackrel{=}{\tau }\in \stackrel{=}{T}\right\}$.

Definition 6.

[31] A CFS is of the structure

$$\stackrel{=}{\mathfrak{M}}=\left\{\left(\stackrel{=}{\tau }, {\stackrel{=}{\Theta }}_{\stackrel{=}{\mathfrak{M}}}(\stackrel{=}{\tau })\right) | \stackrel{=}{\tau }\in \stackrel{=}{T}\right\}=\left\{\left(\stackrel{=}{\tau },{\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{\mathfrak{M}}}(\stackrel{=}{\tau })+\mathfrak{i} {\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{\mathfrak{M}}}(\stackrel{=}{\tau })\right) | \stackrel{=}{\tau }\in \stackrel{=}{T}\right\}$$

where ${\stackrel{=}{\Theta }}_{\stackrel{=}{\mathfrak{M}}}(\stackrel{=}{\tau })$ is a TG and ${\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{\mathfrak{M}}}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{\mathfrak{M}}}(\stackrel{=}{\tau })\in \left[0, 1\right]$, and $\mathfrak{i}=\sqrt{-1}$.

Definition 7.

[31] For two CFSs, $\stackrel{=}{{\mathfrak{M}}_1}=\left\{\left(\stackrel{=}{\tau }, {\stackrel{=}{\Theta }}_{\stackrel{=}{{\mathfrak{M}}_1}}(\stackrel{=}{\tau })\right) | \stackrel{=}{\tau }\in \stackrel{=}{T}\right\}=\left\{\left(\stackrel{=}{\tau },{\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_1}}(\stackrel{=}{\tau })+\mathfrak{i} {\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{{\mathfrak{M}}_1}}(\stackrel{=}{\tau })\right) | \stackrel{=}{\tau }\in \stackrel{=}{T}\right\}$ and $\stackrel{=}{{\mathfrak{M}}_2}=\left\{\left(\stackrel{=}{\tau }, {\stackrel{=}{\Theta }}_{\stackrel{=}{{\mathfrak{M}}_2}}(\stackrel{=}{\tau })\right) | \stackrel{=}{\tau }\in \stackrel{=}{T}\right\}=\left\{\left(\stackrel{=}{\tau },{\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_2}}(\stackrel{=}{\tau })+\mathfrak{i} {\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{{\mathfrak{M}}_2}}(\stackrel{=}{\tau })\right) | \stackrel{=}{\tau }\in \stackrel{=}{T}\right\}$, we have

1. 

${\stackrel{=}{{\mathfrak{M}}_1}}^c=c(\stackrel{=}{{\mathfrak{M}}_1})=\left\{\stackrel{=}{\tau }, \left( 1-{\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_1}}(\stackrel{=}{\tau })+i \left(1-{\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{{\mathfrak{M}}_1}}(\stackrel{=}{\tau })\right)\right) | \stackrel{=}{\tau }\in \stackrel{=}{T}\right\}$;

2. 

$\stackrel{=}{{\mathfrak{M}}_1}{{\displaystyle \cup }}^{}\stackrel{=}{{\mathfrak{M}}_2}=\left\{\stackrel{=}{\tau }, \left(\mathit{max}\left({\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_1}}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_2}}(\stackrel{=}{\tau })\right)+i\mathit{max}\left( {\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{{\mathfrak{M}}_1}}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{{\mathfrak{M}}_2}}(\stackrel{=}{\tau })\right)\right) |\stackrel{=}{\tau }\in \stackrel{=}{T}\right\}$;

3. 

$\stackrel{=}{{\mathfrak{M}}_1}{{\displaystyle \cap }}^{}\stackrel{=}{{\mathfrak{M}}_2}=\left\{\stackrel{=}{\tau }, \left(\mathit{min}\left({\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_1}}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_2}}(\stackrel{=}{\tau })\right)+i\mathit{min}\left( {\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{{\mathfrak{M}}_1}}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{{\mathfrak{M}}_2}}(\stackrel{=}{\tau })\right)\right) |\stackrel{=}{\tau }\in \stackrel{=}{T}\right\}$.

Definition 8.

[41] A BCFS is of the structure

$$\stackrel{=}{\mathfrak{M}}=\left\{\left(\stackrel{=}{\tau }, {\stackrel{=}{\Theta }}_{\stackrel{=}{\mathfrak{M}}}^{+}(\stackrel{=}{\tau }), {\stackrel{=}{\Theta }}_{\stackrel{=}{\mathfrak{M}}}^{-}(\stackrel{=}{\tau })\right) | \stackrel{=}{\tau }\in \stackrel{=}{T}\right\}$$

where ${\stackrel{=}{\Theta }}_{\stackrel{=}{\mathfrak{M}}}^{+}(\stackrel{=}{\tau })={\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{\mathfrak{M}}}^{+}(\stackrel{=}{\tau })+\mathfrak{i}{\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{\mathfrak{M}}}^{+}(\stackrel{=}{\tau })$ identifies the PTG and ${\stackrel{=}{\Theta }}_{\stackrel{=}{\mathfrak{M}}}^{-}(\stackrel{=}{\tau })={\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{\mathfrak{M}}}^{-}(\stackrel{=}{\tau })+\mathfrak{i}{\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{\mathfrak{M}}}^{-}(\stackrel{=}{\tau })$. The values of ${\stackrel{=}{\Theta }}_{\stackrel{=}{\mathfrak{M}}}^{+}(\stackrel{=}{\tau })$ and ${\stackrel{=}{\Theta }}_{\stackrel{=}{\mathfrak{M}}}^{-}(\stackrel{=}{\tau })$ belong to the unit square in a complex plane and ${\mathfrak{f}}_{\stackrel{=}{\mathfrak{M}}}^{+}(\stackrel{=}{\tau }),{\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{\mathfrak{M}}}^{+}(\stackrel{=}{\tau })\in \left[0, 1\right]$ and ${\mathfrak{f}}_{\stackrel{=}{\mathfrak{M}}}^{-}(\stackrel{=}{\tau }),{\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{\mathfrak{M}}}^{-}(\stackrel{=}{\tau })\in \left[-1, 0\right]$.

Definition 9.

[41] For two BCFSs, $\stackrel{=}{{\mathfrak{M}}_1}=\left\{\left(\stackrel{=}{\tau }, {\stackrel{=}{\Theta }}_{\stackrel{=}{{\mathfrak{M}}_1}}^{+}(\stackrel{=}{\tau }), {\stackrel{=}{\Theta }}_{\stackrel{=}{{\mathfrak{M}}_1}}^{-}(\stackrel{=}{\tau })\right) | \stackrel{=}{\tau }\in \stackrel{=}{T}\right\}=\left\{\left(\stackrel{=}{\tau },\left({\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{+}(\stackrel{=}{\tau })+\mathfrak{i}{\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{+}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{-}(\stackrel{=}{\tau })+\mathfrak{i}{\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{-}(\stackrel{=}{\tau })\right)\right) | \stackrel{=}{\tau }\in \stackrel{=}{T}\right\}$ and $\stackrel{=}{{\mathfrak{M}}_2}=\left\{\left(\stackrel{=}{\tau }, {\stackrel{=}{\Theta }}_{\stackrel{=}{{\mathfrak{M}}_2}}^{+}(\stackrel{=}{\tau }), {\stackrel{=}{\Theta }}_{\stackrel{=}{{\mathfrak{M}}_2}}^{-}(\stackrel{=}{\tau })\right) | \stackrel{=}{\tau }\in \stackrel{=}{T}\right\}=\left\{\left(\stackrel{=}{\tau },\left({\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_2}}^{+}(\stackrel{=}{\tau })+\mathfrak{i}{\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{{\mathfrak{M}}_2}}^{+}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_2}}^{-}(\stackrel{=}{\tau })+\mathfrak{i}{\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{{\mathfrak{M}}_2}}^{-}(\stackrel{=}{\tau })\right)\right) | \stackrel{=}{\tau }\in \stackrel{=}{T}\right\}$, we have

1. 

$c(\stackrel{=}{{\mathfrak{M}}_1})={\stackrel{=}{{\mathfrak{M}}_1}}^c=\left\{\left(\stackrel{=}{\tau }, \left(\left[1-{\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{+}(\stackrel{=}{\tau })\right]+\mathfrak{i}\left[1-{\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{+}(\stackrel{=}{\tau })\right],\left[-1-{\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{-}(\stackrel{=}{\tau })\right]+\mathfrak{i}\left[-1-{\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{-}(\stackrel{=}{\tau })\right]\right)\right)| \stackrel{=}{\tau }\in \stackrel{=}{T}\right\}$;

2. 

$\stackrel{=}{{\mathfrak{M}}_1}{{\displaystyle \cup }}^{}\stackrel{=}{{\mathfrak{M}}_2}=\left\{\stackrel{=}{\tau }, \left(\begin{array}{c} \mathit{max}\left({\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{+}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_2}}^{+}(\stackrel{=}{\tau })\right)+i\mathit{max}\left({\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{+}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{{\mathfrak{M}}_2}}^{+}(\stackrel{=}{\tau })\right),\\ \mathit{min}\left({\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{-}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_2}}^{-}(\stackrel{=}{\tau })\right)+i\mathit{min}\left({\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{-}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{{\mathfrak{M}}_2}}^{-}(\stackrel{=}{\tau })\right)\end{array}\right)| \stackrel{=}{\tau }\in \stackrel{=}{T}\right\}$;

3. 

$\stackrel{=}{{\mathfrak{M}}_1}{{\displaystyle \cap }}^{}\stackrel{=}{{\mathfrak{M}}_2}=\left\{\stackrel{=}{\tau }, \left(\begin{array}{c} \mathit{min}\left({\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{+}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_2}}^{+}(\stackrel{=}{\tau })\right)+i\mathit{min}\left({\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{+}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{{\mathfrak{M}}_2}}^{+}(\stackrel{=}{\tau })\right),\\ \mathit{max}\left({\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{-}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{f}}}_{\stackrel{=}{{\mathfrak{M}}_2}}^{-}(\stackrel{=}{\tau })\right)+i\mathit{max}\left({\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{{\mathfrak{M}}_1}}^{-}(\stackrel{=}{\tau }), {\stackrel{=}{\mathfrak{g}}}_{\stackrel{=}{{\mathfrak{M}}_2}}^{-}(\stackrel{=}{\tau })\right)\end{array}\right)| \stackrel{=}{\tau }\in \stackrel{=}{T}\right\}$.

3. The Concept of Bipolar Complex Fuzzy Soft Sets

Here, we have three subsections: in Section 3.1 we introduce the idea of BCFSS, null BCFSS, absolute BCFSS; in Section 3.2, we establish a few elementary operations for BCFSSs such as complement, union, intersection, extended intersection, and related properties; and in Section 3.3, we describe OR and AND operations for BCFSSs.

3.1. Bipolar Complex Fuzzy Soft Sets

Here, we initiate the idea of BCFSS, null BCFSS, absolute BCFSS.

Definition 10.

