Interval-valued bipolar complex fuzzy soft sets and their applications in decision making

There are a lot of problems, which are located in the complexity and vagueness. These issues are properly addressed with the decision-making (DM) approach. When people or organizations have to choose one course of action or alternative from many, decision-making problems arise. Complexity, ambiguity, several conflicting goals, and finite resources are common features of these problems. Here, we'd like to talk about some common problems. The first step is determining how to allocate limited resources, such as time, money, or staff, among numerous projects or activities. The evaluation and management of potential risks and uncertainties connected to several options or conclusions come in second. Thirdly, making decisions that will last a long time in order to achieve specific goals or objectives, like expanding operations or entering new markets. Fourth, predicting future trends or events based on the knowledge and data at hand. Fifth, determining the optimum course of action that, given the constraints and objectives, maximizes advantages or reduces disadvantages. Sixth, moral and ethical considerations are made when making decisions that may have an impact on stakeholders or offer ethical dilemmas. Seventhly, employing a range of tools and techniques for decision-making, such as decision trees, scenario analysis, or cost–benefit analysis, to make reasonable conclusions. Different contexts, such as those involving business, government, healthcare, finances, and personal life, might give rise to these problems. Acquiring relevant information, weighing options, weighing pros and disadvantages, and taking into account the impacts and outcomes of various selections are all necessary for making informed judgments. Numerous researchers, using this tool for solving a lot of issues. Edwards¹ initiated the theory of DM. Slovic et al.² signified DM. Eisenhardt and Zbaracki³ initiated the strategic DM. March⁴ is also propound the primer on DM.

Numerous problems are typically discovered in the ambiguity and intricacy. To overcome these obstacles, the specialists made their best effort. The fuzzy set (FS) was started by Zadeh⁵ under this conception of obstacles. The TG that makes up FS is located between [0, 1]. When academics attempt to address difficulties, new dilemmas develop as a result. There is Atanassov⁶ started intuitionistic FS (IFS), which is made up of TG and non-TG and lies in [0, 1]. The researchers started looking for a solution to these problems. There is a probability that multiple dimensions will be required to solve a problem. A complex fuzzy set (CFS) in its polar version was first introduced by Ramot et al.⁶. After this, Tamir et al.⁷ initiated CFS which is consist of real and unreal part. The addition of real and unreal are lies in [0, 1]. FS is tackled most of the dilemmas but some of them are require positive and negative aspects. Zhang⁸ demonstrated bipolar FS (BFS) which is consist of positive TG (PTG) and negative TG (NTG). The PTG lies in [0, 1] and NTG lies in [− 1, 0]. Later on, Mahmood and Rehman⁹ initiated bipolar complex FS (BCFS). BCFS is consist of \(PTG+\iota PTG\in [\mathrm{0,1}]\), \(NTG+\iota NTG\in [-\mathrm{1,0}]\) and ι = √(− 1) with unit square of complex plane. There are numerous aggregation operators (AOs) on the BCFS like Hamacher AOs demonstrated by Mahmood et al.¹⁰ and Dombi AOs signified by Mahmood and Rehman¹¹.

Moreover, another thing which was more requirement for removing the hurdles in the vagueness and complexity. It’s known as soft set (SS). SS is propounded by Molodtsov¹², which plays an important role in tackles the above dilemmas. SS is the second most prevalent notion after FS. More dilemmas are sort out by combining more than one notions like fuzzy soft set (FSS). The combination of FS and SS are demonstrated by Cagman et al.¹³, which is known as FSS. From this FSS solve more dilemmas which are required of both alternatives and attributes. Cagman and Karataş¹⁴ initiated intuitionistic FSS (IFSS). After that, Arora and Garg initiated a robust aggregation of IFSS. All trials of the researchers are going on and for best solutions. The picture FSS (PFSS) initiated by Dhumras and Bajaj¹⁵.

The efficiency of dimension Thirunavukarasu et al.¹⁶ propounded the theory of complex FSS (CFSS). Additionally, by adding negative aspect to FSS becomes bipolar FSS. For this, Abdullah et al.¹⁷ signified bipolar fuzzy soft sets (BFSS) and its consequences in DM dilemmas. For more attention, Mahmood et al.¹⁸ signified bipolar CFSS (BCFSS) with applications. BCFSS is consist of \(PTG + \iota PTG \in \left[ {0,1} \right]\), \(NTG + \iota NTG \in \left[ { - 1,0} \right]\) and \(\iota = \sqrt { - 1}\) with unit square of the complex plane. Moreover, Jaleel¹⁹ initiated the Dombi AOs in the basis of BCFSS.

By the way, researchers want to improve the prevailing notions. Zadeh²⁰ initiated interval valued FSs (IVFSs). In the interval two values are signified one lower TG (LTG) and upper TG (UTG) belong to [0, 1]. Moreover, researchers are remove the dimensional dilemmas and create a novel notion. For this, Nasir et al.²¹ demonstrated the notion of interval-valued CFSs (IVCFSs). Moreover, Yang et al.²² initiated the combination of IVFS and SS. Additionally, Peng et al.²³ signified interval-valued fuzzy soft (IVFSS) DM approaches through on MABAC, SM and EDAS. Later on Wei et al.²⁴ initiated multiple attribute DM (MADM) with interval-valued BFS (IVBFS). In this notion, researcher add the negative aspect to IVFS and created IVBFS. Most of the dilemmas can handle easily by the above notions but some of them are require attribute aspect. For this, Feng et al.²⁵ signified interval-valued FSSs (IVFSSs) and SS level application in DM and tackled all hurdles which have created by efficiency of attributes. Furthermore, the researcher try to define AOs like, performance of the MADM approach with interval-valued spherical fuzzy Dombi AOs (IVSFDAO) initiated by Hussain et al.²⁶. Liaqat et al.²⁷ initiated Aczel-Alsina AOs based on Interval-valued complex single-valued neutrosophic set and their use in DM dilemmas.

In the “Preliminaries” section, we discussed the basic definitions of the prevailing notions. In the “Interval-valued bipolar complex fuzzy sets and interval-valued bipolar complex fuzzy soft sets” section, we demonstrated the definitions of IVBCFS and IVBCFSS with properties and examples. Here, we invented extended union, extended intersection, restricted union and restricted intersection of IVBCFSS with examples. Here, we also demonstrated OR and AND of IVBCFSS with their examples. In “Interval-valued bipolar complex fuzzy soft AOs” section, we demonstrated IVBCFSAA and IVBCFSGA operators with their properties. In “Multi-attribute decision making technique” section, we utilize DM technique for data and sort out the result. In “Comparative analysis” section, we compare our invented work with prevailing works and show the supremacy and dominance of our work. In “Conclusion” section, we wrap up our findings and conclusions. In the end, we provide an overview of the proposed ideas and provide examples.

In this segment, we demonstrate some basic definitions of the prevailing notions like, FS, SS, BFS, CFS, BCFS, BCFSS, IVFS, and IVFSS.

Definition 1

(March⁴): A FS \(\mathfrak{L}\) over \(\k{\rm U}\) of the shape \(\mathfrak{L} = \left\{ {\left( {\imath , {\rm E}_{ \mathfrak{L} } \left( \imath \right)} \right)|\imath \in \k{\rm U} } \right\}\) where \({\rm E}_{ \mathfrak{L} } \left( \imath \right):\k{\rm U} \to \left[ {0, 1} \right]\) is TG.

Definition 2

(Mahmood and Rehman¹¹): Let \(\k{\rm U}\) be the set of universe, the set of parameter \(\Upsilon\) and \(\Gamma \subseteq \Upsilon\), the pair \(\left( { \mathfrak{L} ,{ }\Gamma } \right)\) is known as SS, where \(\mathfrak{L} :\Gamma \to `{P}\left( {\k{\rm U} } \right)\). \(`{P}\left( {\k{\rm U} } \right)\) is the power set of \(\k{\rm U}\).

Definition 3

(Tamir et al.⁷): A BFS \(\mathfrak{L}\) over \(\k{\rm U}\) is of the shape \(\mathfrak{L} = \left\{ {\left( {\imath , {\rm E}_{ \mathfrak{L} }^{ + } \left( \imath \right), \;{\rm E}_{ \mathfrak{L} }^{ - } \left( \imath \right)} \right)|\imath \in \k{\rm U} } \right\}\) where \({\rm E}_{{\mathfrak{L}}}^{ + } :\k{\rm U} \to \left[ {0, 1} \right]\), \({\rm E}_{{\mathfrak{L}}}^{ - } :\k{\rm U} \to \left[ { - 1, 0} \right]\) are PTG and NTG.

