Prioritization of Strategies of Digital Transformation of Supply Chain Employing Bipolar Complex Fuzzy Linguistic Aggregation Operators

Digitalization has addressed all parts of organizations together with supply chain management. Innovations like installed RFID, GPS, and sensors have assisted organizations with changing their current conventional supply chain framework into more spry, adaptable, open, and cooperative computerized models. A lot of organizations want to get the advantages of digitalization, particularly in the supply chain But here, the biggest problem, that the organizations are facing is the ambiguities, i.e. how to select and evaluate the best strategy for the supply chain digital transformation (SCDT), on which criteria or attribute they can assess the strategies of SCDT, etc. Multi-attribute decision-making (MADM) procedure is the finest technique to evaluate and find out the finest strategy of the SCDT. To handle these ambiguities and for making the MADM technique, this study first establishes the concept of bipolar complex fuzzy linguistic sets (BCFLSs) and its essential properties. Furthermore, we scrutinize average and geometric AOs for BCFLS such as bipolar complex fuzzy linguistic weighted averaging (BCFLWA) and bipolar complex fuzzy linguistic weighted geometric (BCFLWG) operators along with their idempotency, boundedness, and monotonicity properties. After that, to exhibit the usefulness of our established notion and AOs in real life and the selection and evaluation of the strategy of digital transformation of the supply chain, we interpret a DM technique and explore a numerical example. To exhibit the supremacy of the defined BCFLS and defined AOs we compare them with certain prevailing conceptions.

SECTION I. Introduction

In a conventional supply chain, the progression of labor and products includes obtaining and securing unrefined components and parts, planning and creating an item, assessing the interest, orchestrating sale channels, and afterward giving clients perceivability into their orders. In contrast, the digital supply chain gives essentially greater perceivability to the functions of the chain. It is the most common way of incorporating and applying progressed advanced innovations into supply chain activities, from obtainment information, and stock administration to transportation and conveyance. For example, one can get updates on the location of the cargo by utilizing Bluetooth Low Energy resources even if the cargo is in transit. A definitive objective of the digital transformation of the supply chain is to empower experiences for more prominent effectiveness and work with more prominent benefits. Organizations with SCDT can more readily move their assets, people, resources, and stock to where they are required at some random opportunity to diminish costs by answering proactively transport and production threats. The likely settlements of a completely acknowledged SCDT contain saving in various sections, including money, time, and money. Digitizationin the supply chain enables the organization’s preparation, obtaining, and strategies groups to team up, mechanize and influence examination. The few major benefits of the supply chain digital transformation are displayed in Figure 1.

Fig. 1.

The few major benefits of the supply chain digital transformation.

MADM is the technique of taking preference decisions like selection, assessment prioritization, etc., in the presented alternatives which are described by generally multiple different attributes. The MADM issues vary in various areas and the MADM technique is a significant procedure to cope with intricate and tricky data in genuine life problems. To display such sort of intricate and tricky information Zadeh [1] propounded a mathematical structure and named it FS in 1965. FS modified the crisp theory by enlarging the range from a two-point set $\left \{{0,~1}\right \}$ to $\left [{0,~1}\right]$ . Various researchers employed various applications of FS theory [2], [3], [4]. Aggregation operators (AOs) and fuzzy systems modeling was propounded by Yager [5]. Marin et al. [6] initiated a specificity measure relying on the FS similarity. In MADM problems if the information contains positive and negative opinions, then the FSs failed to structure such sort of data. Consequently, Zhang [7] propounded a new mathematical structure, named bipolar FS (BFS) which is the generalization of FS. BFS contains a positive satisfactory degree (PSD) which wrapped in $\left [{0,~1}\right]$ and negative satisfactory degree (NSD) which wrapped in $\left [{-1,\,\,0}\right]$ . Many scholars introduced various techniques in the setting of BFS such as Alghamdi et al. [8] defined multi-criteria decision-making (DM), Akram and Arshad [9] established TOPSIS and ELECTRE-I, Stanujkic et al. [10] diagnosed extension of MULTIMOORA, Alsolame and Alshehri [11] explored the extension of VIKOR, Akram and Al-Kenani [12] diagnosed ELECTRE II. Further, various scholars initiated AOs relying on BFS like Jana et al. [13] described Hamacher AOs, Wei et al. [14] propounded Dombi AOs, and Riaz et al. [15] defined sine trigonometric AOs. Yager and Rybalov [16] explored bipolar aggregation employing uninorms. Graphs are utilized by numerous scholars in the environment of BFS as Akram [17] described bipolar fuzzy (BF) graphs along with applications, Rashmanlou et al. [18] also initiated BF graphs, Akram et al. [19] propounded regular BF graphs, Akram et al. [20] explored BF digraphs. The BF relations and equivalent BF relations were described by Lee and Hur [21] and Dudziak and Pe [22]. The conception of BF soft sets was diagnosed by Abdullah et al. [23].

When in MADM issues if the information contains the $2^{nd}$ dimension i.e. some extra information then the above-stated FS and BFS are failed to structure this kind of data. Thus, Ramot et al. [24] propounded a novel mathematical structure and named it complex FS (CFS) which modified the theory of FS. CFS contains a satisfactory degree in the polar structure, wrapped in the unit disc of a complex plane. Tamir et al. [25] took this work forward and described the satisfactory degree in cartesian structure wrapped in a unit square of the complex plane. The study of graphs in CFS is discussed in [26] and [27] and the study of AOs based on CFS is discussed in [28] and [29]. If the information onthe MADM issues contains positive and negative opinions as well as the $2^{nd}$ dimension i.e. extra information, then above stated FS, BFS, and CFS are failed to structure such type of data. Keeping this in mind, Mahmood and Ur Rehman [30] propounded a novel and wide structure and named it bipolar complex fuzzy set (BCFS), which generalized FS, BFS, and CFS. BCFS contains a PSD wrapped in the first quadrant of the unit square and anNSD wrapped in the third quadrant of the unit square of the complex plane. Because of the rich structure of BCFS, various scholars utilized this conception such as Mahmood et al. [31] propounded Hamacher AOs, Mahmood et al. [32] initiated AOs, and Mahmood and Rehman [33] initiated Dombi AOs. Further, Mahmood et al. [34] propounded bipolar complex fuzzy (BCF) soft sets. Then again, because of the intricacy ofthe information on MADM issues and the vagueness of human discernment, it is hard to structure such sort of information by quantitative values. In such sort of MADM issues and also in some other circumstances various criteria ought to be evaluated in a qualitative structure. For instance, when a reviewer review any manuscript and he/she thinks that the manuscript is excellent because the linguistic assessment is near to human discernment and being assessed by linguistic assessment is reasonable. Thus, firstly, Zadeh [35] propounded the linguistic term (LT) set. Wang et al. [36] propounded intuitionistic linguistic (IL) AOs. Lui and Wang [37] initiated IL power generalized AOs. Lu et al. [38] studied bipolar 2-tuple linguistic AOs.