Suppose that the fixed set be$\stackrel{=}{T}$, the set of parameters be$\stackrel{=}{\mathfrak{P}}$and$\stackrel{=}{\wp }\subseteq \stackrel{=}{\mathfrak{P}}$, then the pair$(\stackrel{=}{\phi }, \stackrel{=}{\wp })$is called BCFSS over$(\stackrel{=}{T})$, where$\stackrel{=}{\phi }: \stackrel{=}{\wp }\to BCFS(\stackrel{=}{T})$,$BCFS(\stackrel{=}{T})$is the family of all BCFSs of$(\stackrel{=}{T})$. It is presented as

$$\begin{gathered} (\stackrel{=}{\phi }, \stackrel{=}{\wp })=\stackrel{=}{\phi }(\stackrel{=}{{\mathfrak{p}}_j})=\left\{\left(\stackrel{=}{{\tau }_k}, {\Theta }_{\stackrel{=}{\phi }}^{+}(\stackrel{=}{{\tau }_k}), {\Theta }_{\stackrel{=}{\phi }}^{-}(\stackrel{=}{{\tau }_k})\right) | \forall \stackrel{=}{{\tau }_k}\in \stackrel{=}{T}, \forall \stackrel{=}{{\mathfrak{p}}_j}\in \stackrel{=}{\wp }\right\} \\ =\left\{\left(\stackrel{=}{{\tau }_k}, {\delta }_{\stackrel{=}{\phi }}^{+}(\stackrel{=}{{\tau }_k})+i {\lambda }_{\stackrel{=}{\phi }}^{+}(\stackrel{=}{{\tau }_k}), {\delta }_{\stackrel{=}{\phi }}^{-}(\stackrel{=}{{\tau }_k})+i {\lambda }_{\stackrel{=}{\phi }}^{-}(\stackrel{=}{{\tau }_k}) \right) | \forall \stackrel{=}{{\tau }_k}\in \stackrel{=}{T}, \forall \stackrel{=}{{\mathfrak{p}}_j}\in \stackrel{=}{\wp }\right\} \end{gathered}$$

Remark 1.

For a set$\stackrel{=}{T}=\{\stackrel{=}{{\tau }_1}, \stackrel{=}{{\tau }_2}, \stackrel{=}{{\tau }_3}, \dots , \stackrel{=}{{\tau }_n}\}$and$\stackrel{=}{\wp }=\{\stackrel{=}{{\mathfrak{p}}_1}, \stackrel{=}{{\mathfrak{p}}_2}, \stackrel{=}{{\mathfrak{p}}_3}, \dots , \stackrel{=}{{\mathfrak{p}}_m}\}\subseteq \stackrel{=}{\mathfrak{P}}$, the tabular representation of BCFSS$(\stackrel{=}{\phi }, \stackrel{=}{\wp })$is signified in

Table 1. The tabular description of BCFSS.

$(\stackrel{=}{\mathit{\phi }}, \stackrel{=}{\mathit{\wp }})$	$\stackrel{=}{{\mathfrak{p}}_1}$	$\stackrel{=}{{\mathfrak{p}}_2}$	$\dots$	$\stackrel{=}{{\mathfrak{p}}_{\mathit{m}}}$
$\stackrel{=}{{\tau }_1}$	${\Theta }_{{\stackrel{=}{\phi }}_{11}}^{+}, {\Theta }_{{\stackrel{=}{\phi }}_{11}}^{-}$	${\Theta }_{{\stackrel{=}{\phi }}_{12}}^{+}, {\Theta }_{{\stackrel{=}{\phi }}_{12}}^{-}$	$\dots$	${\Theta }_{{\stackrel{=}{\phi }}_{1m}}^{+}, {\Theta }_{{\stackrel{=}{\phi }}_{1m}}^{-}$
$\stackrel{=}{{\tau }_2}$	${\Theta }_{{\stackrel{=}{\phi }}_{21}}^{+}, {\Theta }_{{\stackrel{=}{\phi }}_{21}}^{-}$	${\Theta }_{{\stackrel{=}{\phi }}_{22}}^{+}, {\Theta }_{{\stackrel{=}{\phi }}_{22}}^{-}$	$\dots$	${\Theta }_{{\stackrel{=}{\phi }}_{2m}}^{+}, {\Theta }_{{\stackrel{=}{\phi }}_{2m}}^{-}$
$\stackrel{=}{{\tau }_3}$	${\Theta }_{{\stackrel{=}{\phi }}_{31}}^{+}, {\Theta }_{{\stackrel{=}{\phi }}_{31}}^{-}$	${\Theta }_{{\stackrel{=}{\phi }}_{32}}^{+}, {\Theta }_{{\stackrel{=}{\phi }}_{32}}^{-}$	$\dots$	${\Theta }_{{\stackrel{=}{\phi }}_{3m}}^{+}, {\Theta }_{{\stackrel{=}{\phi }}_{3m}}^{-}$
$\dots$	$\dots$	$\dots$	$\dots$	$\dots$
$\stackrel{=}{{\tau }_n}$	${\Theta }_{{\stackrel{=}{\phi }}_{n1}}^{+}, {\Theta }_{{\stackrel{=}{\phi }}_{n1}}^{-}$	${\Theta }_{{\stackrel{=}{\phi }}_{n2}}^{+}, {\Theta }_{{\stackrel{=}{\phi }}_{n2}}^{-}$	$\dots$	${\Theta }_{{\stackrel{=}{\phi }}_{nm}}^{+}, {\Theta }_{{\stackrel{=}{\phi }}_{nm}}^{-}$

.

Example 1.

Suppose that$\stackrel{=}{T}=\{\stackrel{=}{{\tau }_1}, \stackrel{=}{{\tau }_2}, \stackrel{=}{{\tau }_3}, \stackrel{=}{{\tau }_4}\}$is the set of four under attention laptops and$\stackrel{=}{\wp }=\{\stackrel{=}{{\mathfrak{p}}_1}=Battery life, \stackrel{=}{{\mathfrak{p}}_2}=light weight, \stackrel{=}{{\mathfrak{p}}_3}=Costly\}\subseteq \stackrel{=}{\mathfrak{P}}$is the parameters set, then the BCFSS is interpreted as follows

$$\begin{array}{l}\left(\stackrel{=}{\phi }, \stackrel{=}{\wp }\right)\\ =\{\begin{array}{l}\stackrel{=}{\phi }\left(\stackrel{=}{{\mathfrak{p}}_1}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.2+i 0.6 , -0.6-i 0.9\right), \left(\stackrel{=}{{\tau }_2},0.45+i 0.9 , -0.11-i 0.4\right), \\ \left(\stackrel{=}{{\tau }_3},0.7+i 0.3 , -0.23-i 0.56\right),\left(\stackrel{=}{{\tau }_4},0.8+i 0.2 , -0.3-i 0.22\right)\end{array}\right\}\\ \stackrel{=}{\phi }\left(\stackrel{=}{{\mathfrak{p}}_2}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.15+i 0.34 , -0.2-i 0.5\right), \left(\stackrel{=}{{\tau }_2},0.23+i 0.8 , -0.9-i 0.5\right), \\ \left(\stackrel{=}{{\tau }_3},0.35+i 0.6 , -0.46-i 0.28\right),\left(\stackrel{=}{{\tau }_4},0.4+i 0.4 , -0.6-i 0.34\right)\end{array}\right\}\\ \stackrel{=}{\phi }\left(\stackrel{=}{{\mathfrak{p}}_3}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.3+i 0.6 , -0.4-i 1\right), \left(\stackrel{=}{{\tau }_2},0.46+i 0.4 , -0.45-i 0.5\right), \\ \left(\stackrel{=}{{\tau }_3},0.65+i 0.36 , -0.77-i 0.4\right),\left(\stackrel{=}{{\tau }_4},0.8+i 0.8 , -0.4-i 0.68\right)\end{array}\right\}\end{array}& \}\end{array}$$

The tabular description of BCFSS interpreted in example 1 is depicted in

Table 2. The tabular representation of BCFSS described in example 1.

$\left(\stackrel{=}{\mathit{\phi }}, \stackrel{=}{\mathit{\wp }}\right)$	$\stackrel{=}{{\mathfrak{p}}_1}$	$\stackrel{=}{{\mathfrak{p}}_2}$	$\stackrel{=}{{\mathfrak{p}}_3}$
$\stackrel{=}{{\tau }_1}$	$\left(\begin{array}{c}0.2+i 0.6 , \\ -0.6-i 0.9\end{array}\right)$	$\left(\begin{array}{c}0.15+i 0.34 , \\ -0.2-i 0.5\end{array}\right)$	$\left(\begin{array}{c}0.3+i 0.6 , \\ -0.4-i 1\end{array}\right)$
$\stackrel{=}{{\tau }_2}$	$\left(\begin{array}{c}0.45+i 0.9 , \\ -0.11-i 0.4\end{array}\right)$	$\left(\begin{array}{c}0.23+i 0.8 , \\ -0.9-i 0.5\end{array}\right)$	$\left(\begin{array}{c}0.46+i 0.4 , \\ -0.45-i 0.5\end{array}\right)$
$\stackrel{=}{{\tau }_3}$	$\left(\begin{array}{c}0.7+i 0.3 , \\ -0.23-i 0.56\end{array}\right)$	$\left(\begin{array}{c}0.35+i 0.6 , \\ -0.46-i 0.28\end{array}\right)$	$\left(\begin{array}{c}0.65+i 0.36 , \\ -0.77-i 0.4\end{array}\right)$
$\stackrel{=}{{\tau }_4}$	$\left(\begin{array}{c}0.8+i 0.2 , \\ -0.3-i 0.22\end{array}\right)$	$\left(\begin{array}{c}0.4+i 0.4 , \\ -0.6-i 0.34\end{array}\right)$	$\left(\begin{array}{c}0.8+i 0.8 , \\ -0.4-i 0.68\end{array}\right)$

.

Definition 11.

Suppose the fixed set be$\stackrel{=}{T}$, the set of parameters be$\stackrel{=}{\mathfrak{P}}$, then the family of all BCFSS$(\stackrel{=}{T}, \stackrel{=}{\mathfrak{P}})$over a fixed set$\stackrel{=}{T}$with attributes$\stackrel{=}{\mathfrak{P}}$is called bipolar complex fuzzy soft class.

Definition 12.

A BCFSS$(\stackrel{=}{\phi }, \stackrel{=}{\wp })$is called null BCFSS if$\forall \stackrel{=}{\mathfrak{p}}\in \stackrel{=}{\wp }, \stackrel{=}{\phi }( \stackrel{=}{\mathfrak{p}})=\varnothing$and designated by the empty set$\varnothing$, where the value of BCFS will be$(0.0+i 0.0,-0.0-i 0.0)$.

Definition 13.

A BCFSS$(\stackrel{=}{\phi }, \stackrel{=}{\wp })$is called absolute BCFSS if$\forall \stackrel{=}{\mathfrak{p}}\in \stackrel{=}{\wp }$, $\stackrel{=}{\phi }( \stackrel{=}{\mathfrak{p}})=A$, where$A$covers all these mappings which are identical, that is$\stackrel{=}{\phi }( \stackrel{=}{{\mathfrak{p}}_i})=(1+i1,-1-i1)$for all$i$.

Definition 14.

Assume that$(\stackrel{=}{\phi }, \stackrel{=}{\wp })$and$(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$are two BCFSSs over$\stackrel{=}{T}$, then$(\stackrel{=}{\phi }, \stackrel{=}{\wp })$is called a bipolar complex fuzzy soft subset of$(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$if

1. 

$\stackrel{=}{\wp }\subseteq \stackrel{=}{ℛ}$,

2. 