Definition 4

(Zadeh⁵): A CFS \(\mathfrak{L}\) over \(\k{\rm U}\) is of the shape \(\mathfrak{L} = \left\{ {\left( {\imath , {\rm E}_{ \mathfrak{L} } \left( \imath \right)} \right)|\imath \in \k{\rm U} } \right\} = \left\{ {\left( {\imath , {\rm N}_{ \mathfrak{L} } \left( \imath \right) + \iota {\rm O}_{ \mathfrak{L} } \left( \imath \right)} \right)|\imath \in \k{\rm U} } \right\}\), where \({\rm E}_{ \mathfrak{L} } \left( \imath \right)\) is TG , \({\rm N}_{ \mathfrak{L} } \left( \imath \right), {\rm O}_{ \mathfrak{L} } \left( \imath \right) \in \left[ {0, 1} \right]\) and \(\iota = \sqrt { - 1}\).

Definition 5

(Zhang⁸): A BCFS \(\mathfrak{L}\) is of the shape

where \({\rm E}_{ \mathfrak{L} }^{ + } \left( \imath \right) = {\rm N}_{ \mathfrak{L} }^{ + } \left( \imath \right) + \iota {\rm O}_{ \mathfrak{L} }^{ + } \left( \imath \right)\) denote PTG and \({\rm E}_{ \mathfrak{L} }^{ - } \left( \imath \right) = {\rm N}_{ \mathfrak{L} }^{ - } \left( \imath \right) + \iota {\rm O}_{ \mathfrak{L} }^{ - } \left( \imath \right)\) denotes NTG. The values of \({\rm E}_{ \mathfrak{L} }^{ + } \left( \imath \right)\) and \({\rm E}_{ \mathfrak{L} }^{ - } \left( \imath \right)\) can get all values are belong in the unit square of complex plane, \(\iota = \sqrt { - 1}\), \({\rm N}_{ \mathfrak{L} }^{ + } \left( \imath \right), {\rm O}_{ \mathfrak{L} }^{ + } \left( \imath \right) \in \left[ {0, 1} \right]\) and \({\rm N}_{ \mathfrak{L} }^{ - } \left( \imath \right), {\rm O}_{ \mathfrak{L} }^{ - } \left( \imath \right) \in \left[ { - 1, 0} \right]\).

Definition 6

(Abdullah et al.¹⁷): Let \(\k{\rm U}\) be the set universe, the set of parameter \(\Upsilon\) and \(\Gamma \subseteq \Upsilon\), the pair \(\left( {{\mathfrak{L}},{ }\Gamma } \right)\) is known as a BCFSS over \(\k{\rm U}\), where \(\mathfrak{L} :\Gamma \to BCFS\left( {\k{\rm U} } \right)\). \(BCFS\left( {\k{\rm U} } \right)\) is the family of all BCFSs of \(\k{\rm U}\). It is presented as

Definition 7

(Zadeh²⁰): An IVFS \(\mathfrak{L}\) over \(\k{\rm U}\) of the shape \(\mathfrak{L} = \left\{ {\left( {\imath , \left[ {{\rm E}_{ \mathfrak{L} }^{L} \left( \imath \right), {\rm E}_{ \mathfrak{L} }^{U} \left( \imath \right)} \right]} \right)|\imath \in \k{\rm U} } \right\}\) where \({\rm E}_{ \mathfrak{L} }^{L} \left( \imath \right):\k{\rm U} \to \left[ {0, 1} \right]\), and \({\rm E}_{ \mathfrak{L} }^{U} \left( \imath \right):\k{\rm U} \to \left[ {0, 1} \right]\) are the mappings. These mappings are called lower and upper degree membership TG.

Definition 8

(Yang et al.²²): Let \(\k{\rm U}\) be the set of universe, the set of parameter \(\Upsilon\) and \(\Gamma \subseteq \Upsilon\), the pair \(\left( { \mathfrak{L} ,{ }\Gamma } \right)\) is known as an IVFSS over \(\k{\rm U}\), where \(\mathfrak{L} :\Gamma \to IVFS\left( {\k{\rm U} } \right)\). \(IVFS\left( {\k{\rm U} } \right)\) is the family of all IVFSs of \(\k{\rm U}\). It is presented as

In this segment, we initiate the definition of interval-valued bipolar complex fuzzy set (IVBCFS), interval-valued bipolar complex fuzzy soft set (IVBCFSS) and their properties and examples.

Interval-valued bipolar complex fuzzy set

Here, we demonstrate IVBCFS and their properties with examples.

Definition 9

Let \(\k{\rm U}\) be the set of universe and \(\c{D} \left( {\left[ {0, 1} \right]} \right)\) and \(\c{D} \left( {\left[ { - 1, 0} \right]} \right)\) be the set of all closed subintervals of the interval \(\left[ {0, 1} \right]\) and \(\left[ { - 1, 0} \right]\), respectively, then an interval-valued BCFS (IVBCFS) \({\mathfrak{L}}\) is defined as

And \({\rm E}^{ + } \left( \imath \right):\k{\rm U} \to \c{D} \left( {\left[ {0, 1} \right]} \right) + \iota \c{D} \left( {\left[ {0, 1} \right]} \right)\) and \({\rm E}^{ - } \left( \imath \right):\k{\rm U} \to \c{D} \left( {\left[ { - 1, 0} \right]} \right) + \iota \c{D} \left( {\left[ { - 1, 0} \right]} \right)\) are the functions of the PTG and NTG, in that order. Also, \({\rm E}^{ + } \left( \imath \right) = \left[ {{\rm N}^{L + } \left( \imath \right), {\rm O}^{U + } \left( \imath \right)} \right] + \iota \left[ {{\rm N}^{L + } \left( \imath \right), {\rm O}^{U + } \left( \imath \right)} \right]\) denote the (lower and upper) PTG, and \({\rm E}^{ - } \left( \imath \right) = \left[ {{\rm N}^{L - } \left( \imath \right), {\rm O}^{U - } \left( \imath \right)} \right] + \iota \left[ {{\rm N}^{L - } \left( \imath \right), {\rm O}^{U - } \left( \imath \right)} \right]\) denote the (lower and upper) NTG, in that order, where \({\rm E}^{ + } \left( \imath \right) = \left[ {{\rm N}^{L + } \left( \imath \right), {\rm O}^{U + } \left( \imath \right)} \right] + \iota \left[ {{\rm N}^{L + } \left( \imath \right), {\rm O}^{U + } \left( \imath \right)} \right]:\k{\rm U} \to \c{D} \left( {\left[ {0, 1} \right]} \right) + \iota \c{D} \left( {\left[ {0, 1} \right]} \right)\) and \({\rm E}^{ - } \left( \imath \right) = \left[ {{\rm N}^{L - } \left( \imath \right), {\rm O}^{U - } \left( \imath \right)} \right] + \iota \left[ {{\rm N}^{L - } \left( \imath \right), {\rm O}^{U - } \left( \imath \right)} \right]:\k{\rm U} \to \c{D} \left( {\left[ { - 1, 0} \right]} \right) + \iota \c{D} \left( {\left[ { - 1, 0} \right]} \right)\). IVBCFS is represented as

Remark 1

For \({\mathbb{A}},{\mathbb{B}}\subseteq {\mathbb{R}}\), we denote and define as .

Elementary operation on IVBCFSs

Here, we want to establish the operational laws like complement, union and intersection of the IVBCFSs with their examples.

Definition 10

The complement of IVBCFS \(\mathfrak{L}\) is demonstrated and defined as

Example 1

Consider an IVBCFS \(\mathfrak{L}\) over \(k{\rm{U}}\) defined in Example 1. The complement \(\mathfrak{L}\) is designated as trails.

Definition 11

The union and intersection of two IVBCFSs \({\mathfrak{L}}_{1}\) and \({\mathfrak{L}}_{2}\) over \(\k{\rm U}\) is signified as

Example 2

Consider two IVBCFSs \({\mathfrak{L}}_{1}\) and \({\mathfrak{L}}_{2}\) over \(\k{\rm U}\) in Example 2. The union and intersection of \({\mathfrak{L}}_{1}\) and \({\mathfrak{L}}_{2}\) are as follows:

Then their union and intersection defined as

Interval-valued Bipolar complex fuzzy soft set

Here, we demonstrate IVBCFSS and their properties with examples.