Usually, the LT set of 7 terms is identified as $\dot {\acute {\text {S}}}=\left \{{{{\dot {\acute {\text {S}}}_{0}}=extremely\,poor,\dot {\acute {\text {S}}}_{1}}=poor,{\dot {\acute {\text {S}}}_{2}}=marginally\,poor, } {{\dot {\acute {\text {S}}}_{3}}=average,{\dot {\acute {\text {S}}}_{4}}=good,{\dot {\acute {\text {S}}}_{5}}=very\,good,{\dot {\acute {\text {S}}}_{6}} =excellent}\right \}{}$ . The reviewer’s assessment of the manuscript can be signified by ${\dot {\acute {\text {S}}}_{6}}$ , which implies that the manuscript is excellent and a positive satisfactory degree to ${\dot {\acute {\text {S}}}_{6}}$ is 1. Be that as it may, the LT set can’t portray a negative satisfactory degree along with the $2^{nd}$ dimension of the assessment to the LT. This is marginally conflicting with the real condition in certain circumstances, for instance, the reviewer thinks that the PSD of the quality of the manuscript is very good is $0.9+{\iota }\,\,0,8$ , and the NSD of the quality of the manuscript is very good is $-0.4-{\iota }\,\,0.5$ . Then no notion in the literature can model such information and situation. Inspired by this in this study, we introduce bipolar complex fuzzy linguistic set (BCFLS) to model such kind of data which depend on LT set and BCFS. Further, we investigate the MADM technique based on the propounded AOs for BCFLS for the prioritization of the strategies for the digital transformation of the supply chain. This study is organized as: In Section 2, we have revised BCFS, its properties, and the LT set. In Section 3, we propounded the most treasured and expressive conception, named it BCFLS by mixing the conception of BCFSs and LT sets. Moreover, we introduce the essential properties of the defined BCFLS. In Section 4 of the manuscript, we scrutinized average and geometric AOs such as BCFLWA and BCFLWG operators for BCFLNs. Section 5 contained a MADM technique based on BCFLS and a numerical example of the prioritization of strategies for the digital transformation of the supply chain. In Section 6, we did a comparison of the propounded work with prevailing conceptions. The conclusion of this study is established in Section 7.

SECTION II. Preliminaries

In this Section, we are going to revise the conception of BCFS, its properties, and the conception of the LT set. Further, ${\mathfrak {X}_{\mathcal {U}}}$ is the universal set in this analysis.

Definition 1[30]:

The structure of the BCFS on ${\mathfrak {X}_{\mathcal {U}}}$ is spotted as

$$\begin{equation*} {A_{BCFS}}= \big \{{\big ({\mathsf{x}},\big ({{\mu _{A_{BCFS}}^{P}} \big ({\mathsf{x}}}\big), {\mu _{A_{BCFS}}^{N}} \big ({\mathsf{x}}}\big)\big)\big)~|~{\mathsf{x}} \in {\mathfrak {X}_{\mathcal {U}}} \big \}\tag{1}\end{equation*}$$ where, ${\mu _{A_{BCFS}}^{P}}\big ({\mathsf{x}} \big)={\mu _{A_{BCFS}}^{RP}}\big ({\mathsf{x}} \big)+{\iota }{\mu _{A_{BCFS}}^{IP}}\big ({\mathsf{x}} \big)$ is a positive satisfying degree and ${\mu _{A_{BCFS}}^{N}}\left ({\mathsf{x}} \right)={\mu _{A_{BCFS}}^{RN}}\left ({\mathsf{x}} \right)+{\iota }{\mu _{A_{BCFS}}^{IN}}\left ({\mathsf{x}} \right)$ is a negative satisfying degree and ${\mu _{A_{BCFS}}^{RP}}\left ({\mathsf{x}} \right),{\mu _{A_{BCFS}}^{IP}}\left ({\mathsf{x}} \right)\,\, \in \left [{0,1}\right]$ , ${\mu _{A_{BCFS}}^{RN}}\left ({\mathsf{x}} \right),{\mu _{A_{BCFS}}^{IN}}\left ({\mathsf{x}} \right) \in \left [{-1,0}\right]$ . The BCF number (BCFN) would be scrutinized by the set ${A_{BCFS}}=\left ({\mu _{A_{BCFS}}^{P}}\left ({\mathsf{x}} \right),{\mu _{A_{BCFS}}^{N}}\left ({\mathsf{x}} \right)\right)=\left ({{\mu _{A_{BCFS}}^{RP}}\left ({\mathsf{x}} \right)+{\iota }{\mu _{A_{BCFS}}^{IP}}\left ({\mathsf{x}} \right),} {\mu _{A_{BCFS}}^{RN}}\left ({\mathsf{x}} \right)+{\iota }{\mu _{A_{BCFS}}^{IN}}\left ({\mathsf{x}} \right)\right)$ .

Definition 2[31]:

The score value (SV) and accuracy value (AV) for any BCFN ${A_{BCFS}}=\left ({{\mu _{A_{BCFS}}^{P}},\,\,{\mu _{A_{BCFS}}^{N}}}\right)=\left ({{\mu _{A_{BCFS}}^{RP}}+{\iota }\,\,{\mu _{A_{BCFS}}^{IP}},\,\,{\mu _{A_{BCFS}}^{RN}}+{\iota }\,\,{\mu _{A_{BCFS}}^{IN}}}\right)$ is implied in Eq (2) and (3) respectively.

$$\begin{align*} \mathcal {S}\left ({{A_{BCFS}}}\right)=&\frac {1}{4}\left ({2+{\mu _{A_{BCFS}}^{RP}} \!+\! {\mu _{A_{BCFS}}^{IP}} \!+\! {\mu _{A_{BCFS}}^{RN}} \!+\! {\mu _{A_{BCFS}}^{IN}}}\right), \\&\mathcal {S}\left ({{A_{BCFS}}}\right) \in \left [{0,1}\right]\tag{2}\\ \mathcal {H}\left ({{A_{BCFS}}}\right)=&\frac {{\mu _{A_{BCFS}}^{RP}} \!+\! {\mu _{A_{BCFS}}^{IP}} \!+\! {\mu _{A_{BCFS}}^{RN}} \!+\! {\mu _{A_{BCFS}}^{IN}}}{4}, \\&\mathcal {H}\left ({{A_{BCFS}}}\right) \in \left [{0,1}\right]\tag{3}\end{align*}$$

Definition 3[31]:

For two BCFNs ${A_{BCFS-1}}=\Big ({\mu _{A_{BCFS-1}}^{P}},{\mu _{A_{BCFS-1}}^{N}}\Big)=\left ({{\mu _{A_{BCFS-1}}^{RP}}+{\iota } {\mu _{{A_{BCFS}}-1}^{IP}}} { {\mu _{A_{BCFS-1}}^{RN}}+{\iota } {\mu _{A_{BCFS-1}}^{IN}}}\right)$ and ${A_{BCFS-2}}=\left ({{\mu _{A_{BCFS-2}}^{P}}, {\mu _{A_{BCFS-2}}^{N}}}\right) =\Big ({\mu _{A_{BCFS-2}}^{RP}}+{\iota } {\mu _{A_{BCFS-2}}^{IP}}, {\mu _{A_{BCFS-2}}^{RN}}+{\iota } {\mu _{A_{BCFS-2}}^{IN}}\Big)$ , we diagnose the underneath operational laws, where ${\psi }{\geq }0$ ,