$\forall \stackrel{=}{\mathfrak{p}}\in \stackrel{=}{\wp }$,$\stackrel{=}{\phi }(\stackrel{=}{\mathfrak{p}})$is a bipolar complex fuzzy subset of$\stackrel{=}{\varphi }(\stackrel{=}{\mathfrak{p}})$, i.e.,${\delta }_{\stackrel{=}{\phi }}^{+}(\stackrel{=}{{\tau }_j})\le {\delta }_{\stackrel{=}{\varphi }}^{+}(\stackrel{=}{{\tau }_j}), {\lambda }_{\stackrel{=}{\phi }}^{+}(\stackrel{=}{{\tau }_j})\le {\lambda }_{\stackrel{=}{\varphi }}^{+}(\stackrel{=}{{\tau }_j})$,${\delta }_{\stackrel{=}{\phi }}^{-}(\stackrel{=}{{\tau }_j})\ge {\delta }_{\stackrel{=}{\varphi }}^{-}(\stackrel{=}{{\tau }_j}),$and${\lambda }_{\stackrel{=}{\phi }}^{-}(\stackrel{=}{{\tau }_j})\ge {\lambda }_{\stackrel{=}{\varphi }}^{-}(\stackrel{=}{{\tau }_j})$, for all$\in \stackrel{=}{{\tau }_j}\in \stackrel{=}{T}$. and it is signified as$(\stackrel{=}{\phi }, \stackrel{=}{\wp })\subseteq (\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$.

Definition 15.

Assume that$(\stackrel{=}{\phi }, \stackrel{=}{\wp })$and$(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$are two BCFSSs over$\stackrel{=}{T}$, then$(\stackrel{=}{\phi }, \stackrel{=}{\wp })$and$(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$is said to be bipolar complex fuzzy soft equal sets if$(\stackrel{=}{\phi }, \stackrel{=}{\wp })\subseteq (\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$and$(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})\subseteq (\stackrel{=}{\phi }, \stackrel{=}{\wp })$.

3.2. Elementary Operation on BCFSSs

Here, we will establish a few elementary operations for BCFSSs such as a complement, union, intersection, extended intersection, and related properties.

Definition 16.

The complement of a BCFSS$(\stackrel{=}{\phi }, \stackrel{=}{\wp })$is signified and described as

$$\begin{array}{r}{(\stackrel{=}{\phi }, \stackrel{=}{\wp })}^c=\{(\stackrel{=}{{\tau }_j}, 1-{\delta }_{\stackrel{=}{\phi }}^{+}(\stackrel{=}{{\tau }_j})+i \left(1-{\lambda }_{\stackrel{=}{\phi }}^{+}(\stackrel{=}{{\tau }_j})\right),-1-{\delta }_{\stackrel{=}{\phi }}^{-}(\stackrel{=}{{\tau }_j})\\ +i\left(-1-{\lambda }_{\stackrel{=}{\phi }}^{-}(\stackrel{=}{{\tau }_j})\right)) | \forall \stackrel{=}{{\tau }_k}\in \stackrel{=}{T}, \forall \stackrel{=}{{\mathfrak{p}}_j}\in \stackrel{=}{\wp }\}\end{array}$$

Example 2.

Consider a BCFSS$(\stackrel{=}{\phi }, \stackrel{=}{\wp })$over$\stackrel{=}{T}$presented in Example 1. The complement${(\stackrel{=}{\phi }, \stackrel{=}{\wp })}^c$is described as follows

$$\begin{array}{l}{\left(\stackrel{=}{\phi }, \stackrel{=}{\wp }\right)}^c\\ =\{\begin{array}{l}\begin{array}{l}\stackrel{=}{\phi }\left(\stackrel{=}{{\mathfrak{p}}_1}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.8+i 0.4 , -0.4-i 0.1\right), \left(\stackrel{=}{{\tau }_2},0.55+i 0.1 , -0.89-i 0.6\right), \\ \left(\stackrel{=}{{\tau }_3},0.3+i 0.7 , -0.77-i 0.44\right),\left(\stackrel{=}{{\tau }_4},0.2+i 0.8 , -0.7-i 0.78\right)\end{array}\right\}\\ \stackrel{=}{\phi }\left(\stackrel{=}{{\mathfrak{p}}_2}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.85+i 0.66 , -0.8-i 0.5\right), \left(\stackrel{=}{{\tau }_2},0.77+i 0.2 , -0.1-i 0.5\right), \\ \left(\stackrel{=}{{\tau }_3},0.65+i 0.4 , -0.54-i 0.72\right),\left(\stackrel{=}{{\tau }_4},0.6+i 0.6 , -0.4-i 0.66\right)\end{array}\right\}\\ \stackrel{=}{\phi }\left(\stackrel{=}{{\mathfrak{p}}_3}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.7+i 0.4 , -0.6-i 0\right), \left(\stackrel{=}{{\tau }_2},0.54+i 0.6 , -0.55-i 0.5\right), \\ \left(\stackrel{=}{{\tau }_3},0.35+i 0.64 , -0.23-i 0.6\right),\left(\stackrel{=}{{\tau }_4},0.2+i 0.2 , -0.6-i 0.32\right)\end{array}\right\}\end{array}\end{array}& \}\end{array}$$

Definition 17.

The restricted intersection of two BCFSSs$(\stackrel{=}{\phi }, \stackrel{=}{\wp })$and$(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$over$\stackrel{=}{T}$is a BCFSS$(\stackrel{=}{H_R}, \stackrel{=}{\mathfrak{Q}})$, where$\stackrel{=}{\mathfrak{Q}}=\stackrel{=}{\wp }{{\displaystyle \cap }}^{} \stackrel{=}{ℛ}\ne \varnothing$and${\stackrel{=}{H}}_R: \stackrel{=}{\mathfrak{Q}}\to BCFS( \stackrel{=}{T})$is characterized as$\stackrel{=}{H_R}( \stackrel{=}{\mathfrak{p}})=\stackrel{=}{\phi }( \stackrel{=}{\mathfrak{p}}){{\displaystyle \cap }}^{} \stackrel{=}{\varphi }( \stackrel{=}{\mathfrak{p}})$and signified by$(\stackrel{=}{H_R}, \stackrel{=}{\mathfrak{Q}})=(\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cap }}^{}}_R(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$.

Example 3.

Suppose that$\stackrel{=}{T}=\left\{\stackrel{=}{{\tau }_1}, \stackrel{=}{{\tau }_2}, \stackrel{=}{{\tau }_3}, \stackrel{=}{{\tau }_4}\right\}$is the set of four different apartments and$\stackrel{=}{\mathfrak{P}}=\left\{\stackrel{=}{{\mathfrak{p}}_1}=cheap, \stackrel{=}{{\mathfrak{p}}_2}=beautiful, \stackrel{=}{{\mathfrak{p}}_3}=wooden,, \stackrel{=}{{\mathfrak{p}}_4}=modern, \stackrel{=}{{\mathfrak{p}}_5}=expensive\right\}$is the set of parameters,$\stackrel{=}{\wp }=\left\{\stackrel{=}{{\mathfrak{p}}_1}, \stackrel{=}{{\mathfrak{p}}_3}, \stackrel{=}{{\mathfrak{p}}_5}\right\}\subset \stackrel{=}{\mathfrak{P}}$and$\stackrel{=}{ℛ}=\left\{\stackrel{=}{{\mathfrak{p}}_1}, \stackrel{=}{{\mathfrak{p}}_2}, \stackrel{=}{{\mathfrak{p}}_5}\right\}\subset \stackrel{=}{\mathfrak{P}}$, then the two BCFSSs are given as

$$\begin{array}{ccccc}\left(\stackrel{=}{\phi }, \stackrel{=}{\wp }\right)& =& \{& \begin{array}{c}\stackrel{=}{\phi }\left(\stackrel{=}{{\mathfrak{p}}_1}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.4+i 0.4 , -0.4-i 0.7\right), \left(\stackrel{=}{{\tau }_2},0.65+i 0.7 , -0.31-i 0.6\right), \\ \left(\stackrel{=}{{\tau }_3},0.5+i 0.5 , -0.43-i 0.36\right),\left(\stackrel{=}{{\tau }_4},0.6+i 0.4 , -0.5-i 0.44\right)\end{array}\right\}\\ \stackrel{=}{\phi }\left(\stackrel{=}{{\mathfrak{p}}_3}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.34+i 0.23 , -0.5-i 0.3\right), \left(\stackrel{=}{{\tau }_2},0.5+i 0.4 , -0.61-i 0.3\right), \\ \left(\stackrel{=}{{\tau }_3},0.2+i 0.2 , -0.73-i 0.66\right),\left(\stackrel{=}{{\tau }_4},0.3+i 0.7 , -0.8-i 0.74\right)\end{array}\right\}\\ \stackrel{=}{\phi }\left(\stackrel{=}{{\mathfrak{p}}_5}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.5+i 0.4 , -0.6-i 0.8\right), \left(\stackrel{=}{{\tau }_2},0.66+i 0.6 , -0.65-i 0.3\right), \\ \left(\stackrel{=}{{\tau }_3},0.45+i 0.66 , -0.37-i 0.8\right),\left(\stackrel{=}{{\tau }_4},0.6+i 0.5 , -0.3-i 0.8\right)\end{array}\right\}\end{array}& \}\end{array}$$

$$\begin{array}{l}\left(\stackrel{=}{\varphi }, \stackrel{=}{\mathcal{R}}\right)\\ =\{\begin{array}{c}\stackrel{=}{\varphi }\left(\stackrel{=}{{\mathfrak{p}}_1}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.56+i 0.3 , -0.2-i 0.9\right), \left(\stackrel{=}{{\tau }_2},0.7+i 0.6 , -0.21-i 0.4\right), \\ \left(\stackrel{=}{{\tau }_3},0.6+i 0.4 , -0.37-i 0.46\right),\left(\stackrel{=}{{\tau }_4},0.2+i 0.6 , -0.3-i 0.54\right)\end{array}\right\}\\ \stackrel{=}{\varphi }\left(\stackrel{=}{{\mathfrak{p}}_2}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.15+i 0.34 , -0.5-i 0.6\right), \left(\stackrel{=}{{\tau }_2},0.24+i 0.62 , -0.17-i 0.5\right), \\ \left(\stackrel{=}{{\tau }_3},0.65+i 0.4 , -0.6-i 0.8\right),\left(\stackrel{=}{{\tau }_4},0.56+i 0.7 , -0.3-i 0.4\right)\end{array}\right\}\\ \stackrel{=}{\varphi }\left(\stackrel{=}{{\mathfrak{p}}_5}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.35+i 0.54 , -0.4-i 0.3\right), \left(\stackrel{=}{{\tau }_2},0.42+i 0.6 , -0.7-i 0.3\right), \\ \left(\stackrel{=}{{\tau }_3},0.55+i 0.4 , -0.66-i 0.48\right),\left(\stackrel{=}{{\tau }_4},0.6+i 0.6 , -0.4-i 0.54\right)\end{array}\right\}\end{array}& \}\end{array}$$

then their restricted intersection is designated as$(\stackrel{=}{H_R}, \stackrel{=}{\mathfrak{Q}})=(\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cap }}^{}}_R(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$, where$\stackrel{=}{\mathfrak{Q}}=\stackrel{=}{\wp }{{\displaystyle \cap }}^{} \stackrel{=}{ℛ}=\left\{\stackrel{=}{{\mathfrak{p}}_1}, \stackrel{=}{{\mathfrak{p}}_5}\right\}$and is presented as

$$\begin{array}{l}\left(\stackrel{=}{H_R}, \stackrel{=}{\mathfrak{Q}}\right)\\ =\{\begin{array}{c}\stackrel{=}{H_R}\left(\stackrel{=}{{\mathfrak{p}}_1}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.4+i 0.3 , -0.2-i 0.7\right), \left(\stackrel{=}{{\tau }_2},0.65+i 0.6 , -0.21-i 0.4\right), \\ \left(\stackrel{=}{{\tau }_3},0.5+i 0.4 , -0.37-i 0.36\right),\left(\stackrel{=}{{\tau }_4},0.2+i 0.4 , -0.3-i 0.44\right)\end{array}\right\}\\ {\stackrel{=}{H}}_R\left(\stackrel{=}{{\mathfrak{p}}_5}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.35+i 0.4 , -0.4-i 0.3\right), \left(\stackrel{=}{{\tau }_2},0.42+i 0.6 , -0.65-i 0.3\right), \\ \left(\stackrel{=}{{\tau }_3},0.45+i 0.4 , -0.37-i 0.48\right),\left(\stackrel{=}{{\tau }_4},0.6+i 0.5 , -0.3-i 0.54\right)\end{array}\right\}\end{array}\}\end{array}$$

Definition 18.