Definition 12

Suppose that the fixed set \(\k{\rm U}\), the set of parameter \(\Upsilon\) and \(\Gamma \subseteq \Upsilon\), the pair \(\left( { \mathfrak{L} ,{ }\Gamma } \right)\) is called an IVBCFSS over \(\k{\rm U}\), where \(\mathfrak{L} :\Gamma \to IVBCFS\left( {\k{\rm U} } \right)\). \(IVBCFS\left( {\k{\rm U} } \right)\) is the family of all IVBCFSs of \(\k{\rm U}\). It is presented as

Remark 2

For a set \(\k{\rm U} = \left\{ {\imath_{1} , \imath_{2} , \imath_{3} , \ldots , \imath_{{{\tilde{r}} }} } \right\}\) and \(\Gamma = \left\{ {j_{1} , j_{2} , j_{3} , \ldots , j_{{{\tilde{s}} }} } \right\} \subseteq \Upsilon\), the tabular demonstration of IVBCFSS \(\left(\mathfrak{L},\Gamma \right)\) is indicate in Table 1.

Example 3

Suppose that \(\k{\rm U} = \left\{ {\imath_{1} , \imath_{2} , \imath_{3} , \imath_{4} } \right\}\) is the set of four under consideration football players and \(\Gamma =\left\{{j}_{1}=technique, {j}_{2}=Fitness, {j}_{3}=Mindset\right\}\subseteq \Upsilon\) is the set of parameters, then the IVBCFSS is demonstrated as follows

The tabular explanation of IVBCFSS demonstrated in example 1 is indicated in Table 2.

Definition 13

An IVBCFSS \(\left(\mathfrak{L},\Gamma \right)\) is called null IVBCFSS if \(\forall {j}_{\nu }\in \Gamma , \mathfrak{L}\left({j}_{\nu }\right)=\emptyset\) and chosen by empty set \(\emptyset\).

Definition 14

An IVBCFSS \(\left(\mathfrak{L},\Gamma \right)\) is called absolute IVBCFSS if \(\forall j_{\nu } \in \Gamma , \mathfrak{L} \left( {j_{\nu } } \right) = IVBCFS\left( {\k{\rm U} } \right)\).

Definition 15

Assume that \(\left(\mathfrak{L},\Gamma \right)\) and \(\left(\pounds , \text{\yen} \right)\) are two IVBCFSSs over \(\k{\rm U}\), then \(\left(\mathfrak{L},\Gamma \right)\) is is known as IVBCFS subset of \(\left(\pounds , \text{\yen} \right)\) if

1. 1.

   \(\Gamma \subseteq \text{\yen} \),

2. 2.

   \(\forall j\in \Gamma \), \(\mathfrak{L}\left(j\right)\) is IVBCF subset of \(\pounds \left(j\right)\), i.e. for real part

   $$ {\rm N}_{{\mathfrak{L}}}^{L + } \left( {\imath_{\upsilon } } \right) \le {\rm N}_{{\text{\pounds}}}^{L + } \left( {\imath_{\upsilon } } \right),{ }{\rm O}_{{\mathfrak{L}}}^{U + } \left( {\imath_{\upsilon } } \right) \le {\rm O}_{{\text{\pounds}}}^{U + } \left( {\imath_{\upsilon } } \right),{ }{\rm N}_{{\mathfrak{L}}}^{L - } \left( {\imath_{\upsilon } } \right) \ge {\rm N}_{{\text{\pounds}}}^{L - } \left( {\imath_{\upsilon } } \right),{ }{\rm O}_{{\mathfrak{L}}}^{U - } \left( {\imath_{\upsilon } } \right) \ge {\rm O}_{{\text{\pounds}}}^{U - } \left( {\imath_{\upsilon } } \right), $$

   And for imaginary part

   $$ {\rm N}_{{\mathfrak{L}}}^{L + } \left( {\imath_{\upsilon } } \right) \le {\rm N}_{\pounds}^{L + } \left( {\imath_{\upsilon } } \right), {\rm O}_{{\mathfrak{L}}}^{U + } \left( {\imath_{\upsilon } } \right) \le {\rm O}_{\pounds}^{U + } \left( {\imath_{\upsilon } } \right), , {\rm N}_{{\mathfrak{L}}}^{L - } \left( {\imath_{\upsilon } } \right) \ge {\rm N}_{\pounds}^{L - } \left( {\imath_{\upsilon } } \right), {\rm O}_{{\mathfrak{L}}}^{U - } \left( {\imath_{\upsilon } } \right) \ge {\rm O}_{\pounds}^{U - } \left( {\imath_{\upsilon } } \right)\forall \imath_{\upsilon } \in \k{\rm U} . $$

Definition 16

Assume that \(\left( {{\mathfrak{L}}, \Gamma } \right)\) and \(\left( {\pounds, {\text{\yen}} } \right)\) are two IVBCFSSs over \(\k{\rm U}\), then \(\left( {{\mathfrak{L}}, \Gamma } \right)\) and \(\left( {\pounds,{\text{\yen}} } \right)\) are assumed to be IVBCFS equal sets if \(\left( {{\mathfrak{L}}, \Gamma } \right) \subseteq \left( {\pounds, {\text{\yen}} } \right)\) and \(\left( {\pounds, {\text{\yen}} } \right) \subseteq \left( {{\mathfrak{L}}, \Gamma } \right)\).

Elementary operation on IVBCFSSs

Here, we will initiate some elementary operations for IVBCFSSs like complement, union, extended union, intersection, extended intersection and associated properties.

Definition 17

The complement of an IVBCFSS \(\left(\mathfrak{L},\Gamma \right)\) is indicated and described as

Example 4

Consider an IVBCFSS \(\left(\mathfrak{L},\Gamma \right)\) over \(\k{\rm U}\) obtainable in Example 1. The complement of \(\left(\mathfrak{L},\Gamma \right)\) is demonstrated as follows

Definition 18

The restricted intersection of two IVBCFSSs \(\left( {{\mathfrak{L}}, \Gamma } \right)\) and \(\left( {\pounds, {\text{\yen}} } \right)\) over \(\k{\rm U}\) is an IVBCFSS \(\left( {{\text{\AA}}_{R} , {\text{T}}} \right)\), where \({\text{T}} = \Gamma \cap {\text{\yen}} \ne \emptyset\) and \({\text{\AA}}_{R} :{\text{T}} \to IVBCFS\left( {\k{\rm U} } \right)\) considered as \({\text{\AA}}_{R} \left( j \right) = {\mathfrak{L}}\left( j \right) \cap \pounds\left( j \right)\) and indicated by \(\left( {{\text{\AA}}_{R} , {\text{T}}} \right) = \left( {{\mathfrak{L}}, \Gamma } \right) \cap_{R} \left( {\pounds, {\text{\yen}} } \right)\).

Example 5

Suppose that \(\k{\rm U} = \left\{ {\imath_{1} , \imath_{2} , \imath_{3} , \imath_{4} } \right\}\) is the set of dissimilar sights and \(\Upsilon = \left\{ {j_{1} = open\; atmosphere, j_{2} = green \;surroundings, j_{3} = beautiful \;view \;of\;sun, j_{4} = trend\; of\; the \;snowfall} \right\}\) is the set of parameters, \(\Gamma = \left\{ {j_{1} , j_{3} , j_{4} } \right\} \subset \Upsilon\) and \({\text{\yen}} = \left\{ {j_{1} , j_{2} , j_{4} } \right\} \subset \Upsilon\) then the two IVBCFSSs are specified as

Then their restricted intersection is nominated as \(\left( {{\text{\AA}}_{R} , {\text{T}}} \right) = \left( { \mathfrak{L} , \Gamma } \right) \cap_{R} \left( {\pounds , {\text{\yen}} } \right)\), where \({\text{T}} = \Gamma \cap {\text{\yen}} = \left\{ {j_{1} , j_{4} } \right\}\) and presented as

Definition 19

The restricted union of two IVBCFSSs \(\left(\mathfrak{L},\Gamma \right)\) and \(\left( {\pounds ,{\text{\yen}} } \right)\) over \(\k{\rm U}\) is an IVBCFSS \(\left( {\overline{\text{I}}_{R} , {\text{T}}} \right)\), where \({\text{T}} = \Gamma \cap {\text{\yen}} \ne \emptyset\) and \(\overline{\text{I}}_{R} :{\text{T}} \to IVBCFS\left( {\k{\rm U} } \right)\) considered as \(\overline{\text{I}}_{R} \left( j \right) = \mathfrak{L} \left( j \right) \cup \pounds \left( j \right)\) and indicated by \(\left( {\overline{\text{I}}_{R} , {\text{T}}} \right) = \left( { \mathfrak{L} ,{ }\Gamma } \right) \cup_{R} \left( {\pounds ,{\text{\yen}} } \right)\).