1. as shown in the equation at the bottom of the next page.

   $$\begin{align*}&\hspace {-2pc}{A_{BCFS-1}}{\oplus }{A_{BCFS-2}} \\=&\left ({\begin{matrix}{\mu _{A_{BCFS-1}}^{RP}}+{\mu _{A_{BCFS-2}}^{RP}}-{\mu _{A_{BCFS-1}}^{RP}}{\mu _{A_{BCFS-2}}^{RP}}\\ +\, {\iota }\left ({{\mu _{A_{BCFS-1}}^{IP}}+{\mu _{A_{BCFS-2}}^{IP}}-{\mu _{A_{BCFS-1}}^{IP}}{\mu _{A_{BCFS-2}}^{IP}}}\right),\\ -\,\left ({{\mu _{A_{BCFS-1}}^{RN}}{\mu _{A_{BCFS-2}}^{RN}}}\right)+{\iota }\left ({-\left ({{\mu _{A_{BCFS-1}}^{IN}} {\mu _{A_{BCFS-2}}^{IN}}}\right)}\right) \end{matrix}}\right)\end{align*}$$

2. $$\begin{align*}&\hspace {-2pc}{A_{BCFS-1}}{\otimes }{A_{BCFS-2}} \\=&\left ({\begin{matrix} {\mu _{A_{BCFS-1}}^{RP}}{\mu _{A_{BCFS-2}}^{RP}}+{\iota }{\mu _{A_{BCFS-1}}^{IP}}{\mu _{A_{BCFS-2}}^{IP}},\\[3pt] {\mu _{A_{BCFS-1}}^{RN}}+{\mu _{A_{BCFS-2}}^{RN}}+{\mu _{A_{BCFS-1}}^{RN}}{\mu _{A_{BCFS-2}}^{RN}}\\ {\iota }\left ({{\mu _{A_{BCFS-1}}^{IN}}+{\mu _{A_{BCFS-2}}^{IN}}+{\mu _{A_{BCFS-1}}^{IN}}{\mu _{A_{BCFS-2}}^{IN}}}\right) \end{matrix}}\right)\end{align*}$$

3. $$\begin{align*}&\hspace {-2pc}{\psi }{A_{BCFS-1}}\\=&\left ({\begin{matrix} 1\!-\!{\left ({1\!-\!{\mu _{A_{BCFS-1}}^{RP}}}\right)^{\psi }}\!+\!{\iota } \left ({1\!-\!{\left ({1\!-\!{\mu _{A_{BCFS-1}}^{IP}}}\right)^{\psi }}}\right),\\ -\,{\left |{{\mu _{A_{BCFS-1}}^{RN}}}\right |^{\psi }}\!+\!{\iota }\left ({-{\left |{{\mu _{A_{BCFS-1}}^{IN}}}\right |^{\psi }}}\right) \end{matrix}}\right)\end{align*}$$

4. $$\begin{align*}&\hspace {-2pc}{A_{BCFS-1}^{\psi }} \\=&\left ({\!\!\begin{matrix} {\left ({{\mu _{A_{BCFS-1}}^{RP}}}\right)^{\psi }}\!+\!{\iota }{\left ({\!{\mu _{A_{BCFS-1}}^{IP}}\!}\right)^{\psi }},\\ -\,1\!+\!{\left ({\!1\!+\!{\mu _{A_{BCFS-1}}^{RN}}\!}\right)^{\psi }}\!+\!{\iota } \left ({\!-1\!+\!{\left ({\!1\!+\!{\mu _{A_{BCFS-1}}^{IN}}\!}\right)^{\psi }}\!}\right) \end{matrix}\!\!}\right)\end{align*}$$

Definition 4[35]:

A set $\dot {\acute {\text {S}}}$ with odd cardinality $\underline {z}$ i.e. $\dot {\acute {\text {S}}}=\left \{{{\dot {\acute {\text {S}}}_{j}}\,\,|\,\,j=0,\,\,1,\ldots,\underline {z}-1\,\,}\right \}{}$ is said to be LT set, where ${\dot {\acute {\text {S}}}_{j}}$ be a linguistic variable (LV). For instance, $\dot {\acute {\text {S}}}=\left \{{{{\dot {\acute {\text {S}}}_{0}}=extremely\,\,poor,\dot {\acute {\text {S}}}_{1}}=poor, {\dot {\acute {\text {S}}}_{2}}=marginally\,\,poor,} {{\dot {\acute {\text {S}}}_{3}}=average,{\dot {\acute {\text {S}}}_{4}}=good, {\dot {\acute {\text {S}}}_{5}}=very\,\,good,{\dot {\acute {\text {S}}}_{6}}=excellent\,\,}\right \}$ is 7 terms set. Moreover, the set $\dot {\acute {\text {S}}}=\left \{{{\dot {\acute {\text {S}}}_{\gamma }}\,\,|\,\,{\gamma } \in {\mathbb {R} ^{+}}}\right \}{}$ is known as continuous LT sets if it holds the below-given conditions.

1. The ordered set ${\dot {\acute {\text {S}}}_{\gamma }} < {\dot {\acute {\text {S}}}_{\varepsilon }}$ iff ${\gamma } < {\varepsilon }$ ,

2. Negation operator: $Neg\left ({{\dot {\acute {\text {S}}}_{\gamma }}}\right)={\dot {\acute {\text {S}}}_{\varepsilon }}$ such that ${\varepsilon }=\underline {z}-{\gamma }$

3. Max operator: if ${\gamma }\leq {\varepsilon }$ , then $\mathop {\mathrm {max}}\left ({{\dot {\acute {\text {S}}}_{\gamma }},{\dot {\acute {\text {S}}}_{\varepsilon }}}\right)={\dot {\acute {\text {S}}}_{\varepsilon }}$

4. Min operator: if ${\gamma }{\geq }{\varepsilon }$ then $\mathop {\mathrm {min}}\left ({{\dot {\acute {\text {S}}}_{\gamma }},{\dot {\acute {\text {S}}}_{\varepsilon }}}\right)={\dot {\acute {\text {S}}}_{\varepsilon }}$

SECTION III. Bipolar Complex Fuzzy Linguistic Set

In this Section, we propound the most treasured and expressive conception, named BCFLS by mixing the conception of BCFSs and LT sets. Moreover, we introduce the essential properties of the defined BCFLS.