The restricted union of two BCFSSs$(\stackrel{=}{\phi }, \stackrel{=}{\wp })$and$(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$over$\stackrel{=}{T}$is a BCFSS$(\stackrel{=}{I_R}, \stackrel{=}{\mathfrak{Q}})$, where$\stackrel{=}{\mathfrak{Q}}=\stackrel{=}{\wp }{{\displaystyle \cap }}^{} \stackrel{=}{ℛ}\ne \varnothing$and${\stackrel{=}{I}}_R: \stackrel{=}{\mathfrak{Q}}\to BC\phi S( \stackrel{=}{T})$is characterized as$\stackrel{=}{I_R}( \stackrel{=}{\mathfrak{p}})=\stackrel{=}{\phi }(\stackrel{=}{\mathfrak{p}}){{\displaystyle \cup }}^{}\stackrel{=}{\varphi }(\stackrel{=}{\mathfrak{p}})$and signified by$(\stackrel{=}{I_R}, \stackrel{=}{\mathfrak{Q}})=(\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cup }}^{}}_R(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$.

Example 4

Consider the two BCFSSs$(\stackrel{=}{\phi }, \stackrel{=}{\wp })$and$(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$over$\stackrel{=}{T}$presented in Example 3. Their restricted union is designated as$(\stackrel{=}{I_R}, \stackrel{=}{\mathfrak{Q}})=(\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cup }}^{}}_R(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$, where$\stackrel{=}{\mathfrak{Q}}=\stackrel{=}{\wp }{{\displaystyle \cap }}^{} \stackrel{=}{ℛ}=\left\{ \stackrel{=}{{\mathfrak{p}}_1}, \stackrel{=}{{\mathfrak{p}}_5}\right\}$and is presented as

$$\left(\stackrel{=}{I_R}, \stackrel{=}{\mathfrak{Q}}\right)\begin{array}{cccc}& & & \\ =& \{& \begin{array}{c}\stackrel{=}{I_R}\left(\stackrel{=}{{\mathfrak{p}}_1}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.56+i 0.4 , -0.4-i 0.9\right), \left(\stackrel{=}{{\tau }_2},0.7+i 0.7 , -0.31-i 0.6\right), \\ \left(\stackrel{=}{{\tau }_3},0.6+i 0.5 , -0.43-i 0.46\right),\left(\stackrel{=}{{\tau }_4},0.6+i 0.6 , -0.5-i 0.54\right)\end{array}\right\}\\ \stackrel{=}{I_R}\left(\stackrel{=}{{\mathfrak{p}}_5}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.5+i 0.54 , -0.6-i 0.8\right), \left(\stackrel{=}{{\tau }_2},0.66+i 0.6 , -0.7-i 0.3\right), \\ \left(\stackrel{=}{{\tau }_3},0.55+i 0.66 , -0.66-i 0.8\right),\left(\stackrel{=}{{\tau }_4},0.6+i 0.6 , -0.4-i 0.8\right)\end{array}\right\}\end{array}& \}\end{array}$$

Definition 19.

The extended intersection of two BCFSSs$(\stackrel{=}{\phi }, \stackrel{=}{\wp })$and$(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$over$\stackrel{=}{T}$is a BCFSS$(\stackrel{=}{H_{ℰ}}, \stackrel{=}{\mathfrak{Q}})$, where$\stackrel{=}{\mathfrak{Q}}=\stackrel{=}{\wp }{{\displaystyle \cup }}^{} \stackrel{=}{ℛ}$,

$$\stackrel{=}{H_{ℰ}}( \stackrel{=}{\mathfrak{p}})=\{\begin{array}{cc}\stackrel{=}{\phi }( \stackrel{=}{\mathfrak{p}})& \mathrm{if} \stackrel{=}{\mathfrak{p}}\in \stackrel{=}{\wp }-\stackrel{=}{ℛ}\hfill \\ \stackrel{=}{\varphi }( \stackrel{=}{\mathfrak{p}})& \mathrm{if} \stackrel{=}{\mathfrak{p}}\in \stackrel{=}{ℛ}-\stackrel{=}{\wp }\hfill \\ \stackrel{=}{\phi }( \stackrel{=}{\mathfrak{p}}){{\displaystyle \cap }}^{}\stackrel{=}{\varphi }( \stackrel{=}{\mathfrak{p}})& \mathrm{if} \stackrel{=}{\mathfrak{p}}\in \stackrel{=}{\wp }{{\displaystyle \cap }}^{} \stackrel{=}{ℛ}\hfill \end{array}$$

and signified by$(\stackrel{=}{H_{ℰ}}, \stackrel{=}{\mathfrak{Q}})=(\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cap }}^{}}_{ℰ}(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$.

Example 5.

Consider the two BCFSSs$(\stackrel{=}{\phi }, \stackrel{=}{\wp })$and$(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$over$\stackrel{=}{T}$presented in Example 3. Their extended intersection is designated as$(\stackrel{=}{H_{ℰ}}, \stackrel{=}{\mathfrak{Q}})=(\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cap }}^{}}_{ℰ}(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$, where$\stackrel{=}{\mathfrak{Q}}=\stackrel{=}{\wp }{{\displaystyle \cup }}^{} \stackrel{=}{ℛ}=\{ \stackrel{=}{{\mathfrak{p}}_1},\stackrel{=}{{\mathfrak{p}}_2}, \stackrel{=}{{\mathfrak{p}}_3}, \stackrel{=}{{\mathfrak{p}}_5}\}$and is presented as

$$\begin{array}{l}\left(\stackrel{=}{H_{\mathcal{E}}}, \stackrel{=}{\mathfrak{Q}}\right)\\ =\{\begin{array}{c}\stackrel{=}{H_{\mathcal{E}}}\left(\stackrel{=}{{\mathfrak{p}}_1}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.4+i 0.3 , -0.2-i 0.7\right), \left(\stackrel{=}{{\tau }_2},0.65+i 0.6 , -0.21-i 0.4\right), \\ \left(\stackrel{=}{{\tau }_3},0.5+i 0.4 , -0.37-i 0.36\right),\left(\stackrel{=}{{\tau }_4},0.2+i 0.4 , -0.3-i 0.44\right)\end{array}\right\}\\ \stackrel{=}{H_{\mathcal{E}}}\left(\stackrel{=}{{\mathfrak{p}}_2}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.15+i 0.34 , -0.5-i 0.6\right), \left(\stackrel{=}{{\tau }_2},0.24+i 0.62 , -0.17-i 0.5\right), \\ \left(\stackrel{=}{{\tau }_3},0.65+i 0.4 , -0.6-i 0.8\right),\left(\stackrel{=}{{\tau }_4},0.56+i 0.7 , -0.3-i 0.4\right)\end{array}\right\}\\ \stackrel{=}{H_{\mathcal{E}}}\left(\stackrel{=}{{\mathfrak{p}}_3}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.34+i 0.23 , -0.5-i 0.3\right), \left(\stackrel{=}{{\tau }_2},0.5+i 0.4 , -0.61-i 0.3\right), \\ \left(\stackrel{=}{{\tau }_3},0.2+i 0.2 , -0.73-i 0.66\right),\left(\stackrel{=}{{\tau }_4},0.3+i 0.7 , -0.8-i 0.74\right)\end{array}\right\}\\ \stackrel{=}{H_{\mathcal{E}}}\left(\stackrel{=}{{\mathfrak{p}}_5}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.35+i 0.4 , -0.4-i 0.3\right), \left(\stackrel{=}{{\tau }_2},0.42+i 0.6 , -0.65-i 0.3\right), \\ \left(\stackrel{=}{{\tau }_3},0.45+i 0.4 , -0.37-i 0.48\right),\left(\stackrel{=}{{\tau }_4},0.6+i 0.5 , -0.3-i 0.54\right)\end{array}\right\}\end{array}\}\end{array}$$

Proposition 1.

The BCFSSs$(\stackrel{=}{\phi }, \stackrel{=}{\wp }), (\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$and$(\stackrel{=}{L}, \stackrel{=}{\mathcal{D}})$over$\stackrel{=}{T}$satisfies the following properties

1. 

$(\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cup }}^{}}_{ℛ}(\stackrel{=}{\phi }, \stackrel{=}{\wp })=(\stackrel{=}{\phi }, \stackrel{=}{\wp })$,$(\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cap }}^{}}_{ℛ}(\stackrel{=}{\phi }, \stackrel{=}{\wp })=(\stackrel{=}{\phi }, \stackrel{=}{\wp })$

2. 

$(\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cup }}^{}}_{ℛ}\varnothing =(\stackrel{=}{\phi }, \stackrel{=}{\wp })$,$(\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cap }}^{}}_{ℛ}\varnothing =\varnothing$where$\varnothing$is the null BCFSS

3. 

$(\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cup }}^{}}_{ℛ}\left((\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cap }}^{}}_{ℛ}(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})\right)=(\stackrel{=}{\phi }, \stackrel{=}{\wp })$,$(\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cap }}^{}}_{ℛ}\left((\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cup }}^{}}_{ℛ}(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})\right)=(\stackrel{=}{\phi }, \stackrel{=}{\wp })$

4. 

$(\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cup }}^{}}_{ℛ}(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})=(\stackrel{=}{\varphi }, \stackrel{=}{ℛ}){{{\displaystyle \cup }}^{}}_{ℛ}(\stackrel{=}{\phi }, \stackrel{=}{\wp })$,$(\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cap }}^{}}_{ℛ}(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})=(\stackrel{=}{\varphi }, \stackrel{=}{ℛ}){{{\displaystyle \cap }}^{}}_{ℛ}(\stackrel{=}{\phi }, \stackrel{=}{\wp })$

5. 

$(\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cup }}^{}}_{ℛ}\left((\stackrel{=}{\varphi }, \stackrel{=}{ℛ}){{{\displaystyle \cup }}^{}}_{ℛ}(\stackrel{=}{L}, \stackrel{=}{\mathcal{D}})\right)=\left((\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cup }}^{}}_{ℛ}(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})\right){{{\displaystyle \cup }}^{}}_{ℛ}(\stackrel{=}{L}, \stackrel{=}{\mathcal{D}})$,

$(\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cap }}^{}}_{ℛ}\left((\stackrel{=}{\varphi }, \stackrel{=}{ℛ}){{{\displaystyle \cap }}^{}}_{ℛ}(\stackrel{=}{L}, \stackrel{=}{\mathcal{D}})\right)=\left((\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cap }}^{}}_{ℛ}(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})\right){{{\displaystyle \cap }}^{}}_{ℛ}(\stackrel{=}{L}, \stackrel{=}{\mathcal{D}})$

6. 