Example 6

Consider two IVBCFSSs \(\left(\mathfrak{L},\Gamma \right)\) and \(\left( {\pounds , {\text{\yen}} } \right)\) over \(\k{\rm U}\) is obtainable in the Example 3. Their restricted union is nominated as \(\left( {\overline{\text{I}}_{R} , {\text{T}}} \right) = \left( { \mathfrak{L} ,{ }\Gamma } \right) \cup_{R} \left( {\pounds , {\text{\yen}} } \right)\), where \({\text{T}}=\Gamma \cap \text{\yen} =\left\{{j}_{1}, {j}_{4}\right\}\) and presented as

Definition 20

The extended intersection of two IVBCFSSs \(\left(\mathfrak{L},\Gamma \right)\) and \(\left(\pounds , \text{\yen} \right)\) over \(\k{\rm U}\) is an IVBCFSS \(\left({{\text{\AA}}}_{E}, {\text{T}}\right)\), where \({\text{T}}=\Gamma \cup \text{\yen} \),

and indicated by \(\left({{\text{\AA}}}_{E}, {\text{T}}\right)=\left(\mathfrak{L},\Gamma \right){\cap }_{E}\left(\pounds , \text{\yen} \right)\).

Example 7

Consider the two IVBCFSSs \(\left( {{\mathfrak{L}}, \Gamma } \right)\) and \(\left( {\pounds, {\text{\yen}} } \right)\) over \(\k{\rm U}\) is obtainable in the example 3. Their extended intersection is nominated as \(\left( {{\text{\AA}}_{E} , {\text{T}}} \right) = \left( {{\mathfrak{L}}, \Gamma } \right) \cap_{R} \left( {\pounds, {\text{\yen}} } \right)\), where \({\text{T}} = \Gamma \cup {\text{\yen}} = \left\{ {j_{1} , j_{2} , j_{3} , j_{4} } \right\}\) and is presented as

Definition 21

The extended union of two IVBCFSSs \(\left(\mathfrak{L},\Gamma \right)\) and \(\left(\pounds , \text{\yen} \right)\) over \(\k{\rm U}\) is an IVBCFSS \(\left({ {\bar{\text{I}}} }_{E}, {\text{T}}\right)\), where \({\text{T}}=\Gamma \cup \text{\yen} \),

and indicated by \(\left({ {\bar{\text{I}}} }_{E}, {\text{T}}\right)=\left(\mathfrak{L},\Gamma \right){\cup }_{E}\left(\pounds , \text{\yen} \right)\).

Example 8

Consider the two IVBCFSSs \(\left(\mathfrak{L},\Gamma \right)\) and \(\left(\pounds , \text{\yen} \right)\) over \(\k{\rm U}\) is accessible in the Example 3. Their extended union is nominated as \(\left({ {\bar{\text{I}}} }_{E}, {\text{T}}\right)=\left(\mathfrak{L},\Gamma \right){\cup }_{E}\left(\pounds , \text{\yen} \right)\), where \({\text{T}}=\Gamma \cup \text{\yen} =\left\{{j}_{1}, {j}_{2}, {j}_{3}, {j}_{5}\right\}\) and is presented as

Proposition 1

The IVBCFSSs \(\left(\mathfrak{L},\Gamma \right)\), \(\left(\pounds , \text{\yen} \right)\) and \(\left(/\kern-9pt{\rm E}, {-}{\kern-9pt}{\rm D}\right)\) over \(\k{\rm U}\) satisfies the following properties

1. 1.

   \(\left(\mathfrak{L},\Gamma \right){\cup }_{E}\left(\mathfrak{L},\Gamma \right)=\left(\mathfrak{L},\Gamma \right)\), \(\left(\mathfrak{L},\Gamma \right){\cap }_{E}\left(\mathfrak{L},\Gamma \right)=\left(\mathfrak{L},\Gamma \right)\)

2. 2.

   \(\left(\mathfrak{L},\Gamma \right){\cup }_{E}\emptyset=\left(\mathfrak{L},\Gamma \right)\), \(\left(\mathfrak{L},\Gamma \right){\cap }_{E}\emptyset=\emptyset\)

3. 3.

   \(\left(\mathfrak{L},\Gamma \right){\cup }_{E}\left(\left(\mathfrak{L},\Gamma \right){\cap }_{E}\left(\pounds , \text{\yen} \right)\right)=\left(\mathfrak{L},\Gamma \right)\), \(\left(\mathfrak{L},\Gamma \right){\cap }_{E}\left(\left(\mathfrak{L},\Gamma \right){\cup }_{E}\left(\pounds , \text{\yen} \right)\right)=\left(\mathfrak{L},\Gamma \right)\)

4. 4.

   \(\left(\mathfrak{L},\Gamma \right){\cup }_{E}\left(\pounds , \text{\yen} \right)=\left(\pounds , \text{\yen} \right){\cup }_{E}\left(\mathfrak{L},\Gamma \right)\), \(\left(\mathfrak{L},\Gamma \right){\cap }_{E}\left(\pounds , \text{\yen} \right)=\left(\pounds , \text{\yen} \right){\cap }_{E}\left(\mathfrak{L},\Gamma \right)\)

5. 5.

   \(\left(\mathfrak{L},\Gamma \right){\cup }_{E}\left(\left(\pounds , \text{\yen} \right){\cup }_{E}\left(/\kern-9pt{\rm E}, {-}{\kern-9pt}{\rm D}\right)\right)=\left(\left(\mathfrak{L},\Gamma \right){\cup }_{E}\left(\pounds , \text{\yen} \right)\right){\cup }_{E}\left(/\kern-9pt{\rm E}, {-}{\kern-9pt}{\rm D}\right)\) ,

   \(\left(\mathfrak{L},\Gamma \right){\cap }_{E}\left(\left(\pounds , \text{\yen} \right){\cap }_{E}\left(/\kern-9pt{\rm E}, {-}{\kern-9pt}{\rm D}\right)\right)=\left(\left(\mathfrak{L},\Gamma \right){\cap }_{E}\left(\pounds , \text{\yen} \right)\right){\cap }_{E}\left(/\kern-9pt{\rm E}, {-}{\kern-9pt}{\rm D}\right)\)

6. 6.

   \(\left(\mathfrak{L},\Gamma \right){\cap }_{E}\left(\left(\pounds , \text{\yen} \right){\cup }_{E}\left(/\kern-9pt{\rm E}, {-}{\kern-9pt}{\rm D}\right)\right)=\left(\left(\mathfrak{L},\Gamma \right){\cap }_{E}\left(\pounds , \text{\yen} \right)\right){\cup }_{E}\left(\left(\mathfrak{L},\Gamma \right){\cap }_{E}\left(/\kern-9pt{\rm E}, {-}{\kern-9pt}{\rm D}\right)\right)\), \(\left(\mathfrak{L},\Gamma \right){\cup }_{E}\left(\left(\pounds , \text{\yen} \right){\cap }_{E}\left(/\kern-9pt{\rm E}, {-}{\kern-9pt}{\rm D}\right)\right)=\left(\left(\mathfrak{L},\Gamma \right){\cup }_{E}\left(\pounds , \text{\yen} \right)\right){\cap }_{E}\left(\left(\mathfrak{L},\Gamma \right){\cup }_{E}\left(/\kern-9pt{\rm E}, {-}{\kern-9pt}{\rm D}\right)\right)\)

7. 7.

   \({\left(\left(\mathfrak{L},\Gamma \right){\cup }_{E}\left(\pounds , \text{\yen} \right)\right)}^{c}={\left(\mathfrak{L},\Gamma \right)}^{c}{{\cap }_{E}\left(\pounds , \text{\yen} \right)}^{c}\), \({\left(\left(\mathfrak{L},\Gamma \right){\cap }_{E}\left(\pounds , \text{\yen} \right)\right)}^{c}={\left(\mathfrak{L},\Gamma \right)}^{c}{{\cup }_{E}\left(\pounds , \text{\yen} \right)}^{c}\)

Proposition 2

The IVBCFSSs \(\left(\mathfrak{L},\Gamma \right)\), \(\left(\pounds , \text{\yen} \right)\) and \(\left(/\kern-9pt{\rm E}, {-}{\kern-9pt}{\rm D}\right)\) over \(\k{\rm U}\) satisfies the following properties

1. 1.