Definition 5:

The structure of the BCFLS on ${\mathfrak {X}_{\mathcal {U}}}$ is spotted as

$$\begin{align*} {A_{BCFLS}}=\left \{\begin{matrix} \left ({x},\left (\dot {\acute {\text {S}}}_{\gamma \left ({\mathsf{x}}\right)}, \left ({\mu _{A_{BCFLS}}^{P}}\left ({\mathsf{x}}\right), {\mu _{A_{BCFLS}}^{N}}\left ({\mathsf{x}}\right)\right)\right)\right)\\ | {\mathsf{x}} \in {\mathfrak {X}_{\mathcal {U}}}\end{matrix}\right \}\tag{4}\end{align*}$$ where, ${\dot {\acute {\text {S}}}_{\gamma \left ({\mathsf{x}} \right)}} \in \dot {\acute {\text {S}}}$ , ${\mu _{A_{BCFLS}}^{P}}\left ({\mathsf{x}} \right)={\mu _{A_{BCFLS}}^{RP}}\left ({\mathsf{x}} \right)+{\iota }\,\,{\mu _{A_{BCFLS}}^{IP}}\left ({\mathsf{x}} \right)$ is a positive satisfying degree and ${\mu _{A_{BCFLS}}^{N}}\left ({\mathsf{x}} \right)={\mu _{A_{BCFLS}}^{RN}}\left ({\mathsf{x}} \right)+{\iota }\,\,{\mu _{A_{BCFLS}}^{IN}}\left ({\mathsf{x}} \right)$ is a negative satisfying degree and ${\mu _{A_{BCFLS}}^{RP}}\left ({\mathsf{x}} \right),{\mu _{A_{BCFLS}}^{IP}}\left ({\mathsf{x}} \right) \in \left [{0,\,\,1}\right]$ , ${\mu _{A_{BCFLS}}^{RN}}\left ({\mathsf{x}} \right),{\mu _{A_{BCFLS}}^{IN}}\left ({\mathsf{x}} \right)\,\, \in \left [{-1,\,\,0}\right]$ , of each element $x \in {\mathfrak {X}_{\mathcal {U}}}$ to the linguistic term ${\dot {\acute {\text {S}}}_{\gamma \left ({\mathsf{x}} \right)}}$ . The bipolar complex fuzzy linguistic (BCFL) number (BCFLN) would be analyzed by the set ${A_{BCFLS}}=\left ({{\dot {\acute {\text {S}}}_{\gamma \left ({\mathsf{x}} \right)}},\left ({{\mu _{A_{BCFLS}}^{P}}\left ({\mathsf{x}} \right),\,\,{\mu _{A_{BCFLS}}^{N}}\left ({\mathsf{x}} \right)}\right)}\right)=\left ({{\dot {\acute {\text {S}}}_{\gamma \left ({\mathsf{x}} \right)}},\left ({{\mu _{A_{BCFLS}}^{RP}}\left ({\mathsf{x}} \right)+{\iota }\,\,{\mu _{A_{BCFLS}}^{IP}}\left ({\mathsf{x}} \right),\,\,{\mu _{A_{BCFLS}}^{RN}}\left ({\mathsf{x}} \right)}+{{\iota } {\mu _{A_{BCFLS}}^{IN}}\left ({\mathsf{x}} \right)}\right)}\right)$ .

Definition 6:

For two BCFLNs ${A_{BCFLS-1}}=\left ({{\dot {\acute {\text {S}}}_{\gamma _{1}}},} {\left ({{\mu _{A_{BCFLS-1}}^{P}},\,\,{\mu _{A_{BCFLS-1}}^{N}}}\right)}\right) =\left ({{\dot {\acute {\text {S}}}_{\gamma _{1}}},\left ({{\mu _{A_{BCFLS-1}}^{RP}} }\right.}\right.\,\,\Big.\Big.+{\iota }\,\,{\mu _{{A_{BCFLS}}-1}^{IP}},{\mu _{A_{BCFLS-1}}^{RN}}+{\iota }\,\,{\mu _{A_{BCFLS-1}}^{IN}}\Big)\Big)$ and ${A_{BCFLS-2}}=\Big ({\dot {\acute {\text {S}}}_{\gamma _{2}}},\Big ({\mu _{A_{BCFLS-2}}^{P}},{\mu _{A_{BCFLS-2}}^{N}}\Big)\Big)\!=\!\left ({{\dot {\acute {\text {S}}}_{{\gamma _{2}}\left ({\mathsf{x}} \right)}},\left ({{\mu _{A_{BCFLS-2}}^{RP}}}\right.}\right.\,\,\Big.\Big.+{\iota }\,\,{\mu _{A_{BCFLS-2}}^{IP}}, {\mu _{A_{BCFLS-2}}^{RN}}+{\iota }\,\,{\mu _{A_{BCFLS-2}}^{IN}}\Big)\Big)$ , we diagnose the underneath operational laws, where ${\psi }{\geq }0$ ,

1. as shown in the equation at the bottom of the next page.

   $$\begin{align*}&\hspace {-2pc}{A_{BCFLS-1}}{\oplus }~{A_{BCFLS-2}} \\=&\left ({\begin{matrix} {\dot {\acute {\text {S}}}_{{\gamma _{1}}+{\gamma _{1}}-\frac {{\gamma _{1}}{\gamma _{2}}}{\underline {z}}}},\\[0.5pc] \left ({\begin{matrix}{\mu _{A_{BCFLS-1}}^{RP}}+{\mu _{A_{BCFLS-2}}^{RP}}-{\mu _{A_{BCFLS-1}}^{RP}}{\mu _{A_{BCFLS-2}}^{RP}}\\[0.5pc] +\,{\iota }\left ({{\mu _{A_{BCFLS-1}}^{IP}}+{\mu _{A_{BCFLS-2}}^{IP}}-{\mu _{A_{BCFLS-1}}^{IP}}{\mu _{A_{BCFLS-2}}^{IP}}}\right),\\[0.5pc] -\,\left ({{\mu _{A_{BCFLS-1}}^{RN}}{\mu _{A_{BCFLS-2}}^{RN}}}\right)+{\iota }\left ({-\left ({{\mu _{A_{BCFLS-1}}^{IN}}{\mu _{A_{BCFLS-2}}^{IN}}}\right)}\right) \end{matrix}}\right) \end{matrix}}\right)\end{align*}$$

2. as shown in the equation at the bottom of the next page.

   $$\begin{align*}&\hspace {-2pc}{A_{BCFLS-1}}{\otimes }~{A_{BCFLS-2}} \\=&\left ({\begin{matrix} {\dot {\acute {\text {S}}}_{\frac {{\gamma _{1}}{\gamma _{2}}}{\underline {z}}}},\\[0.5pc] \left ({\begin{matrix}{\mu _{A_{BCFLS-1}}^{RP}}{\mu _{A_{BCFLS-2}}^{RP}}+{\iota }{\mu _{A_{BCFLS-1}}^{IP}}{\mu _{A_{BCFLS-2}}^{IP}},\\[0.5pc] {\mu _{A_{BCFLS-1}}^{RN}}+{\mu _{A_{BCFLS-2}}^{RN}}+{\mu _{A_{BCFLS-1}}^{RN}}{\mu _{A_{BCFLS-2}}^{RN}}\\[0.5pc] {\iota }\left ({{\mu _{A_{BCFLS-1}}^{IN}}+{\mu _{A_{BCFLS-2}}^{IN}}+{\mu _{A_{BCFLS-1}}^{IN}}{\mu _{A_{BCFLS-2}}^{IN}}}\right) \end{matrix}}\right)\end{matrix}}\right)\end{align*}$$