$(\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cap }}^{}}_{ℛ}\left((\stackrel{=}{\varphi }, \stackrel{=}{ℛ}){{{\displaystyle \cup }}^{}}_{ℛ}(\stackrel{=}{L}, \stackrel{=}{\mathcal{D}})\right)=\left((\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cap }}^{}}_{ℛ}(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})\right){{{\displaystyle \cup }}^{}}_{ℛ}\left((\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cap }}^{}}_{ℛ}(\stackrel{=}{L}, \stackrel{=}{\mathcal{D}})\right)$

$(\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cup }}^{}}_{ℛ}\left((\stackrel{=}{\varphi }, \stackrel{=}{ℛ}){{{\displaystyle \cap }}^{}}_{ℛ}(\stackrel{=}{L}, \stackrel{=}{\mathcal{D}})\right)=\left((\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cup }}^{}}_{ℛ}(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})\right){{{\displaystyle \cap }}^{}}_{ℛ}\left((\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cup }}^{}}_{ℛ}(\stackrel{=}{L}, \stackrel{=}{\mathcal{D}})\right)$

7. 

${\left((\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cup }}^{}}_{ℛ}(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})\right)}^c={(\stackrel{=}{\phi }, \stackrel{=}{\wp })}^c{{{\displaystyle \cap }}^{}}_{ℛ}{(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})}^c$,${\left((\stackrel{=}{\phi }, \stackrel{=}{\wp }){{{\displaystyle \cap }}^{}}_{ℛ}(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})\right)}^c={(\stackrel{=}{\phi }, \stackrel{=}{\wp })}^c{{{\displaystyle \cup }}^{}}_{ℛ}{(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})}^c$

3.3. OR and AND operations on BCFSSs

Here, we will describe OR and AND operations for BCFSSs.

Definition 20.

Suppose that$(\stackrel{=}{\phi }, \stackrel{=}{\wp })$and$(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$are two BCFSSs over$\stackrel{=}{T}$, then the OR operation is a BCFSS and described by$(\stackrel{=}{\phi }, \stackrel{=}{\wp })\vee (\stackrel{=}{\varphi }, \stackrel{=}{ℛ})=(\stackrel{=}{I}, \stackrel{=}{\wp }\times \stackrel{=}{ℛ})$, where$\stackrel{=}{I}(\stackrel{=}{𝒶}, \stackrel{=}{𝒷})=\stackrel{=}{\phi }(\stackrel{=}{𝒶}){{\displaystyle \cup }}^{}\stackrel{=}{\varphi }(\stackrel{=}{𝒷}) \forall (\stackrel{=}{𝒶},\stackrel{=}{𝒷})\in \stackrel{=}{\wp }\times \stackrel{=}{ℛ}$.

Example 6.

Consider the two BCFSSs$(\stackrel{=}{\phi }, \stackrel{=}{\wp })$and$(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$over$\stackrel{=}{T}$presented in Example 3. The OR operation is designated as$(\stackrel{=}{\phi }, \stackrel{=}{\wp })\vee (\stackrel{=}{\varphi }, \stackrel{=}{ℛ})=(\stackrel{=}{I}, \stackrel{=}{\wp }\times \stackrel{=}{ℛ})$, where$\stackrel{=}{\wp }\times \stackrel{=}{ℛ}=\{\stackrel{=}{{\mathfrak{p}}_1}, \stackrel{=}{{\mathfrak{p}}_3}, \stackrel{=}{{\mathfrak{p}}_5}\}\times \{\stackrel{=}{{\mathfrak{p}}_1}, \stackrel{=}{{\mathfrak{p}}_2}, \stackrel{=}{{\mathfrak{p}}_5}\}=\{(\stackrel{=}{{\mathfrak{p}}_1},\stackrel{=}{{\mathfrak{p}}_1}), (\stackrel{=}{{\mathfrak{p}}_1},\stackrel{=}{{\mathfrak{p}}_2}), (\stackrel{=}{{\mathfrak{p}}_1},\stackrel{=}{{\mathfrak{p}}_5}), (\stackrel{=}{{\mathfrak{p}}_3},\stackrel{=}{{\mathfrak{p}}_1}), (\stackrel{=}{{\mathfrak{p}}_3},\stackrel{=}{{\mathfrak{p}}_2}),(\stackrel{=}{{\mathfrak{p}}_3},\stackrel{=}{{\mathfrak{p}}_5}), (\stackrel{=}{{\mathfrak{p}}_5},\stackrel{=}{{\mathfrak{p}}_1}), (\stackrel{=}{{\mathfrak{p}}_5},\stackrel{=}{{\mathfrak{p}}_2}), (\stackrel{=}{{\mathfrak{p}}_5},\stackrel{=}{{\mathfrak{p}}_5})\}$and is presented as

$$\begin{array}{l}\left(\stackrel{=}{I}, \stackrel{=}{\wp }\times \stackrel{=}{\mathcal{R}}\right)\\ =\{\begin{array}{c}\stackrel{=}{I}\left(\stackrel{=}{{\mathfrak{p}}_1},\stackrel{=}{{\mathfrak{p}}_1} \right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.56+i 0.4 , -0.4-i 0.9\right), \left(\stackrel{=}{{\tau }_2},0.7+i 0.7 , -0.31-i 0.6\right), \\ \left(\stackrel{=}{{\tau }_3},0.6+i 0.5 , -0.43-i 0.46\right),\left(\stackrel{=}{{\tau }_4},0.6+i 0.6 , -0.5-i 0.54\right)\end{array}\right\}\\ \stackrel{=}{I}\left(\stackrel{=}{{\mathfrak{p}}_1},\stackrel{=}{{\mathfrak{p}}_2}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.4+i 0.4 , -0.5-i 0.7\right), \left(\stackrel{=}{{\tau }_2},0.65+i 0.7 , -0.31-i 0.6\right), \\ \left(\stackrel{=}{{\tau }_3},0.65+i 0.5 , -0.6-i 0.8\right),\left(\stackrel{=}{{\tau }_4},0.6+i 0.7 , -0.5-i 0.5\right)\end{array}\right\}\\ \stackrel{=}{I}\left(\stackrel{=}{{\mathfrak{p}}_1},\stackrel{=}{{\mathfrak{p}}_5}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.4+i 0.54 , -0.4-i 0.7\right), \left(\stackrel{=}{{\tau }_2},0.65+i 0.7 , -0.7-i 0.6\right), \\ \left(\stackrel{=}{{\tau }_3},0.55+i 0.5 , -0.66-i 0.48\right),\left(\stackrel{=}{{\tau }_4},0.6+i 0.6 , -0.5-i 0.54\right)\end{array}\right\}\\ \stackrel{=}{I}\left(\stackrel{=}{{\mathfrak{p}}_3},\stackrel{=}{{\mathfrak{p}}_1} \right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.56+i 0.3 , -0.5-i 0.9\right), \left(\stackrel{=}{{\tau }_2},0.7+i 0.6 , -0.61-i 0.4\right), \\ \left(\stackrel{=}{{\tau }_3},0.6+i 0.4, -0.73-i 0.66\right),\left(\stackrel{=}{{\tau }_4},0.3+i 0.7 , -0.8-i 0.74\right)\end{array}\right\}\\ \stackrel{=}{I}\left(\stackrel{=}{{\mathfrak{p}}_3},\stackrel{=}{{\mathfrak{p}}_2} \right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.34+i 0.34 , -0.5-i 0.6\right), \left(\stackrel{=}{{\tau }_2},0.5+i 0.62 , -0.61-i 0.5\right), \\ \left(\stackrel{=}{{\tau }_3},0.65+i 0.4 , -0.73-i 0.8\right),\left(\stackrel{=}{{\tau }_4},0.56+i 0.7 , -0.8-i 0.74\right)\end{array}\right\}\\ \stackrel{=}{I}\left(\stackrel{=}{{\mathfrak{p}}_3},\stackrel{=}{{\mathfrak{p}}_5} \right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.35+i 0.54 , -0.5-i 0.3\right), \left(\stackrel{=}{{\tau }_2},0.5+i 0.6, -0.7-i 0.3\right), \\ \left(\stackrel{=}{{\tau }_3},0.55+i 0.4 , -0.73-i 0.66\right),\left(\stackrel{=}{{\tau }_4},0.6+i 0.7 , -0.8-i 0.74\right)\end{array}\right\}\\ \stackrel{=}{I}\left(\stackrel{=}{{\mathfrak{p}}_5},\stackrel{=}{{\mathfrak{p}}_1} \right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.56+i 0.4 , -0.6-i 0.9\right), \left(\stackrel{=}{{\tau }_2},0.7+i 0.6 , -0.65-i 0.4\right), \\ \left(\stackrel{=}{{\tau }_3},0.6+i 0.66 , -0.37-i 0.8\right),\left(\stackrel{=}{{\tau }_4},0.6+i 0.6 , -0.3-i 0.8\right)\end{array}\right\}\\ \stackrel{=}{I}\left(\stackrel{=}{{\mathfrak{p}}_5},\stackrel{=}{{\mathfrak{p}}_2} \right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.5+i 0.4 , -0.6-i 0.8\right), \left(\stackrel{=}{{\tau }_2},0.66+i 0.62 , -0.65-i 0.5\right), \\ \left(\stackrel{=}{{\tau }_3},0.65+i 0.66 , -0.6-i 0.8\right),\left(\stackrel{=}{{\tau }_4},0.6+i 0.7 , -0.3-i 0.8\right)\end{array}\right\}\\ \stackrel{=}{I}\left(\stackrel{=}{{\mathfrak{p}}_5},\stackrel{=}{{\mathfrak{p}}_5} \right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.5+i 0.54 , -0.6-i 0.8\right), \left(\stackrel{=}{{\tau }_2},0.66+i 0.6 , -0.7-i 0.3\right), \\ \left(\stackrel{=}{{\tau }_3},0.55+i 0.4 , -0.66-i 0.8\right),\left(\stackrel{=}{{\tau }_4},0.6+i 0.6 , -0.4-i 0.8\right)\end{array}\right\}\end{array}\}\end{array}$$

Definition 21.

Suppose that$(\stackrel{=}{\phi }, \stackrel{=}{\wp })$and$(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$are two BCFSSs over$\stackrel{=}{T}$, then the AND operation is a BCFSS and described by$(\stackrel{=}{\phi }, \stackrel{=}{\wp })\wedge (\stackrel{=}{\varphi }, \stackrel{=}{ℛ})=(\stackrel{=}{H}, \stackrel{=}{\wp }\times \stackrel{=}{ℛ})$, where$\stackrel{=}{H}(\stackrel{=}{𝒶}, \stackrel{=}{𝒷})=\stackrel{=}{\phi }(\stackrel{=}{𝒶}){{\displaystyle \cap }}^{}\stackrel{=}{\varphi }(\stackrel{=}{𝒷}) \forall (\stackrel{=}{𝒶},\stackrel{=}{𝒷})\in \stackrel{=}{\wp }\times \stackrel{=}{ℛ}$.

Example 7.