   \(\left( {{\mathfrak{L}}, \Gamma } \right) \cup_{R} \left( {{\mathfrak{L}}, \Gamma } \right) = \left( {{\mathfrak{L}}, \Gamma } \right)\), \(\left( {{\mathfrak{L}}, \Gamma } \right) \cap_{R} \left( {{\mathfrak{L}}, \Gamma } \right) = \left( {{\mathfrak{L}}, \Gamma } \right)\)

2. 2.

   \(\left( {{\mathfrak{L}}, \Gamma } \right) \cup_{R} \emptyset = \left( {{\mathfrak{L}}, \Gamma } \right)\), \(\left( {{\mathfrak{L}}, \Gamma } \right) \cap_{R} \emptyset = \emptyset\)

3. 3.

   \(\left( {{\mathfrak{L}}, \Gamma } \right) \cup_{R} \left( {\left( {{\mathfrak{L}}, \Gamma } \right) \cap_{R} \left( {\pounds, {\text{\yen}} } \right)} \right) = \left( {{\mathfrak{L}}, \Gamma } \right)\), \(\left( {{\mathfrak{L}}, \Gamma } \right) \cap_{R} \left( {\left( {{\mathfrak{L}}, \Gamma } \right) \cup_{R} \left( {\pounds, {\text{\yen}} } \right)} \right) = \left( {{\mathfrak{L}}, \Gamma } \right)\)

4. 4.

   \(\left( {{\mathfrak{L}}, \Gamma } \right) \cup_{R} \left( {\pounds, {\text{\yen}} } \right) = \left( {\pounds, {\text{\yen}} } \right) \cup_{R} \left( {{\mathfrak{L}}, \Gamma } \right)\), \(\left( {{\mathfrak{L}}, \Gamma } \right) \cap_{R} \left( {\pounds, {\text{\yen}} } \right) = \left( {\pounds, {\text{\yen}} } \right) \cap_{R} \left( {{\mathfrak{L}}, \Gamma } \right)\)

5. 5.

   \(\left( {{\mathfrak{L}}, \Gamma } \right) \cup_{R} \left( {\left( {\pounds, {\text{\yen}} } \right) \cup_{R} \left( {/\kern-9pt{\rm E} , {-}{\kern-9pt}{\rm D}} \right)} \right) = \left( {\left( {{\mathfrak{L}}, \Gamma } \right) \cup_{R} \left( {\pounds, {\text{\yen}} } \right)} \right) \cup_{R} \left( {/\kern-9pt{\rm E} , {-}{\kern-9pt}{\rm D}} \right)\),

   \(\left( {{\mathfrak{L}}, \Gamma } \right) \cap_{R} \left( {\left( {\pounds, {\text{\yen}} } \right) \cap_{R} \left( {/\kern-9pt{\rm E} , {-}{\kern-9pt}{\rm D}} \right)} \right) = \left( {\left( {{\mathfrak{L}}, \Gamma } \right) \cap_{R} \left( {\pounds, {\text{\yen}} } \right)} \right) \cap_{R} \left( {/\kern-9pt{\rm E} , {-}{\kern-9pt}{\rm D}} \right)\)

6. 6.

   \( \left( {{\mathfrak{L}}, \Gamma } \right) \cap_{R} \left( {\left( {\pounds, {\text{\yen}} } \right) \cup_{R} \left( {/\kern-9pt{\rm E} , {-}{\kern-9pt}{\rm D}} \right)} \right) = \left( {\left( {{\mathfrak{L}}, \Gamma } \right) \cap_{R} \left( {\pounds, {\text{\yen}} } \right)} \right) \cup_{R} \left( {\left( {{\mathfrak{L}}, \Gamma } \right) \cap_{R} \left( {/\kern-9pt{\rm E} , {-}{\kern-9pt}{\rm D}} \right)} \right), \)

   \(\left( {{\mathfrak{L}}, \Gamma } \right) \cup_{R} \left( {\left( {\pounds, {\text{\yen}} } \right) \cap_{R} \left( {/\kern-9pt{\rm E} , {-}{\kern-9pt}{\rm D}} \right)} \right) = \left( {\left( {{\mathfrak{L}}, \Gamma } \right) \cup_{R} \left( {\pounds, {\text{\yen}} } \right)} \right) \cap_{R} \left( {\left( {{\mathfrak{L}}, \Gamma } \right) \cup_{R} \left( {/\kern-9pt{\rm E} , {-}{\kern-9pt}{\rm D}} \right)} \right)\)

7. 7.

   \(\left( {\left( {{\mathfrak{L}}, \Gamma } \right) \cup_{R} \left( {\pounds, {\text{\yen}} } \right)} \right)^{c} = \left( {{\mathfrak{L}}, \Gamma } \right)^{c} \cap_{R} \left( {\pounds, {\text{\yen}} } \right)^{c}, \; \left( {\left( {{\mathfrak{L}}, \Gamma } \right) \cap_{R} \left( {\pounds, {\text{\yen}} } \right)} \right)^{c} = \left( {{\mathfrak{L}}, \Gamma } \right)^{c} \cup_{R} \left( {\pounds, {\text{\yen}} } \right)^{c}\)

OR and AND operations on IVBCFSSs

Here, we will signify OR and AND operations for IVBCFSSs.

Definition 22

Let that two IVBCFSSs \(\left(\mathfrak{L},\Gamma \right)\) and \(\left(\pounds , \text{\yen} \right)\) over \(\k{\rm U}\), then the OR operation is an IVBCFSS and demonstrated by \(\left(\mathfrak{L},\Gamma \right)\vee \left(\pounds , \text{\yen} \right)=\left( {\bar{\text{I}}} , \Gamma \times \text{\yen} \right)\), where \( {\bar{\text{I}}} \left(\mathfrak{a},\mathfrak{b}\right)=\mathfrak{L}\left(\mathfrak{a}\right)\cup \pounds \left(\mathfrak{b}\right) \forall \left(\mathfrak{a},\mathfrak{b}\right)\in \Gamma \times \text{\yen} \).

Example 9

Consider the two IVBCFSSs \(\left(\mathfrak{L},\Gamma \right)\) and \(\left(\pounds , \text{\yen} \right)\) over \(\k{\rm U}\) interpreted in example 3. The OR operation is signified as \(\left( {{\mathfrak{L}}, \Gamma } \right) \vee \left( {\pounds, {\text{\yen}} } \right) = \left( {\overline{\text{I}}, \Gamma \times {\text{\yen}} } \right)\) where \(\Gamma \times {\text{\yen}} = \left\{ {j_{1} , j_{3} , j_{4} } \right\} \times \left\{ {j_{1} , j_{2} , j_{4} } \right\} = \left\{ {\left( {j_{1} , j_{1} } \right), \left( {j_{1} , j_{2} } \right), \left( {j_{1} , j_{4} } \right), \left( {j_{3} , j_{1} } \right), \left( {j_{3} , j_{2} } \right), \left( {j_{3} , j_{4} } \right), \left( {j_{4} , j_{1} } \right), \left( {j_{4} , j_{2} } \right), \left( {j_{4} , j_{4} } \right)} \right\}\) and is interpreted as

Definition 23

Suppose that two IVBCFSSs \(\left( {{\mathfrak{L}},{ }\Gamma } \right)\) and \(\left( {\pounds, {\text{\yen}} } \right)\) over \(\k{\rm U}\), then the AND operation is an IVBCFSS and demonstrated by \(\left( {{\mathfrak{L}},{ }\Gamma } \right) \wedge \left( {\pounds,{\text{\yen}} } \right) = \left( {{\hat{\text{H}}}, \Gamma \times {\text{\yen}} } \right)\), where \(\widehat{{\text{H}}}\left( {{\mathfrak{a}},{\mathfrak{b}}} \right) = {\mathfrak{L}}\left( {\mathfrak{a}} \right) \cap \pounds\left( {\mathfrak{b}} \right)\;\forall \;\left( {{\mathfrak{a}},{\mathfrak{b}}} \right) \in \Gamma \times {\text{\yen}}\).