3. as shown in the equation at the bottom of the next page.

   $$\begin{align*}&\hspace {-2pc}{\psi }{A_{BCFLS-1}} \\=&\left ({\begin{matrix} {\dot {\acute {\text {S}}}_{\underline {z}-\underline {z}{\left ({1 -\frac {\gamma _{1}}{\underline {z}}}\right)^{\psi }}}},\\[0.5pc] \left ({\begin{matrix} 1-{\left ({1-{\mu _{A_{BCFLS-1}}^{RP}}}\right)^{\psi }}+{\iota } \left ({1-{\left ({1-{\mu _{A_{BCFLS-1}}^{IP}}}\right)^{\psi }}}\right),\\[0.5pc] -{\left |{{\mu _{A_{BCFLS-1}}^{RN}}}\right |^{\psi }}+{\iota }\left ({-{\left |{{\mu _{A_{BCFLS-1}}^{IN}}}\right |^{\psi }}}\right) \end{matrix}}\right) \end{matrix}}\right)\end{align*}$$

4. as shown in the equation at the bottom of the next page.

   $$\begin{align*}&\hspace {-2pc}{A_{BCFLS-1}^{\psi }} \\=&\left ({\begin{matrix} {\dot {\acute {\text {S}}}_{\underline {z}{\left ({\frac {\gamma _{1}}{\underline {z}}}\right)^{\psi }}}},\\[0.5pc] \left ({\begin{matrix} {\left ({{\mu _{A_{BCFLS-1}}^{RP}}}\right)^{\psi }}+{\iota }{\left ({{\mu _{A_{BCFLS-1}}^{IP}}}\right)^{\psi }},\\[0.5pc] -1+{\left ({1+{\mu _{A_{BCFLS-1}}^{RN}}}\right)^{\psi }}+{\iota }\left ({-1+{\left ({1+{\mu _{A_{BCFLS-1}}^{IN}}}\right)^{\psi }}}\right) \end{matrix}}\right) \end{matrix}}\right)\end{align*}$$

Definition 7:

The score value for any BCFLN ${A_{BCFLS}}=\left ({{\dot {\acute {\text {S}}}_{\gamma }},\left ({{\mu _{A_{BCFLS}}^{P}},{\mu _{A_{BCFLS}}^{N}}}\right)}\right) =\left ({{\dot {\acute {\text {S}}}_{\gamma }},\left ({{\mu _{A_{BCFLS}}^{RP}}+{\iota }{\mu _{A_{BCFLS}}^{IP}},} {{\mu _{A_{BCFLS}}^{RN}} +{\iota }\,\,{\mu _{A_{BCFLS}}^{IN}}}\right)}\right)$ is implied as

$$\begin{align*}&\hspace {-1.7pc}{\mathcal {S}_{BCFLS}}\left ({{A_{BCFLS}}}\right) \\=&\frac {1}{4}\left ({2\!+\!{\mu _{A_{BCFLS}}^{RP}}\!+\!{\mu _{A_{BCFLS}}^{IP}} \!+\!{\mu _{A_{BCFLS}}^{RN}}\!+\!{\mu _{A_{BCFLS}}^{IN}}}\right)\!\times \! {\dot {\acute {\text {S}}}_{\gamma }}\tag{5}\end{align*}$$

Definition 8:

The accuracy value for any BCFLN ${A_{BCFLS}}=\left ({{\dot {\acute {\text {S}}}_{\gamma }},\left ({{\mu _{A_{BCFLS}}^{P}},\,\,{\mu _{A_{BCFLS}}^{N}}}\right)}\right) =\left ({{\dot {\acute {\text {S}}}_{\gamma }},\left ({{\mu _{A_{BCFLS}}^{RP}}} {+{\iota }\,\,{\mu _{A_{BCFLS}}^{IP}},\,\,{\mu _{A_{BCFLS}}^{RN}} +{\iota }{\mu _{A_{BCFLS}}^{IN}}}\right)}\right)$ is implied as

$$\begin{align*}&\hspace {-0.5pc}{\mathcal {S}_{BCFLS}}\left ({{A_{BCFLS}}}\right) \\&=\,\frac {{\mu _{A_{BCFLS}}^{RP}}\!+\!{\mu _{A_{BCFLS}}^{IP}}\!+\!{\mu _{A_{BCFLS}}^{RN}} \!+\!{\mu _{A_{BCFLS}}^{IN}}}{4}\!\times \!{\dot {\acute {\text {S}}}_{\gamma }}\tag{6}\end{align*}$$

Theorem 1:

For two BCFLNs ${A_{BCFLS-1}}=\left ({{\dot {\acute {\text {S}}}_{\gamma _{1}}},} {\left ({{\mu _{A_{BCFLS-1}}^{P}}, {\mu _{A_{BCFLS-1}}^{N}}}\right)}\right)\!=\!\left ({{\dot {\acute {\text {S}}}_{\gamma _{1}}},\left ({{\mu _{A_{BCFLS-1}}^{RP}} \!+{\iota }\,\,{\mu _{{A_{BCFLS}}-1}^{IP}},} {{\mu _{A_{BCFLS-1}}^{RN}}+{\iota }\,\,{\mu _{A_{BCFLS-1}}^{IN}}}\right)}\right)$ and ${A_{BCFLS-2}}=\left ({{\dot {\acute {\text {S}}}_{\gamma _{2}}},} {\left ({{\mu _{A_{BCFLS-2}}^{P}},{\mu _{A_{BCFLS-2}}^{N}}}\right)}\right) =\left ({{\dot {\acute {\text {S}}}_{{\gamma _{2}}\left ({\mathsf{x}} \right)}},\left ({{\mu _{A_{BCFLS-2}}^{RP}}}\right.}\right.\,\,\Big.\Big.+{\iota }\,\,{\mu _{A_{BCFLS-2}}^{IP}}, {\mu _{A_{BCFLS-2}}^{RN}}+{\iota }\,\,{\mu _{A_{BCFLS-2}}^{IN}}\Big)\Big)$ , and ${\psi _{1}},~{\psi _{2}}{\geq }0$ , we have

1. ${A_{BCFLS-1}}{\oplus }{A_{BCFLS-2}}={A_{BCFLS-2}}{\oplus }{A_{BCFLS-1}}$

2. ${A_{BCFLS-1}}{\otimes }{A_{BCFLS-2}}={A_{BCFLS-2}}{\otimes }{A_{BCFLS-1}}$

3. ${\psi _{1}}\left ({{A_{BCFLS-1}}{\oplus }{A_{BCFLS-2}}}\right)={\psi _{1}}{A_{BCFLS-1}}{\oplus }{\psi _{1}}{A_{BCFLS-2}}$