Consider the two BCFSSs$(\stackrel{=}{\phi }, \stackrel{=}{\wp })$and$(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$over$\stackrel{=}{T}$presented in Example 3. The AND operation is designated as$(\stackrel{=}{\phi }, \stackrel{=}{\wp })\wedge (\stackrel{=}{\varphi }, \stackrel{=}{ℛ})=(\stackrel{=}{H}, \stackrel{=}{\wp }\times \stackrel{=}{ℛ})$, where$\stackrel{=}{\wp }\times \stackrel{=}{ℛ}=\{\stackrel{=}{{\mathfrak{p}}_1}, \stackrel{=}{{\mathfrak{p}}_3}, \stackrel{=}{{\mathfrak{p}}_5}\}\times \{\stackrel{=}{{\mathfrak{p}}_1}, \stackrel{=}{{\mathfrak{p}}_2}, \stackrel{=}{{\mathfrak{p}}_5}\}=\{ (\stackrel{=}{{\mathfrak{p}}_1},\stackrel{=}{{\mathfrak{p}}_1}), (\stackrel{=}{{\mathfrak{p}}_1},\stackrel{=}{{\mathfrak{p}}_2}), (\stackrel{=}{{\mathfrak{p}}_1},\stackrel{=}{{\mathfrak{p}}_5}), (\stackrel{=}{{\mathfrak{p}}_3},\stackrel{=}{{\mathfrak{p}}_1}), (\stackrel{=}{{\mathfrak{p}}_3},\stackrel{=}{{\mathfrak{p}}_2}),(\stackrel{=}{{\mathfrak{p}}_3},\stackrel{=}{{\mathfrak{p}}_5}), (\stackrel{=}{{\mathfrak{p}}_5},\stackrel{=}{{\mathfrak{p}}_1}), (\stackrel{=}{{\mathfrak{p}}_5},\stackrel{=}{{\mathfrak{p}}_2}), (\stackrel{=}{{\mathfrak{p}}_5},\stackrel{=}{{\mathfrak{p}}_5})\}$and is presented as

$$\begin{array}{l}\left(\stackrel{=}{H}, \stackrel{=}{\wp }\times \stackrel{=}{\mathcal{R}}\right)\\ =\left\{\begin{array}{c}\stackrel{=}{H}\left(\stackrel{=}{{\mathfrak{p}}_1},\stackrel{=}{{\mathfrak{p}}_1} \right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.4+i 0.3 , -0.2-i 0.7\right), \left(\stackrel{=}{{\tau }_2},0.65+i 0.6 , -0.21-i 0.4\right), \\ \left(\stackrel{=}{{\tau }_3},0.5+i 0.4 , -0.37-i 0.36\right),\left(\stackrel{=}{{\tau }_4},0.2+i 0.4 , -0.3-i 0.44\right)\end{array}\right\}\\ \stackrel{=}{H}\left(\stackrel{=}{{\mathfrak{p}}_1},\stackrel{=}{{\mathfrak{p}}_2}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.15+i 0.34 , -0.4-i 0.6\right), \left(\stackrel{=}{{\tau }_2},0.24+i 0.62, -0.17-i 0.5\right), \\ \left(\stackrel{=}{{\tau }_3},0.5+i 0.4 , -0.43-i 0.36\right),\left(\stackrel{=}{{\tau }_4},0.56+i 0.4 , -0.3-i 0.4\right)\end{array}\right\}\\ \stackrel{=}{H}\left(\stackrel{=}{{\mathfrak{p}}_1},\stackrel{=}{{\mathfrak{p}}_5}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.35+i 0.4 , -0.4-i 0.3\right), \left(\stackrel{=}{{\tau }_2},0.42+i 0.6 , -0.31-i 0.3\right), \\ \left(\stackrel{=}{{\tau }_3},0.5+i 0.4, -0.43-i 0.36\right),\left(\stackrel{=}{{\tau }_4},0.6+i 0.4 , -0.4-i 0.44\right)\end{array}\right\}\\ \stackrel{=}{H}\left(\stackrel{=}{{\mathfrak{p}}_3},\stackrel{=}{{\mathfrak{p}}_1} \right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.34+i 0.23 , -0.2-i 0.3\right), \left(\stackrel{=}{{\tau }_2},0.5+i 0.4 , -0.21-i 0.3\right), \\ \left(\stackrel{=}{{\tau }_3},0.2+i 0.2 , -0.37-i 0.46\right),\left(\stackrel{=}{{\tau }_4},0.2+i 0.6, -0.3-i 0.54\right)\end{array}\right\}\\ \stackrel{=}{H}\left(\stackrel{=}{{\mathfrak{p}}_3},\stackrel{=}{{\mathfrak{p}}_2} \right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.15+i 0.23 , -0.5-i 0.3\right), \left(\stackrel{=}{{\tau }_2},0.24+i 0.4 , -0.17-i 0.3\right), \\ \left(\stackrel{=}{{\tau }_3},0.2+i 0.2 , -0.5-i 0.66\right),\left(\stackrel{=}{{\tau }_4},0.3+i 0.7 , -0.3-i 0.4\right)\end{array}\right\}\\ \stackrel{=}{H}\left(\stackrel{=}{{\mathfrak{p}}_3},\stackrel{=}{{\mathfrak{p}}_5} \right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.34+i 0.23 , -0.4-i 0.3\right), \left(\stackrel{=}{{\tau }_2},0.42+i 0.4 , -0.61-i 0.3\right), \\ \left(\stackrel{=}{{\tau }_3},0.2+i 0.2 , -0.66-i 0.48\right),\left(\stackrel{=}{{\tau }_4},0.3+i 0.6, -0.4-i 0.54\right)\end{array}\right\}\\ \stackrel{=}{H}\left(\stackrel{=}{{\mathfrak{p}}_5},\stackrel{=}{{\mathfrak{p}}_1} \right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.5+i 0.3 , -0.2-i 0.8\right), \left(\stackrel{=}{{\tau }_2},0.66+i 0.6 , -0.21-i 0.3\right), \\ \left(\stackrel{=}{{\tau }_3},0.45+i 0.4, -0.37-i 0.46\right),\left(\stackrel{=}{{\tau }_4},0.2+i 0.5 , -0.3-i 0.54\right)\end{array}\right\}\\ \stackrel{=}{H}\left(\stackrel{=}{{\mathfrak{p}}_5},\stackrel{=}{{\mathfrak{p}}_2} \right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.15+i 0.4 , -0.5-i 0.6\right), \left(\stackrel{=}{{\tau }_2},0.24+i 0.6 , -0.17-i 0.3\right), \\ \left(\stackrel{=}{{\tau }_3},0.45+i 0.4, -0.37-i 0.8\right),\left(\stackrel{=}{{\tau }_4},0.56+i 0.5 , -0.3-i 0.4\right)\end{array}\right\}\\ \stackrel{=}{H}\left(\stackrel{=}{{\mathfrak{p}}_5},\stackrel{=}{{\mathfrak{p}}_5} \right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.35+i 0.4 , -0.4-i 0.3\right), \left(\stackrel{=}{{\tau }_2},0.42+i 0.6 , -0.65-i 0.3\right), \\ \left(\stackrel{=}{{\tau }_3},0.45+i 0.4, -0.37-i 0.48\right),\left(\stackrel{=}{{\tau }_4},0.6+i 0.5 , -0.3-i 0.54\right)\end{array}\right\}\end{array}\right\}\end{array}$$

Proposition 2.

The BCFSSs$(\stackrel{=}{\phi }, \stackrel{=}{\wp })$and$(\stackrel{=}{\varphi }, \stackrel{=}{ℛ})$over$\stackrel{=}{T}$satisfies the idempotent property, i.e.,

1. 

$(\stackrel{=}{\phi }, \stackrel{=}{\wp })\vee (\stackrel{=}{\phi }, \stackrel{=}{\wp })=(\stackrel{=}{\phi }, \stackrel{=}{\wp })$

2. 

$(\stackrel{=}{\phi }, \stackrel{=}{\wp })\wedge (\stackrel{=}{\phi }, \stackrel{=}{\wp })=(\stackrel{=}{\phi }, \stackrel{=}{\wp })$.

Proof 1.

Let $(\stackrel{=}{\phi }, \stackrel{=}{\wp })\vee (\stackrel{=}{\phi }, \stackrel{=}{\wp })=(\stackrel{=}{I}, \stackrel{=}{\wp }\times \stackrel{=}{\wp })$ and $\stackrel{=}{𝒶}\in \stackrel{=}{\wp }$. Then, by definition (21) we have $\stackrel{=}{I}(\stackrel{=}{𝒶}, \stackrel{=}{𝒶})=\stackrel{=}{\phi }(\stackrel{=}{𝒶}){{\displaystyle \cup }}^{}\stackrel{=}{\phi }(\stackrel{=}{𝒶})$, since $\stackrel{=}{\phi }(\stackrel{=}{𝒶}){{\displaystyle \cup }}^{}\stackrel{=}{\phi }(\stackrel{=}{𝒶})=\stackrel{=}{\phi }(\stackrel{=}{𝒶})$ $\Rightarrow$ $\stackrel{=}{I}(\stackrel{=}{𝒶}, \stackrel{=}{𝒶})=\stackrel{=}{\phi }(\stackrel{=}{𝒶})$ $\Rightarrow (\stackrel{=}{I}, \stackrel{=}{\wp }\times \stackrel{=}{\wp })=(\stackrel{=}{\phi }, \stackrel{=}{\wp })$. Hence, $(\stackrel{=}{\phi }, \stackrel{=}{\wp })\vee (\stackrel{=}{\phi }, \stackrel{=}{\wp })=(\stackrel{=}{\phi }, \stackrel{=}{\wp })$. 2. Let $(\stackrel{=}{\phi }, \stackrel{=}{\wp })\wedge (\stackrel{=}{\phi }, \stackrel{=}{\wp })=(\stackrel{=}{H}, \stackrel{=}{\wp }\times \stackrel{=}{\wp })$ and $\stackrel{=}{𝒶}\in \stackrel{=}{\wp }$. Then, by definition (22) we have $\stackrel{=}{H}(\stackrel{=}{𝒶}, \stackrel{=}{𝒶})=\stackrel{=}{\phi }(\stackrel{=}{𝒶}){{\displaystyle \cap }}^{}\stackrel{=}{\phi }(\stackrel{=}{𝒶})$, since $\stackrel{=}{\phi }(\stackrel{=}{𝒶}){{\displaystyle \cap }}^{}\stackrel{=}{\phi }(\stackrel{=}{𝒶})=\stackrel{=}{\phi }(\stackrel{=}{𝒶})$ $\Rightarrow$ $\stackrel{=}{H}(\stackrel{=}{𝒶}, \stackrel{=}{𝒶})=\stackrel{=}{\phi }(\stackrel{=}{𝒶})$ $\Rightarrow (\stackrel{=}{H}, \stackrel{=}{\wp }\times \stackrel{=}{\wp })=(\stackrel{=}{\phi }, \stackrel{=}{\wp })$. Hence, $(\stackrel{=}{\phi }, \stackrel{=}{\wp })\wedge (\stackrel{=}{\phi }, \stackrel{=}{\wp })=(\stackrel{=}{\phi }, \stackrel{=}{\wp })$. □

4. Multi-Attribute Decision-Making Technique

BCFSS has numerous applications to cope with the vagueness and ambiguities which we face in our various daily life issues. In this section, we present a DM procedure for solving information in the setting of BCFSSs and apply it to real-life DM problems.

4.1. DM Procedure

• Utilize $\stackrel{=}{T}$ as a fixed set and $\stackrel{=}{\wp }\subseteq \stackrel{=}{\mathfrak{P}}$ as a set of parameters.
• Take the BCFSS in the tabular representation
• Construct the separate tabular form for both PTG and NTG
• Determine the comparison tables for PTG and NTG by the comparison method presented in [27]. In this article, we interpret the lexicographical order to compare two BCFNs.
• Determine the PTG score and NTG score.
• Determine the final score by subtracting the PTG score from the NTG score.
• Get the highest score, if it appears in $j\text{-}th$ row, then $\stackrel{=}{{\tau }_j}$ will be the best optimal.