Example 10

Consider the two IVBCFSSs \(\left( {{\mathfrak{L}},{ }\Gamma } \right)\) and \(\left( {\pounds, {\text{\yen}} } \right)\) over \(\k{\rm U}\) interpreted in example 3. The AND operation is signified as \(\left( {{\mathfrak{L}},\Gamma } \right) \wedge \left( {\pounds,{\text{\yen}} } \right) = \left( {{\hat{\text{H}}},\Gamma \times {\text{\yen}} } \right)\) where \(\Gamma \times {\text{\yen}} = \left\{ {j_{1} , j_{3} , j_{4} } \right\} \times \left\{ {j_{1} , j_{2} , j_{4} } \right\} = \left\{ {\left( {j_{1} , j_{1} } \right), \left( {j_{1} , j_{2} } \right), \left( {j_{1} , j_{4} } \right), \left( {j_{3} , j_{1} } \right), \left( {j_{3} , j_{2} } \right), \left( {j_{3} , j_{4} } \right), \left( {j_{4} , j_{1} } \right), \left( {j_{4} , j_{2} } \right), \left( {j_{4} , j_{4} } \right)} \right\}\) and is interpreted as

Proposition 3

The IVBCFSSs \(\left(\mathfrak{L},\Gamma \right)\), and \(\left( {\pounds,{\text{\yen}} } \right)\) over \(\k{\rm U}\) satisfies the idempotent property i.e.

1. 1.

   \(\left(\mathfrak{L},\Gamma \right)\vee \left(\mathfrak{L},\Gamma \right)=\left(\mathfrak{L},\Gamma \right)\)

2. 2.

   \(\left(\mathfrak{L},\Gamma \right)\wedge \left(\mathfrak{L},\Gamma \right)=\left(\mathfrak{L},\Gamma \right)\)

Proof

1. 1.

   Let \(\left( {{\mathfrak{L}}, \Gamma } \right) \vee \left( {{\mathfrak{L}}, \Gamma } \right) = \left( {\overline{\text{I}}, \Gamma \times \Gamma } \right)\) and \({\mathfrak{a}} \in \Gamma\). Then \(\overline{\text{I}}\left( {{\mathfrak{a}}, {\mathfrak{a}}} \right) = {\mathfrak{L}}\left( {\mathfrak{a}} \right) \cup {\mathfrak{L}}\left( {\mathfrak{a}} \right)\), since \({\mathfrak{L}}\left( {\mathfrak{a}} \right) \cup {\mathfrak{L}}\left( {\mathfrak{a}} \right) = {\mathfrak{L}}\left( {\mathfrak{a}} \right) \Rightarrow \overline{\text{I}}\left( {{\mathfrak{a}}, {\mathfrak{a}}} \right) = {\mathfrak{L}}\left( {\mathfrak{a}} \right) \Rightarrow \left( {\overline{\text{I}}, \Gamma \times \Gamma } \right) = \left( {{\mathfrak{L}}, \Gamma } \right)\). Hence \(\left( {{\mathfrak{L}}, \Gamma } \right) \vee \left( {{\mathfrak{L}}, \Gamma } \right) = \left( {{\mathfrak{L}}, \Gamma } \right)\).

2. 2.

   Let \(\left( {{\mathfrak{L}},{ }\Gamma } \right) \wedge \left( {{\mathfrak{L}},{ }\Gamma } \right) = \left( {{\hat{\text{H}}},{ }\Gamma \times \Gamma } \right)\) and \({\mathfrak{a}} \in \Gamma\). Then \({\hat{\text{H}}}\left( {{\mathfrak{a}},{ }{\mathfrak{a}}} \right) = {\mathfrak{L}}\left( {\mathfrak{a}} \right) \cap {\mathfrak{L}}\left( {\mathfrak{a}} \right)\), since \({\mathfrak{L}}\left( {\mathfrak{a}} \right) \cap {\mathfrak{L}}\left( {\mathfrak{a}} \right) = {\mathfrak{L}}\left( {\mathfrak{a}} \right) \Rightarrow {\hat{\text{H}}}\left( {{\mathfrak{a}},{ }{\mathfrak{a}}} \right) = {\mathfrak{L}}\left( {\mathfrak{a}} \right) \Rightarrow \left( {{\hat{\text{H}}},{ }\Gamma \times \Gamma } \right) = \left( {{\mathfrak{L}},{ }\Gamma } \right)\). Hence \(\left( {{\mathfrak{L}},{ }\Gamma } \right) \wedge \left( {{\mathfrak{L}},{ }\Gamma } \right) = \left( {{\mathfrak{L}},{ }\Gamma } \right)\).

In this segment, we demonstrate IVBCFSAAOs and IVBCFSGAOs with properties.

Interval-valued bipolar complex fuzzy soft average aggregation operator

Definition 24

Suppose

be the assortment of IVBCFSNs and let \(n_{\upsilon , } t_{\nu }\) are the weight vectors (WVs) for experts \(\imath_{\upsilon } ^{\prime}s\) and \(j_{\nu }^{\prime}s\) respectively, holding \(n_{\upsilon } \ge 0, t_{\nu } \ge 0\) such that \(\mathop \sum \nolimits_{\upsilon = 1}^{{{\tilde{r}}}} n_{\upsilon } = 1\) and \(\mathop \sum \nolimits_{\nu = 1}^{{{\tilde{s}} }} t_{\nu } = 1\). The IVBCFS weighted averaging aggregation (IVBCFSWAA) operator is the function \(IVBCFSWAA:{\mathfrak{L}}^{n} \to {\mathfrak{L}}\) such that

Theorem 1

Suppose

be the collection of IVBCFSNs, then aggregated value of them usage the IVBCFSWAA operator is also a IVBCFSN, and

Proof

For \({\tilde{r}} = 1\), we get \(n_{1} = 1\),

Again, for \({\tilde{s}} = 1\), we get \(t_{1} = 1\) and hence,

Thus, (8) is true for \({\tilde{r}} = 1\) and \({\tilde{s}} = 1\).

Assume that (8) is true for \({\tilde{s}} = {\bar{a}}_{1} + 1\), \({\tilde{r}} = {\tilde{b}}_{1}\) and \({\tilde{s}} = {\bar{a}}_{1}\), \({\tilde{r}} = {\tilde{b}}_{1} + 1\), then it is follows that

Also,

Now for \({\tilde{s}} = {\bar{a}}_{1} + 1\), \({\tilde{r}} = {\tilde{b}}_{1} + 1\), we got

Hence, (8) is true \({\tilde{s}} = {\bar{a}}_{1} + 1\), \({\tilde{r}} = {\tilde{b}}_{1} + 1\), consequently for induction the results are satisfied \(\forall {\tilde{r}} ,{\tilde{s}} \ge\)\(1\).

Now we desire to explain the following properties by the usage the operator IVBCFSWAA.

Theorem 2

(Idempotency): Suppose

be the assortment of IVBCFSNs, then using the IVBCFSWAA operator is also a IVBCFSNs that are all equal, i.e., \({\mathfrak{L}}_{\upsilon \nu }=\mathfrak{L} \forall \upsilon , \nu \) then

Proof

Since \({\mathfrak{L}}_{\upsilon \nu }=\mathfrak{L}=\left({\rm E}_{\mathfrak{L}}^{+}, {\rm E}_{\mathfrak{L}}^{-}\right)=\left(\begin{array}{c}\left[{\rm N}_{\mathfrak{L}}^{L+}, {\rm O}_{\mathfrak{L}}^{U+}\right]+\iota \left[{\rm N}_{\mathfrak{L}}^{L+}, {\rm O}_{\mathfrak{L}}^{U+}\right], \\ \left[{\rm N}_{\mathfrak{L}}^{L-}, {\rm O}_{\mathfrak{L}}^{U-}\right]+\iota \left[{\rm N}_{\mathfrak{L}}^{L-}, {\rm O}_{\mathfrak{L}}^{U-}\right]\end{array}\right) \forall \upsilon , \nu \) then,

Theorem 3

(Boundedness): Suppose

be the assortment of IVBCFSNs.