4. ${\left ({{A_{BCFLS-1}}{\otimes }{A_{BCFLS-2}}}\right)^{\psi _{1}}} ={A_{BCFLS-1}^{\psi _{1}}}{\otimes }{A_{BCFLS-2}^{\psi _{1}}}$

5. ${\psi _{1}}{A_{BCFLS-1}}{\oplus }{\psi _{2}}{A_{BCFLS-1}}=\left ({{\psi _{1}}+{\psi _{2}}}\right){A_{BCFLS-1}}$

6. ${A_{BCFLS-1}^{\psi _{1}}}{\otimes }{A_{BCFLS-1}^{\psi _{2}}}={A_{BCFLS-1}^{{\psi _{1}}+{\psi _{2}}}}$

Proof:

Trivial

SECTION IV. Bipolar Complex Fuzzy Linguistic Aggregation Operators

In this part of the manuscript, we scrutinize average and geometric AOs such as BCFLWA and BCFLWG operators for BCFLNs. Let ${{\mathsf{w}} _{wv}}=\left ({{\mathsf{w}}_{wv-1}},{\mathsf{w}}_{wv-2},\ldots,{\mathsf{w}}_{wv-n}\right)$ be a weight vector (WV) with the properties that ${\mathsf{w}}_{wv-j} \in \left [{0,1}\right]$ and $\mathop {\sum }_{j=1}^{n}{\mathsf{w}}_{wv-j}=1$ in this study.

Definition 9:

By considering the family of BCFLNs i.e. ${A_{BCFLS-j}}=\left ({{\dot {\acute {\text {S}}}_{\gamma _{j}}},\left ({{\mu _{A_{BCFLS-j}}^{P}},\,\,{\mu _{A_{BCFLS-j}}^{N}}}\right)}\right) =\left ({{\dot {\acute {\text {S}}}_{\gamma _{j}}},\left ({{\mu _{A_{BCFLS-j}}^{RP}} +{\iota } {\mu _{{A_{BCFLS}}-j}^{IP}},\,\,{\mu _{A_{BCFLS-j}}^{RN}}+{\iota }\,\,{\mu _{A_{BCFLS-j}}^{IN}}}\right)}\right),\,\,j=1,\,\,2,\ldots,\,\,n$ , the BCFLWA operator is scrutinized as

$$\begin{align*}&\hspace {-2pc}BCFLWA\left ({{A_{BCFLS-1}},~{A_{BCFLS-2}},\ldots,~{A_{BCFLS-n}}}\right) \\=&{\begin{array}{c} n\\ {\oplus }\\ j=1\end{array}}{{\mathsf{w}}_{wv-j}}{A_{BCFLS-j}} \\=&{{\mathsf{w}}_{wv-1}}{A_{BCFLS-1}}{\oplus } {{\mathsf{w}}_{wv-2}}{A_{BCFLS-2}} \\&{\oplus }\cdots {\oplus }{{\mathsf{w}}_{wv-n}}{A_{BCFLS-n}}\tag{7}\end{align*}$$

Theorem 2:

By aggregating Eq (7), we achieved the outcome again in the structure of BCFLN and

$$\begin{align*}&\hspace {-2pc}BCFLWA\left ({{A_{BCFLS-1}},{A_{BCFLS-2}},\ldots,{A_{BCFLS-n}}}\right) \\=&\left ({\begin{matrix} {\dot {\acute {\text {S}}}_{\underline {z}-\underline {z}\mathop {\prod }_{j=1}^{n}{\left ({1 -\frac {\gamma _{j}}{\underline {z}}}\right)^{{\mathsf{w}}_{wv-j}}}}},\\ \left ({\begin{matrix} 1-\mathop {\prod }\limits _{j=1}^{n}{\left ({1-{\mu _{A_{BCFLS-j}}^{RP}}}\right)^{{\mathsf{w}}_{wv-j}}}\\ +\,{\iota }~\left ({1-\mathop {\prod }\limits _{j=1}^{n} {\left ({1-{\mu _{A_{BCFLS-j}}^{IP}}}\right)^{{\mathsf{w}}_{wv-j}}}}\right),\\ -\mathop {\prod }\limits _{j=1}^{n}{\left |{{\mu _{A_{BCFLS-j}}^{RN}}}\right |^{{\mathsf{w}}_{wv-j}}}\\ +\,{\iota }~\left ({-\mathop {\prod }\limits _{j=1}^{n} {\left |{{\mu _{A_{BCFLS-j}}^{IN}}}\right |^{{\mathsf{w}}_{wv-j}}}}\right)\end{matrix}}\right) \end{matrix}}\right)\tag{8}\end{align*}$$

Proof:

We would prove this by utilizing the procedure of mathematical induction. Suppose that $n=2$ , then as shown in the equation at the top of the next page,