4.2. Illustrated Example (Scenario 1)

Suppose a person wants to purchase a new laptop for his business work and considers four different laptops, i.e., $\stackrel{=}{T}=\left\{\stackrel{=}{{\tau }_1}, \stackrel{=}{{\tau }_2}, \stackrel{=}{{\tau }_3}, \stackrel{=}{{\tau }_4}\right\}$ along with the three different attributes (parameters), i.e., $\stackrel{=}{A}=\left\{\stackrel{=}{{\mathfrak{p}}_1}=Battery life , \stackrel{=}{{\mathfrak{p}}_2}=light weight, \stackrel{=}{{\mathfrak{p}}_3}=Costly\right\}\subseteq \stackrel{=}{\mathfrak{P}}$. Based on these attributes he wants to find the best laptop to purchase. The information is interpreted in the setting of BCFSSs as follows

$$\begin{array}{cccc}\left(\stackrel{=}{\phi }, \stackrel{=}{\wp }\right)& & & \\ =& \{& \begin{array}{c}\stackrel{=}{\phi }\left(\stackrel{=}{{\mathfrak{p}}_1}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.2+i 0.6 , -0.6-i 0.9\right), \left(\stackrel{=}{{\tau }_2},0.45+i 0.9 , -0.11-i 0.4\right), \\ \left(\stackrel{=}{{\tau }_3},0.7+i 0.3 , -0.23-i 0.56\right),\left(\stackrel{=}{{\tau }_4},0.8+i 0.2 , -0.3-i 0.22\right)\end{array}\right\}\\ \stackrel{=}{\phi }\left(\stackrel{=}{{\mathfrak{p}}_2}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.15+i 0.34 , -0.2-i 0.5\right), \left(\stackrel{=}{{\tau }_2},0.23+i 0.8 , -0.9-i 0.5\right), \\ \left(\stackrel{=}{{\tau }_3},0.35+i 0.6 , -0.46-i 0.28\right),\left(\stackrel{=}{{\tau }_4},0.4+i 0.4 , -0.6-i 0.34\right)\end{array}\right\}\\ \stackrel{=}{\phi }\left(\stackrel{=}{{\mathfrak{p}}_3}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.3+i 0.6 , -0.4-i 1\right), \left(\stackrel{=}{{\tau }_2},0.46+i 0.4 , -0.45-i 0.5\right), \\ \left(\stackrel{=}{{\tau }_3},0.65+i 0.36 , -0.77-i 0.4\right),\left(\stackrel{=}{{\tau }_4},0.8+i 0.8 , -0.4-i 0.68\right)\end{array}\right\}\end{array}& \}\end{array}$$

The tabular description of BCFSS interpreted in scenario 1 is depicted in

Table 3. The tabular representation of BCFSS in scenario 1.

$(\stackrel{=}{\mathit{\phi }}, \stackrel{=}{\mathit{\wp }})$	$\stackrel{=}{{\mathfrak{p}}_1}$	$\stackrel{=}{{\mathfrak{p}}_2}$	$\stackrel{=}{{\mathfrak{p}}_3}$
$\stackrel{=}{{\tau }_1}$	$\left(\begin{array}{c}0.2+i 0.6 , \\ -0.6-i 0.9\end{array}\right)$	$\left(\begin{array}{c}0.15+i 0.34 , \\ -0.2-i 0.5\end{array}\right)$	$\left(\begin{array}{c}0.3+i 0.6 , \\ -0.4-i 1\end{array}\right)$
$\stackrel{=}{{\tau }_2}$	$\left(\begin{array}{c}0.45+i 0.9 , \\ -0.11-i 0.4\end{array}\right)$	$\left(\begin{array}{c}0.23+i 0.8 , \\ -0.9-i 0.5\end{array}\right)$	$\left(\begin{array}{c}0.46+i 0.4 , \\ -0.45-i 0.5\end{array}\right)$
$\stackrel{=}{{\tau }_3}$	$\left(\begin{array}{c}0.7+i 0.3 , \\ -0.23-i 0.56\end{array}\right)$	$\left(\begin{array}{c}0.35+i 0.6 , \\ -0.46-i 0.28\end{array}\right)$	$\left(\begin{array}{c}0.65+i 0.36 , \\ -0.77-i 0.4\end{array}\right)$
$\stackrel{=}{{\tau }_4}$	$\left(\begin{array}{c}0.8+i 0.2 , \\ -0.3-i 0.22\end{array}\right)$	$\left(\begin{array}{c}0.4+i 0.4 , \\ -0.6-i 0.34\end{array}\right)$	$\left(\begin{array}{c}0.8+i 0.8 , \\ -0.4-i 0.68\end{array}\right)$

.

Here, our goal is to find the best laptop. The tabular form for both PTG and NTG are given in

Table 4. The tabular description PTG scenario 1.

$.$	$\stackrel{=}{{\mathfrak{p}}_1}$	$\stackrel{=}{{\mathfrak{p}}_2}$	$\stackrel{=}{{\mathfrak{p}}_3}$
$\stackrel{=}{{\tau }_1}$	$0.2+i 0.6$	$0.15+i 0.34$	$0.3+i 0.6$
$\stackrel{=}{{\tau }_2}$	$0.45+i 0.9$	$0.23+i 0.8$	$0.46+i 0.4$
$\stackrel{=}{{\tau }_3}$	$0.7+i 0.3$	$0.35+i 0.6$	$0.65+i 0.36$
$\stackrel{=}{{\tau }_4}$	$0.8+i 0.2$	$0.4+i 0.4$	$0.8+i 0.8$

and

Table 5. The tabular description of NTG scenario 1.

$.$	$\stackrel{=}{{\mathfrak{p}}_1}$	$\stackrel{=}{{\mathfrak{p}}_2}$	$\stackrel{=}{{\mathfrak{p}}_3}$
$\stackrel{=}{{\tau }_1}$	$-0.6-i 0.9$	$-0.2-i 0.5$	$-0.4-i 1$
$\stackrel{=}{{\tau }_2}$	$-0.11-i 0.4$	$-0.9-i 0.5$	$-0.45-i 0.5$
$\stackrel{=}{{\tau }_3}$	$-0.23-i 0.56$	$-0.46-i 0.28$	$-0.77-i 0.4$
$\stackrel{=}{{\tau }_4}$	$-0.3-i 0.22$	$-0.6-i 0.34$	$-0.4-i 0.68$

, respectively.

The comparison tables for both PTG and NTG are in

Table 6. The comparison table for PTG scenario 1.

$.$	$\stackrel{=}{{\mathit{\tau }}_1}$	$\stackrel{=}{{\mathit{\tau }}_2}$	$\stackrel{=}{{\mathit{\tau }}_3}$	$\stackrel{=}{{\mathit{\tau }}_4}$
$\stackrel{=}{{\tau }_1}$	$3$	$0$	$0$	$0$
$\stackrel{=}{{\tau }_2}$	$3$	$3$	$0$	$0$
$\stackrel{=}{{\tau }_3}$	$3$	$3$	$3$	$0$
$\stackrel{=}{{\tau }_4}$	$3$	$3$	$3$	$3$

and

Table 7. The comparison table for NTG scenario 1.

$.$	$\stackrel{=}{{\mathit{\tau }}_1}$	$\stackrel{=}{{\mathit{\tau }}_2}$	$\stackrel{=}{{\mathit{\tau }}_3}$	$\stackrel{=}{{\mathit{\tau }}_4}$
$\stackrel{=}{{\tau }_1}$	$3$	$1$	$1$	$2$
$\stackrel{=}{{\tau }_2}$	$2$	$3$	$1$	$2$
$\stackrel{=}{{\tau }_3}$	$2$	$2$	$3$	$1$
$\stackrel{=}{{\tau }_4}$	$1$	$1$	$1$	$3$

, respectively.

The PTG score and NTG score are described in

Table 8. PTG score table scenario 1.

$.$	Row Sum $\stackrel{=}{\mathit{R}}\stackrel{=}{\mathit{s}}$	Column Sum $\stackrel{=}{\mathit{C}}\stackrel{=}{\mathit{s}}$	Membership Score $\stackrel{=}{\mathit{R}}\stackrel{=}{\mathit{s}}-\stackrel{=}{\mathit{C}}\stackrel{=}{\mathit{s}}$
$\stackrel{=}{{\tau }_1}$	$3$	$12$	$-9$
$\stackrel{=}{{\tau }_2}$	$6$	$9$	$-3$
$\stackrel{=}{{\tau }_3}$	$9$	$6$	$3$
$\stackrel{=}{{\tau }_4}$	$12$	$3$	$9$

and

Table 9. NTG score table scenario 1.

$.$	Row Sum $\stackrel{=}{\mathit{R}}\stackrel{=}{\mathit{s}}$	Column Sum $\stackrel{=}{\mathit{C}}\stackrel{=}{\mathit{s}}$	Membership Score $\stackrel{=}{\mathit{R}}\stackrel{=}{\mathit{s}}-\stackrel{=}{\mathit{C}}\stackrel{=}{\mathit{s}}$
$\stackrel{=}{{\tau }_1}$	$7$	$8$	$-1$
$\stackrel{=}{{\tau }_2}$	$8$	$7$	$1$
$\stackrel{=}{{\tau }_3}$	$8$	$6$	$2$
$\stackrel{=}{{\tau }_4}$	$6$	$8$	$-2$

.

Decision 1.

From

Table 10. Final score table scenario 1.

$.$	PTG Score $\stackrel{=}{\mathit{P}}\stackrel{=}{\mathit{s}}$	NTG Score $\stackrel{=}{\mathit{N}}\stackrel{=}{\mathit{s}}$	Final Score $\stackrel{=}{\mathit{P}}\stackrel{=}{\mathit{s}}-\stackrel{=}{\mathit{N}}\stackrel{=}{\mathit{s}}$
$\stackrel{=}{{\tau }_1}$	$-9$	$-1$	$-8$
$\stackrel{=}{{\tau }_2}$	$-3$	$1$	$-4$
$\stackrel{=}{{\tau }_3}$	$3$	$2$	$1$
$\stackrel{=}{{\tau }_4}$	$9$	$-2$	$11$

. The person will purchase the laptop$\stackrel{=}{{\tau }_4}$. If, for some reasons, he does not want to purchase the laptop$\stackrel{=}{{\tau }_4}$, he will purchase$\stackrel{=}{{\tau }_3}$because$\stackrel{=}{{\tau }_3}$is the second best alternative.

4.3. Illustrated Example (Scenario 2, DM in Human Resources)

Imagine a person is the CEO of an e-commerce start-up. His work is growing and requires recruiting the right assets to assist him with understanding the vision of making a leading online retail platform. He would have to recruit people who are capable and skilled in their fields, including procurement, software development, logistics, and operations. Here, suppose that he wants to hire two persons for his business in the shortlisted four applicants, i.e., $\stackrel{=}{T}=\left\{\stackrel{=}{{\tau }_1}, \stackrel{=}{{\tau }_2}, \stackrel{=}{{\tau }_3}, \stackrel{=}{{\tau }_4}\right\}$. He is considering the three important characteristics (Parameters) of these four applicants, i.e., $\stackrel{=}{A}=\left\{\stackrel{=}{{\mathfrak{p}}_1}=Professionalism , \stackrel{=}{{\mathfrak{p}}_2}=Honesty and integrity, \stackrel{=}{{\mathfrak{p}}_3}=Innovative ideas\right\}\subseteq \stackrel{=}{\mathfrak{P}}$. Based on these attributes he wants to select the best two applicants. The information of the applicants based on their characteristics are interpreted in the environment of BCFSSs as follows

$$\begin{array}{l}\left(\stackrel{=}{\phi }, \stackrel{=}{\wp }\right)\\ =\left\{\begin{array}{c}\stackrel{=}{\phi }\left(\stackrel{=}{{\mathfrak{p}}_1}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.9+i 1 , -0.5-i 0.5\right), \left(\stackrel{=}{{\tau }_2},0.2+i 0.1 , -0.3-i 0.4\right), \\ \left(\stackrel{=}{{\tau }_3},0.5+i 0.8 , -0.43-i 0.5\right),\left(\stackrel{=}{{\tau }_4},0.3+i 0.2, -0.3-i 0.3\right)\end{array}\right\}\\ \stackrel{=}{\phi }\left(\stackrel{=}{{\mathfrak{p}}_2}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.5+i 0.8 , -0.4-i 0.66\right), \left(\stackrel{=}{{\tau }_2},0.2+i 0.4, -0.6-i 0.5\right), \\ \left(\stackrel{=}{{\tau }_3},0.5+i 0.9 , -0.6-i 0.8\right),\left(\stackrel{=}{{\tau }_4},0.23+i 0.45 , -0.8-i 0.4\right)\end{array}\right\}\\ \stackrel{=}{\phi }\left(\stackrel{=}{{\mathfrak{p}}_3}\right)=\left\{\begin{array}{c}\left(\stackrel{=}{{\tau }_1},0.65+i 0.75 , -0.2-i 0.5\right), \left(\stackrel{=}{{\tau }_2},0.26+i 0.7, -0.4-i 0.25\right), \\ \left(\stackrel{=}{{\tau }_3},0.65+i 0.36 , -0.77-i 0.4\right),\left(\stackrel{=}{{\tau }_4},0.11+i 0.22, -0.33-i 0.44\right)\end{array}\right\}\end{array}\right\}\end{array}$$

The tabular description of BCFSS interpreted in scenario 2 is depicted in

Table 11. The tabular representation of BCFSS in scenario 2.