And \({\mathfrak{L}}_{\upsilon \nu }^{+}=\left(\underset{\nu }{{\text{max}}}\,\underset{\upsilon }{{\text{max}}}\left\{{\rm E}_{{\mathfrak{L}}_{\upsilon \nu }}^{+}\right\}, \underset{\nu }{{\text{min}}}\,\underset{\upsilon }{{\text{min}}}\,\left\{{\rm E}_{{\mathfrak{L}}_{\upsilon \nu }}^{-}\right\}\right)=\left(\begin{array}{c}\left[\underset{\nu }{{\text{max}}}\,\underset{\upsilon }{{\text{max}}}\left\{{\rm N}_{{\mathfrak{L}}_{\upsilon \nu }}^{L+}\right\}, \underset{\mathit{\nu }}{{\text{max}}}\underset{\upsilon }{{\text{max}}}\left\{{\rm O}_{{\mathfrak{L}}_{\upsilon \nu }}^{U+}\right\}\right]+\\ \iota \left[\underset{\nu }{{\text{max}}}\underset{\upsilon }{{\text{max}}}\left\{{\rm N}_{{\mathfrak{L}}_{\upsilon \nu }}^{L+}\right\}, \underset{\mathit{\nu }}{{\text{max}}}\underset{\upsilon }{{\text{max}}}\left\{{\rm O}_{{\mathfrak{L}}_{\upsilon \nu }}^{U+}\right\}\right], \\ \left[\underset{\nu }{{\text{min}}}\,\underset{\upsilon }{{\text{min}}}\,\left\{{\rm N}_{{\mathfrak{L}}_{\upsilon \nu }}^{L-}\right\}, \underset{\nu }{{\text{min}}}\,\underset{\upsilon }{{\text{min}}}\,\left\{{\rm O}_{{\mathfrak{L}}_{\upsilon \nu }}^{U-}\right\}\right]+\\ \iota \left[\underset{\nu }{{\text{min}}}\,\underset{\upsilon }{{\text{min}}}\,\left\{{\rm N}_{{\mathfrak{L}}_{\upsilon \nu }}^{L-}\right\}, \underset{\nu }{{\text{min}}}\,\underset{\upsilon }{{\text{min}}}\,\left\{{\rm O}_{{\mathfrak{L}}_{\upsilon \nu }}^{U-}\right\}\right]\end{array}\right)\)

Then,

Proof

Since, \({\mathfrak{L}}_{\upsilon \nu }=\left({\rm E}_{{\mathfrak{L}}_{\upsilon \nu }}^{+}, {\rm E}_{{\mathfrak{L}}_{\upsilon \nu }}^{-}\right)\) is a IVBCFSN then

Which implies that

Therefore,

Again,

Hence we obtain,

Suppose

then from Eqs. (11) and (13),

And

Then by definition score function

Here, we have three cases be demonstrated:

Case 1 If \({\text{\yen}} \left( {{\mathfrak{L}}_{\upsilon \nu } } \right) < {\text{\yen}} \left( {{\mathfrak{L}}_{\upsilon \nu }^{ + } } \right)\) and \({\text{\yen}} \left( {{\mathfrak{L}}_{\upsilon \nu } } \right) > {\text{\yen}} \left( {{\mathfrak{L}}_{\upsilon \nu }^{ - } } \right)\), by comparison of two IVBCFSNs, we have

Case 2 If \({\text{\yen}} \left( {{\mathfrak{L}}_{\upsilon \nu } } \right) = {\text{\yen}} \left( {{\mathfrak{L}}_{\upsilon \nu }^{ + } } \right)\), i.e., \({\rm E}_{\pounds}^{ + } + {\rm E}_{\pounds}^{ - } = \mathop {\max }\limits_{\nu } \mathop {\max }\limits_{\upsilon } \left\{ {{\rm E}_{{{\mathfrak{L}}_{\upsilon \nu } }}^{ + } } \right\} + \mathop {\min }\limits_{\nu } \mathop {\min }\limits_{\upsilon } \left\{ {{\rm E}_{{{\mathfrak{L}}_{\upsilon \nu } }}^{ - } } \right\}\) then by above inequalities

Thus,

\(\theta = {\rm E}_{{\text{\pounds}}}^{ + } - {\rm E}_{{\text{\pounds}}}^{ - } = \mathop {\max }\limits_{\nu } \mathop {\max }\limits_{\upsilon } \left\{ {{\rm E}_{{{\mathfrak{L}}_{\upsilon \nu } }}^{ + } } \right\} - \mathop {\min }\limits_{\nu } \mathop {\min }\limits_{\upsilon } \left\{ {{\rm E}_{{{\mathfrak{L}}_{\upsilon \nu } }}^{ - } } \right\} = \theta \left( {{\mathfrak{L}}_{\upsilon \nu }^{ + } } \right)\),

Then by comparison of two IVBCFSNs, we have

Case 3 If \({\text{\yen}} \left( { \mathfrak{L}_{\upsilon \nu } } \right) = {\text{\yen}} \left( { \mathfrak{L}_{\upsilon \nu }^{ - } } \right)\), i.e., \({\rm E}_{\pounds}^{ + } + {\rm E}_{\pounds}^{ - } = \mathop {\min }\limits_{\nu } \mathop {\min }\limits_{\upsilon } \left\{ {{\rm E}_{{ \mathfrak{L}_{\upsilon \nu } }}^{ + } } \right\} + \mathop {\max }\limits_{\nu } \mathop {\max }\limits_{\upsilon } \left\{ {{\rm E}_{{ \mathfrak{L}_{\upsilon \nu } }}^{ - } } \right\}\), then by above inequalities.

Thus,

Then by comparison of two IVBCFSNs, we have

Theorem 4

(Shift-invariance property): If

be another IVBCFSN, then

Proof

Since \({\mathfrak{L}}\) and \({\mathfrak{L}}_{\upsilon \nu }\) are IVBCFSNs. Then, we get

Hence,

Theorem 5

(Homogeneity property): For any real number \(\chi > 0\), we have

Proof

Suppose

be the collection of IVBCFSNs and \(\chi > 0\) for any real value. Then,

Thus,

Interval-valued bipolar complex fuzzy soft geometric aggregation operator

Definition 25

Suppose

be the collection of IVBCFSNs and let \({n}_{\upsilon , }{t}_{\nu }\) are the weight vectors (WVs) for experts \({\iota }_{\upsilon }^{\prime}s\) and \({j}_{\nu }^{\prime}s\) respectively, holding \(n_{\upsilon } \ge 0, t_{\nu } \ge 0\) such that \(\mathop \sum \nolimits_{\upsilon = 1}^{{{\tilde{r}} }} n_{\upsilon } = 1\) and \(\mathop \sum \nolimits_{\nu = 1}^{{{\tilde{s}} }} t_{\nu } = 1\). The IVBCFS weighted geometric aggregation (IVBCFSWGA) operator is the function \(IVBCFSWGA:{\mathfrak{L}}^{n}\to \mathfrak{L}\) such that

Theorem 6

Suppose

be the assortment of IVBCFSNs, then aggregated value of them usage the IVBCFSWAA operator is also a IVBCFSN, and

Proof

As like to Theorem 1.

Now we desire to explain the following properties by the usage the operator IVBCFSWGA.

Theorem 7

(Idempotency): Suppose

be the collection of IVBCFSNs, then using the IVBCFSWGA operator is also a IVBCFSNs that are all equal, i.e., \({\mathfrak{L}}_{\upsilon \nu } = {\mathfrak{L}} \forall \upsilon , \nu\) then

Proof

As like to Theorem 2.

Theorem 8

(Boundedness): Suppose

be the collection of IVBCFSNs.

And \({\mathfrak{L}}_{\upsilon \nu }^{+}=\left(\underset{\nu }{{\text{max}}}\,\underset{\upsilon }{{\text{max}}}\left\{{\rm E}_{{\mathfrak{L}}_{\upsilon \nu }}^{+}\right\}, \underset{\nu }{{\text{min}}}\,\underset{\upsilon }{{\text{min}}}\,\left\{{\rm E}_{{\mathfrak{L}}_{\upsilon \nu }}^{-}\right\}\right)=\left(\begin{array}{c}\left[\underset{\nu }{{\text{max}}}\,\underset{\upsilon }{{\text{max}}}\left\{{\rm N}_{{\mathfrak{L}}_{\upsilon \nu }}^{L+}\right\}, \underset{\mathit{\nu }}{{\text{max}}}\,\underset{\upsilon }{{\text{max}}}\left\{{\rm O}_{{\mathfrak{L}}_{\upsilon \nu }}^{U+}\right\}\right]+\\ \iota \left[\underset{\nu }{{\text{max}}}\,\underset{\upsilon }{{\text{max}}}\left\{{\rm N}_{{\mathfrak{L}}_{\upsilon \nu }}^{L+}\right\}, \underset{\mathit{\nu }}{{\text{max}}}\,\underset{\upsilon }{{\text{max}}}\left\{{\rm O}_{{\mathfrak{L}}_{\upsilon \nu }}^{U+}\right\}\right], \\ \left[\underset{\nu }{{\text{min}}}\,\underset{\upsilon }{{\text{min}}}\,\left\{{\rm N}_{{\mathfrak{L}}_{\upsilon \nu }}^{L-}\right\}, \underset{\nu }{{\text{min}}}\,\underset{\upsilon }{{\text{min}}}\,\left\{{\rm O}_{{\mathfrak{L}}_{\upsilon \nu }}^{U-}\right\}\right]+\\ \iota \left[\underset{\nu }{{\text{min}}}\,\underset{\upsilon }{{\text{min}}}\,\left\{{\rm N}_{{\mathfrak{L}}_{\upsilon \nu }}^{L-}\right\}, \underset{\nu }{{\text{min}}}\,\underset{\upsilon }{{\text{min}}}\,\left\{{\rm O}_{{\mathfrak{L}}_{\upsilon \nu }}^{U-}\right\}\right]\end{array}\right)\)

Then,

Proof

As like to Theorem 3.