$$\begin{align*}&\hspace {-2pc}{{{\mathsf{w}}_{wv-1}}A_{BCFLS-1}} \\=&\left ({{\dot {\acute {\text {S}}}_{\underline {z}-\underline {z}{\left ({1 -\frac {\gamma _{1}}{\underline {z}}}\right)^{{\mathsf{w}}_{wv-1}}}}},~\left ({\begin{matrix} 1-{\left ({1-{\mu _{A_{BCFLS-1}}^{RP}}}\right)^{{\mathsf{w}}_{wv-1}}}\\ +\,{\iota }~\left ({1-{\left ({1-{\mu _{A_{BCFLS-1}}^{IP}}}\right)^{{\mathsf{w}}_{wv-1}}}}\right),\\ -{\left |{{\mu _{A_{BCFLS-1}}^{RN}}}\right |^{{\mathsf{w}}_{wv-1}}}\\ +\,{\iota }~\left ({-{\left |{{\mu _{A_{BCFLS-1}}^{IN}}}\right |^{{\mathsf{w}}_{wv-1}}}}\right) \end{matrix}}\right)}\right)\\&\hspace {-2pc}{{{\mathsf{w}}_{wv-2}}A_{BCFLS-2}} \\=&\left ({{\dot {\acute {\text {S}}}_{\underline {z}-\underline {z}{\left ({1 -\frac {\gamma _{2}}{\underline {z}}}\right)^{{\mathsf{w}}_{wv-2}}}}}, \left ({\begin{matrix} 1-{\left ({1-{\mu _{A_{BCFLS-2}}^{RP}}}\right)^{{\mathsf{w}}_{wv-2}}}\\ +\,{\iota }~\left ({1-{\left ({1-{\mu _{A_{BCFLS-2}}^{IP}}}\right)^{{\mathsf{w}}_{wv-2}}}}\right),\\ -{\left |{{\mu _{A_{BCFLS-2}}^{RN}}}\right |^{{\mathsf{w}}_{wv-2}}}\\ +\,{\iota }~\left ({-{\left |{{\mu _{A_{BCFLS-2}}^{IN}}}\right |^{{\mathsf{w}}_{wv-2}}}}\right)\end{matrix}}\right)}\right)\end{align*}$$ and as shown in the equation at the bottom of the next page, $$\begin{align*}&\hspace {-2pc}{{{\mathsf{w}}_{wv-1}}A_{BCFLS-1}}{\oplus } {{{\mathsf{w}}_{wv-2}}A_{BCFLS-2}} \\=&\left ({{\dot {\acute {\text {S}}}_{\underline {z}-\underline {z}{\left ({1 -\frac {\gamma _{1}}{\underline {z}}}\right)^{{\mathsf{w}}_{wv-1}}}}},~\left ({\begin{matrix} 1-{\left ({1-{\mu _{A_{BCFLS-1}}^{RP}}}\right)^{{\mathsf{w}}_{wv-1}}}\\ +\,{\iota }~\left ({1-{\left ({1-{\mu _{A_{BCFLS-1}}^{IP}}}\right)^{{\mathsf{w}}_{wv-1}}}}\right),\\ -{\left |{{\mu _{A_{BCFLS-1}}^{RN}}}\right |^{{\mathsf{w}}_{wv-1}}}\\ +\,{\iota }~\left ({-{\left |{{\mu _{A_{BCFLS-1}}^{IN}}}\right |^{{\mathsf{w}}_{wv-1}}}}\right) \end{matrix}}\right)}\right) \\&\oplus \,\left ({{\dot {\acute {\text {S}}}_{\underline {z}-\underline {z}{\left ({1 -\frac {\gamma _{2}}{\underline {z}}}\right)^{{\mathsf{w}}_{wv-2}}}}},~\left ({\begin{matrix} 1-{\left ({1-{\mu _{A_{BCFLS-2}}^{RP}}}\right)^{{\mathsf{w}}_{wv-2}}}\\ +\,{\iota }~\left ({1-{\left ({1-{\mu _{A_{BCFLS-2}}^{IP}}}\right)^{{\mathsf{w}}_{wv-2}}}}\right),\\ -{\left |{{\mu _{A_{BCFLS-2}}^{RN}}}\right |^{{\mathsf{w}}_{wv-2}}}\\ +\,{\iota }~\left ({-{\left |{{\mu _{A_{BCFLS-2}}^{IN}}}\right |^{{\mathsf{w}}_{wv-2}}}}\right) \end{matrix}}\right)}\right)\\=&\left ({\begin{matrix} {\dot {\acute {\text {S}}}_{\underline {z}-\underline {z}{\left ({1 -\frac {\gamma _{1}}{\underline {z}}}\right)^{{\mathsf{w}}_{wv-1}}}{\left ({1 -\frac {\gamma _{2}}{\underline {z}}}\right)^{{\mathsf{w}}_{wv-2}}}}},\\ \left ({\begin{matrix} \begin{matrix} 1-{\left ({1-{\mu _{A_{BCFLS-1}}^{RP}}}\right)^{{\mathsf{w}}_{wv-1}}} {\left ({1-{\mu _{A_{BCFLS-2}}^{RP}}}\right)^{{\mathsf{w}}_{wv-2}}}\\ +\,{\iota }~\left ({1-{\left ({1-{\mu _{A_{BCFLS-1}}^{IP}}}\right)^{{\mathsf{w}}_{wv-1}}} {\left ({1-{\mu _{A_{BCFLS-2}}^{IP}}}\right)^{{\mathsf{w}}_{wv-2}}}}\right)\\ -\left ({\left ({-{\left |{{\mu _{A_{BCFLS-1}}^{RN}}}\right |^{{\mathsf{w}}_{wv-1}}}}\right) \left ({-{\left |{{\mu _{A_{BCFLS-2}}^{RN}}}\right |^{{\mathsf{w}}_{wv-2}}}}\right)}\right)\end{matrix},\\ +\,{\iota }~\left ({-\left ({\left ({-{\left |{{\mu _{A_{BCFLS-1}}^{IN}}}\right |^{{\mathsf{w}}_{wv-1}}}}\right) \left ({-{\left |{{\mu _{A_{BCFLS-2}}^{IN}}}\right |^{{\mathsf{w}}_{wv-2}}}}\right)}\right)}\right) \end{matrix}}\right)\end{matrix}}\right)\\=&\left ({\begin{matrix} {\dot {\acute {\text {S}}}_{\underline {z}-\underline {z}\mathop {\prod }_{j=1}^{2}{\left ({1 -\frac {\gamma _{j}}{\underline {z}}}\right)^{{\mathsf{w}}_{wv-j}}}}},\\ \left ({\begin{matrix} 1-\mathop {\prod }\limits _{j=1}^{2}{\left ({1-{\mu _{A_{BCFLS-j}}^{RP}}}\right)^{{\mathsf{w}}_{wv-j}}}\\ +\,{\iota }~\left ({1-\mathop {\prod }\limits _{j=1}^{2} {\left ({1-{\mu _{A_{BCFLS-j}}^{IP}}}\right)^{{\mathsf{w}}_{wv-j}}}}\right),\\ -\mathop {\prod }\limits _{j=1}^{2}{\left |{{\mu _{A_{BCFLS-j}}^{RN}}}\right |^{{\mathsf{w}}_{wv-j}}}\\ +\,{\iota }~\left ({-\mathop {\prod }\limits _{j=1}^{2}{\left |{{\mu _{A_{BCFLS-j}}^{IN}} }\right |^{{\mathsf{w}}_{wv-j}}}}\right)\end{matrix}}\right) \end{matrix}}\right)\end{align*}$$ $\Rightarrow$ Eq. (8) holds for $n=2$ . Next,consider that Eq. (8) holds for $n =\mathcal {E}$ , then $$\begin{align*}&\hspace {-2pc}BCFLWA\left ({{A_{BCFLS-1}},~{A_{BCFLS-2}},\ldots,~{A_{BCFLS-\mathcal {E}}}}\right) \\=&\left ({\begin{matrix} {\dot {\acute {\text {S}}}_{\underline {z}-\underline {z}\mathop {\prod }_{j=1}^{\mathcal {E}}{\left ({1 -\frac {\gamma _{j}}{\underline {z}}}\right)^{{\mathsf{w}}_{wv-j}}}}},\\ \left ({\begin{matrix} 1-\mathop {\prod }\limits _{j=1}^{\mathcal {E}} {\left ({1-{\mu _{A_{BCFLS-j}}^{RP}}}\right)^{{\mathsf{w}}_{wv-j}}}\\ +\,{\iota }~\left ({1-\mathop {\prod }\limits _{j=1}^{\mathcal {E}} {\left ({1-{\mu _{A_{BCFLS-j}}^{IP}}}\right)^{{\mathsf{w}}_{wv-j}}}}\right),\\ -\mathop {\prod }\limits _{j=1}^{\mathcal {E}} {\left |{{\mu _{A_{BCFLS-j}}^{RN}}}\right |^{{\mathsf{w}}_{wv-j}}}\\ +\,{\iota }~\left ({-\mathop {\prod }\limits _{j=1}^{\mathcal {E}} {\left |{{\mu _{A_{BCFLS-j}}^{IN}}}\right |^{{\mathsf{w}}_{wv-j}}}}\right) \end{matrix}}\right) \end{matrix}}\right)\end{align*}$$ Next to show that Eq. (8) holds for $n =\mathcal {E}+ 1$ , $$\begin{align*}&\hspace {-1.3pc}BCFLWA \\[-1pt]&\times \,\left ({{A_{BCFLS-1}},{A_{BCFLS-2}},\ldots,{A_{BCFLS-\mathcal {E}}},{A_{BCFLS-\mathcal {E}+1}}}\right) \\[-1pt]=&\left ({\begin{matrix} {\dot {\acute {\text {S}}}_{\underline {z}-\underline {z}\mathop {\prod }_{j=1}^{\mathcal {E}}{\left ({1 -\frac {\gamma _{j}}{\underline {z}}}\right)^{{\mathsf{w}}_{wv-j}}}}},\\[-1pt] \left ({\begin{matrix} 1-\mathop {\prod }\limits _{j=1}^{\mathcal {E}}{\left ({1-{\mu _{A_{BCFLS-j}}^{RP}}}\right)^{{\mathsf{w}}_{wv-j}}}\\[-1pt] +\,{\iota }~\left ({1-\mathop {\prod }\limits _{j=1}^{\mathcal {E}}{\left ({1-{\mu _{A_{BCFLS-j}}^{IP}}}\right)^{{\mathsf{w}}_{wv-j}}}}\right),\\ -\mathop {\prod }\limits _{j=1}^{\mathcal {E}}{\left |{{\mu _{A_{BCFLS-j}}^{RN}}}\right |^{{\mathsf{w}}_{wv-j}}}\\[-1pt] +\,{\iota }~\left ({-\mathop {\prod }\limits _{j=1}^{\mathcal {E}} {\left |{{\mu _{A_{BCFLS-j}}^{IN}}}\right |^{{\mathsf{w}}_{wv-j}}}}\right) \end{matrix}}\right) \end{matrix}}\right) \\[-1pt]&\oplus \,\left ({\begin{matrix} {\dot {\acute {\text {S}}}_{\underline {z}-\underline {z}{\left ({1 -\frac {\gamma _{\mathcal {E}+1}}{\underline {z}}}\right)^{{\mathsf{w}}_{wv-\mathcal {E}+1}}}}},\\[-1pt] \left ({\begin{matrix} 1-{\left ({1-{\mu _{A_{BCFLS-\mathcal {E}+1}}^{RP}}}\right)^{{\mathsf{w}}_{wv-\mathcal {E}+1}}}\\[-1pt] +\,{\iota }~\left ({1-{\left ({1-{\mu _{A_{BCFLS-\mathcal {E}+1}}^{IP}}}\right)^{{\mathsf{w}}_{wv-\mathcal {E}+1}}}}\right),\\[-1pt] -{\left |{{\mu _{A_{BCFLS-\mathcal {E}+1}}^{RN}}}\right |^{{\mathsf{w}}_{wv-\mathcal {E}+1}}}\\[-1pt] +\,{\iota }~\left ({-{\left |{{\mu _{A_{BCFLS-\mathcal {E}+1}}^{IN}}}\right |^{{\mathsf{w}}_{wv-\mathcal {E}+1}}} }\right)\end{matrix}}\right)\end{matrix}}\right)\\[-1pt]=&\left ({\begin{matrix} {\dot {\acute {\text {S}}}_{\underline {z}-\underline {z}\mathop {\prod }_{j=1}^{\mathcal {E}+1}{\left ({1 -\frac {\gamma _{j}}{\underline {z}}}\right)^{{\mathsf{w}}_{wv-j}}}}},\\[-1pt] \left ({\begin{matrix} 1-\mathop {\prod }\limits _{j=1}^{\mathcal {E}+1} {\left ({1-{\mu _{A_{BCFLS-j}}^{RP}}}\right)^{{\mathsf{w}}_{wv-j}}}\\[-1pt] +\,{\iota }~\left ({1-\mathop {\prod }\limits _{j=1}^{\mathcal {E}+1} {\left ({1-{\mu _{A_{BCFLS-j}}^{IP}}}\right)^{{\mathsf{w}}_{wv-j}}}}\right),\\[-1pt] -\mathop {\prod }\limits _{j=1}^{\mathcal {E}+1} {\left |{{\mu _{A_{BCFLS-j}}^{RN}}}\right |^{{\mathsf{w}}_{wv-j}}}\\[-1pt] +\,{\iota }~\left ({-\mathop {\prod }\limits _{j=1}^{\mathcal {E}+1} {\left |{{\mu _{A_{BCFLS-j}}^{IN}}}\right |^{{\mathsf{w}}_{wv-j}}}}\right)\end{matrix}}\right)\end{matrix}}\right)\end{align*}$$ Consequently, Eq. (8) is valid ${\forall }~n$ .