$\left(\stackrel{=}{\mathit{\phi }}, \stackrel{=}{\mathit{\wp }}\right)$	$\stackrel{=}{{\mathfrak{p}}_1}$	$\stackrel{=}{{\mathfrak{p}}_2}$	$\stackrel{=}{{\mathfrak{p}}_3}$
$\stackrel{=}{{\tau }_1}$	$\left(\begin{array}{c}0.9+i 1, \\ -0.5-i 0.5\end{array}\right)$	$\left(\begin{array}{c}0.5+i 0.8, \\ -0.4-i 0.66\end{array}\right)$	$\left(\begin{array}{c}0.65+i 0.75, \\ -0.2-i 0.5\end{array}\right)$
$\stackrel{=}{{\tau }_2}$	$\left(\begin{array}{c}0.2+i 0.1, \\ -0.3-i 0.4\end{array}\right)$	$\left(\begin{array}{c}0.2+i 0.4, \\ -0.6-i 0.5\end{array}\right)$	$\left(\begin{array}{c}0.26+i 0.7, \\ -0.4-i 0.25\end{array}\right)$
$\stackrel{=}{{\tau }_3}$	$\left(\begin{array}{c}0.5+i 0.8, \\ -0.43-i 0.5\end{array}\right)$	$\left(\begin{array}{c}0.5+i 0.9, \\ -0.6-i 0.8\end{array}\right)$	$\left(\begin{array}{c}0.65+i 0.36, \\ -0.77-i 0.4\end{array}\right)$
$\stackrel{=}{{\tau }_4}$	$\left(\begin{array}{c}0.3+i 0.2, \\ -0.3-i 0.3\end{array}\right)$	$\left(\begin{array}{c}0.23+i 0.45, \\ -0.8-i 0.4\end{array}\right)$	$\left(\begin{array}{c}0.11+i 0.22 , \\ -0.33-i 0.44\end{array}\right)$

.

Here, our goal is to find out the two best applicants. The tabular form for both PTG and NTG for scenario 2 are given in

Table 12. The tabular description PTG of scenario 2.

$.$	$\stackrel{=}{{\mathfrak{p}}_1}$	$\stackrel{=}{{\mathfrak{p}}_2}$	$\stackrel{=}{{\mathfrak{p}}_3}$
$\stackrel{=}{{\tau }_1}$	$0.9+i 1$	$0.5+i 0.8$	$0.65+i 0.75$
$\stackrel{=}{{\tau }_2}$	$0.2+i 0.1$	$0.2+i 0.4$	$0.26+i 0.7$
$\stackrel{=}{{\tau }_3}$	$0.5+i 0.8$	$0.5+i 0.9$	$0.65+i 0.36$
$\stackrel{=}{{\tau }_4}$	$0.3+i 0.2$	$0.23+i 0.45$	$0.11+i 0.22$

and

Table 13. The tabular description of NTG of scenario 2.

$.$	$\stackrel{=}{{\mathfrak{p}}_1}$	$\stackrel{=}{{\mathfrak{p}}_2}$	$\stackrel{=}{{\mathfrak{p}}_3}$
$\stackrel{=}{{\tau }_1}$	$-0.5-i 0.5$	$-0.4-i 0.66$	$-0.2-i 0.5$
$\stackrel{=}{{\tau }_2}$	$-0.3-i 0.4$	$-0.6-i 0.5$	$-0.4-i 0.25$
$\stackrel{=}{{\tau }_3}$	$-0.43-i 0.5$	$-0.6-i 0.8$	$-0.77-i 0.4$
$\stackrel{=}{{\tau }_4}$	$-0.3-i 0.3$	$-0.8-i 0.4$	$-0.33-i 0.44$

, respectively.

The comparison tables for both PTG and NTG of scenario 2, are

Table 14. The comparison table for PTG of scenario 2.

$.$	$\stackrel{=}{{\mathit{\tau }}_1}$	$\stackrel{=}{{\mathit{\tau }}_2}$	$\stackrel{=}{{\mathit{\tau }}_3}$	$\stackrel{=}{{\mathit{\tau }}_4}$
$\stackrel{=}{{\tau }_1}$	$3$	$3$	$2$	$3$
$\stackrel{=}{{\tau }_2}$	$0$	$3$	$0$	$1$
$\stackrel{=}{{\tau }_3}$	$1$	$3$	$3$	$3$
$\stackrel{=}{{\tau }_4}$	$0$	$2$	$0$	$3$

and

Table 15. The comparison table for NTG of scenario 2.

$.$	$\stackrel{=}{{\mathit{\tau }}_1}$	$\stackrel{=}{{\mathit{\tau }}_2}$	$\stackrel{=}{{\mathit{\tau }}_3}$	$\stackrel{=}{{\mathit{\tau }}_4}$
$\stackrel{=}{{\tau }_1}$	$3$	$1$	$1$	$1$
$\stackrel{=}{{\tau }_2}$	$2$	$3$	$0$	$2$
$\stackrel{=}{{\tau }_3}$	$2$	$3$	$3$	$2$
$\stackrel{=}{{\tau }_4}$	$2$	$1$	$1$	$3$

, respectively.

The PTG score and NTG score are described in

Table 16. PTG score table of scenario 2.

$.$	Row Sum $\stackrel{=}{\mathit{R}}\stackrel{=}{\mathit{s}}$	Column Sum $\stackrel{=}{\mathit{C}}\stackrel{=}{\mathit{s}}$	Membership Score $\stackrel{=}{\mathit{R}}\stackrel{=}{\mathit{s}}-\stackrel{=}{\mathit{C}}\stackrel{=}{\mathit{s}}$
$\stackrel{=}{{\tau }_1}$	$11$	$4$	$7$
$\stackrel{=}{{\tau }_2}$	$4$	$11$	$-7$
$\stackrel{=}{{\tau }_3}$	$10$	$5$	$5$
$\stackrel{=}{{\tau }_4}$	$5$	$10$	$-5$

and

Table 17. NTG score table of scenario 2.

$.$	Row Sum $\stackrel{=}{\mathit{R}}\stackrel{=}{\mathit{s}}$	Column Sum $\stackrel{=}{\mathit{C}}\stackrel{=}{\mathit{s}}$	Membership Score $\stackrel{=}{\mathit{R}}\stackrel{=}{\mathit{s}}-\stackrel{=}{\mathit{C}}\stackrel{=}{\mathit{s}}$
$\stackrel{=}{{\tau }_1}$	$6$	$9$	$-3$
$\stackrel{=}{{\tau }_2}$	$7$	$8$	$-1$
$\stackrel{=}{{\tau }_3}$	$10$	$5$	$5$
$\stackrel{=}{{\tau }_4}$	$7$	$8$	$-1$

of scenario 2.

Decision 2.

From

Table 18. Final score table of scenario 2.

$.$	PTG Score $\stackrel{=}{\mathit{P}}\stackrel{=}{\mathit{s}}$	NTG Score $\stackrel{=}{\mathit{N}}\stackrel{=}{\mathit{s}}$	Final Score $\stackrel{=}{\mathit{P}}\stackrel{=}{\mathit{s}}-\stackrel{=}{\mathit{N}}\stackrel{=}{\mathit{s}}$
$\stackrel{=}{{\tau }_1}$	$7$	$-3$	$10$
$\stackrel{=}{{\tau }_2}$	$-7$	$-1$	$-6$
$\stackrel{=}{{\tau }_3}$	$5$	$5$	$0$
$\stackrel{=}{{\tau }_4}$	$-5$	$-1$	$-4$

, we found out that$\stackrel{=}{{\tau }_1}$and$\stackrel{=}{{\tau }_3}$are the best applicants because they have the highest scores, so the CEO will hire applicants$\stackrel{=}{{\tau }_1}$and$\stackrel{=}{{\tau }_3}$.

5. Comparative Study and Advantages

When a decision analyst collects data or information in the shape of BCFSSs then there does not exist any kind of prevailing notion which can handle this information or data. The above-defined concept of BCFSSs is the only tool to solve this kind of information and help decision analysts to make a decision. Thus, our invented idea is the generalization of prevailing theories such as SS [4], FS [1], FSS [19], BFS [26], CFS [32], CFSS [34], BCFS [41] as follows:

• If we employ just one parameter, then the invented BCFSS will degenerate to BCFS.
• If we overlook the NTG, then the invented BCFSS will degenerate to CFSS.
• When we utilize just one parameter and overlook the NTG, then the BCFSS will degenerate to CFS.
• When we overlook the NTG and let the unreal part equal to zero in the PTG, then the BCFSS will condense to FSS.
• When we consider the single parameter and let the unreal part of both PTG and NTG equal to zero, then the established BCFSS will condense to BFS.
• When we consider the single parameter, overlook the NTG, and let the unreal part equal to zero in the PTG, then the established BCFSS will condense to FS.
• When we overlook the NTG and let the unreal part equal to zero in the PTG, then the BCFSS will condense to FSS and if the fuzzy value set of every parameter becomes a crisp set, then the BCFSS degenerates to SS.

If a decision analyst collects the data in the shape of the above prevailing notions, our invented idea can handle it and can be more helpful in modeling problems. To acquire satisfaction and efficiency, the BCFSS is a significant and valuable tool to handle hurdles and ambiguous notions in genuine life troubles. We fuse three ideas, i.e., parameterization, complex fuzziness, and bipolarity, which make the mathematical structure of BCFSS more modified and desirable as compared to prevailing ideas, as discussed above.

6. Conclusions

In this article, we invented the concept of BCFSS which is the combination of BCFS and SS. The concept of BCFS modifies many prevailing notions such as SS, FS, FSS, BFS, CFS, CFSS, and BCFS which we discussed in the comparative section. We also interpreted the tabular form of the invented BCFSS. Additionally, in this study, we invented the elementary operations for BCFSS, such as a complement, union, intersection, extended intersection, and related properties, and illustrated them with the assistance of examples. The OR and AND operations for the bipolar complex fuzzy soft set are presented in this study, along with examples. Furthermore, we interpreted the decision-making algorithm and real-life examples (scenario 1 and scenario 2) for BCFSS to display the success and useability of the invented BCFSS. Finally, a comparative study of invented notions with some prevailing ideas is also discussed in this article to demonstrate the efficiency and powerfulness of the invented work.

In the future, we will employ the invented work in various domains such as CHFS [44] and picture fuzzy N-SS [45].