Theorem 9

(Shift-invariance property): If

be another IVBCFSN, then

Proof

As like to Theorem 4.

Theorem 10

(Homogeneity property): For any real number \(\chi >0\), we have

Proof

As like to Theorem 5.

IVBCFSS has a wide range of uses to deal with the ambiguity and uncertainty that we express in our various real-world problems. In this section, we represent a DM method for elaborating on material in the context of IVBCFSSs and put it to use solving real-world DM issues.

DM process

1. 1.

   By using \(\k{\rm U}\) is the set of universe and \({\mathring{\text{A}}} \subseteq \Upsilon\) be the set of attribute.

2. 2.

   Take IVBCFSS in the presentation of tabular form

3. 3.

   Make single table for every LPTG, UPTG, LNTG, and NTG

4. 4.

   Analyze the comparison tables for LPTG, UPTG, LNTG, and UNTG

5. 5.

   Explore the score of each LPTG, UPTG, LNTG, and UNTG table

6. 6.

   Compute the ultimate score by subtracting the PTG score from NTG score

7. 7.

   Select the greater score, if it appears in \(\mathfrak{a}{\text{th}}\) row, then \({\iota }_{\mathfrak{a}}\) will be the best ideal.

Demonstrated example (set-up 1)

Suppose a man want to buy a innovative mobile for his trade and consider four dissimilar mobiles i.e. \(\k{\rm U} = \left\{ {\imath_{1} , \imath_{2} , \imath_{3} , \imath_{4} } \right\}\) further with the three dissimilar attributes i.e. \({\mathring{\text{A}}} =\left\{{j}_{1}=battery\; timing, {j}_{2}=operating \; system, {j}_{3}=storage\; space\right\}\subseteq \Upsilon\). He desires to purchase the best mobile device under these criteria. The information was presented in the context of the IVBCFSSs below

The tabular representation of IVBCFSS interpreted in set-up 1 is depicted in Table 3.

Here, our purpose to select best mobile. The tabular form for four LPTG, UPTG, LNTG, and UNTG are specified in Tables 4, 5, 6, and 7 respectively.

The comparison tables for four LPTG, UPTG, LNTG and UNTG are specified in Tables 8, 9, 10, and 11 respectively.

The scores of LPTG, UPTG, LNTG and UNTG are specified in Tables 12, 13, 14, and 15 respectively.

In Table 16, we demonstrate final score of Tables 12 and 13 respectively.

In Table 17, we demonstrate final score of Tables 14 and 15 respectively.

Decision the man will purchase the mobile \({\iota }_{2}\) is best option in Table 16. In the rest of some reasons if he doesn’t desire to buy the mobile \({\iota }_{2}\), he will buy \({\iota }_{1}\) or \({\iota }_{4}\) because \({\iota }_{1}\) and \({\iota }_{4}\) are the further best options in view of Table 16.

The man will purchase the mobile \({\iota }_{2}\) in Table 17. In the rest of some reasons if he doesn’t desire to buy the mobile \({\iota }_{2}\), he will buy \({\iota }_{1}\) or \({\iota }_{4}\) because buy \({\iota }_{1}\) and \({\iota }_{4}\) are the further best options in view of Table 17. By the inspection of above both tables should be decide the best option of \({\iota }_{2}\) and further 2^nd options are same in above both Tables 16 and 17.

Demonstrated example (set-up 2)

Suppose a professor want to recruit a research assistant for helping in his research area and consider four dissimilar applicants i.e. \(\k{\rm U} = \left\{ {\imath_{1} , \imath_{2} , \imath_{3} , \imath_{4} } \right\}\) further with the three dissimilar attributes i.e. \({\mathring{\text{A}}} =\left\{{j}_{1}=creative person, {j}_{2}=hardworking person, {j}_{3}=regular person\right\}\subseteq \Upsilon\). He desires to select the best applicant under these criteria. The information was presented in the context of the IVBCFSSs below

The tabular representation of IVBCFSS interpreted in set-up 2 is depicted in Table 18.

Here, our purpose to select best applicant. The tabular form for four LPTG, UPTG, LNTG and UNTG are specified in Tables 19, 20, 21, and 22 respectively.

The comparison tables for four LPTG, UPTG, LNTG and UNTG are specified in Tables 23, 24, 25, and 26 respectively.

The scores of LPTG, UPTG, LNTG and UNTG are specified in Tables 27, 28, 29, and 30 respectively.

In Table 31, we demonstrate final score of Tables 27 and 28 respectively.

In Table 32, we demonstrate final score of Tables 29 and 30 respectively.

Decision the professor will select the applicant \({\iota }_{2}\) is the best option in Table 31. In the rest of some reasons if he doesn’t desire to select applicant \({\iota }_{2}\), he will select \({\iota }_{3}\) because \({\iota }_{3}\) is the further best applicant in view of Table 31.

The professor will applicant \({\iota }_{2}\) is the best option in Table 32. In the rest of some reasons if he doesn’t desire to select \({\iota }_{2}\), he will select \({\iota }_{1}\) or \({\iota }_{4}\) because \({\iota }_{1}\) and \({\iota }_{4}\) are the further best applicant’s options and are also applicable for selection in view of Table 32. By the inspection of above both tables should be decide the best applicant \({\iota }_{2}\), which is approved from above both result tables.

Firstly, the expert arrange data or materials in the form of IVBCFSSs then there doesn’t satisfied the prevailing concepts which can hold this materials or data. The above demonstrated idea IVBCFSSs are exist the data and provided a lot of information for the decision maker. Decision maker can easily sort out all hurdles which are facing in prevailing ideas. In this idea, we signified the generalization of prevailing notions like SS¹¹, FS⁴, FSS¹², BFSS¹⁶ and BCFSSs¹⁷ as follows.

1. 1.

   If we remove the upper and lower approximation then the demonstrated IVBCFSSs will degenerated to BCFSSs.

2. 2.

   If we remove the upper, lower approximation and NTG then the demonstrated IVBCFSSs will degenerated to CFSSs.

3. 3.

   If we remove the upper, lower approximation, parameter and NTG then the demonstrated IVBCFSSs will degenerated to CFSs.

4. 4.

   If we remove the upper, lower approximation, parameter, amplitude term and NTG then the demonstrated IVBCFSSs will degenerated to FSs.

5. 5.

   If we remove the upper, lower approximation, amplitude term, NTG and FS then the demonstrated IVBCFSSs will degenerated to SS.

Our suggested approach can satisfy this kind of data and be more beneficial in problem modelling if a decision maker gathers the data in the form of the aforementioned prevalent notions. The IVBCFSS is a crucial and effective tool for gaining satisfaction and efficiency in order to grasp difficult and ambiguous types of theory in everyday problems. As a result of the fusion of four concepts—parameterization, complex fuzziness, bipolarity, and interval aspects—the mathematical structure of the IVBCFSS is more altered and appealing than it would be if the concepts were used alone, as was the case with the prevailing concepts listed above.

This article officially confirms two terms like IVBCFS and IVBCFSS. Those are the mixture of IVBFS, and CFSS. Here, we also match the established work with dominant notions, like FS, SS, CFS, FSS, BFS, BFSS, BCFS, and BCFSS. We invented IVBCFS and IVBCFSS by adding lower and upper approximations to BCFS and BCFSS. Moreover, BCFSS is required more extension in the novel concept. In the notion of IVBCFS, we demonstrated some operations like complement, union and intersection of IVBCFS with their examples. In the concept of IVBCFSS, we illustrated the significance of IVBCFSS and its applications. In addition, we included examples for the basic operations complement, extended union, extended intersection, restricted union, and restricted intersection. With worked examples, we also presented the OR and AND operations here. Moreover, we also demonstrated some fundamental aggregation operators (AOs) like IVBCFS average aggregation (IVBCFSAA), IVBCFS geometric (IVBCFSGA) operators and with their properties. Additionally, we interpreted the DM method and common instances (set-up 1 and set-up 2) of IVBCFSS to highlight the success and use of the exhibited IVBCFSS. Finally, this study also interprets the efficiency and potency of the established work by a relative investigation of established thought with prevalent ideas. In the IVBCFSSs, we can improve this by applying Dombi AOs, Hamacher AOs and similarity measures and so on.