Moreover, the idempotency, monotonicity, and boundedness are satisfied by the diagnosed BCFLWA operators and are presented as follows

• Idempotency: By considering the family of BCFLNs i.e. ${A_{BCFLS-j}}=\left ({{\dot {\acute {\text {S}}}_{\gamma _{j}}},\left ({{\mu _{A_{BCFLS-j}}^{P}}, {\mu _{A_{BCFLS-j}}^{N}}}\right)}\right)=\left ({{\dot {\acute {\text {S}}}_{\gamma _{j}}},\left ({{\mu _{A_{BCFLS-j}}^{RP}} \!+\!{\iota }{\mu _{{A_{BCFLS}}-j}^{IP}},{\mu _{A_{BCFLS-j}}^{RN}}\!+\!{\iota }{\mu _{A_{BCFLS-j}}^{IN}}}\right)}\right)$ , $j=1,2,\ldots,n$ , if ${A_{BCFLS-j}}={A_{BCFLS}}$ for all $j$ , then

  $$\begin{align*}&\hspace {-2pc}BCFLWA\left ({{A_{BCFLS-1}}, {A_{BCFLS-2}},\ldots, {A_{BCFLS-n}}}\right) \\&\qquad \qquad \qquad \qquad \qquad =\,{A_{BCFLS}}\tag{9}\end{align*}$$

• Monotonicity: By considering two families of BCFLNs i.e. ${A_{BCFLS-j}}=\left ({{\dot {\acute {\text {S}}}_{\gamma _{j}}},\left ({{\mu _{A_{BCFLS-j}}^{P}}, {\mu _{A_{BCFLS-j}}^{N}}}\right)}\right)=\left ({{\dot {\acute {\text {S}}}_{\gamma _{j}}},\left ({{\mu _{A_{BCFLS-j}}^{RP}}+{\iota } {\mu _{{A_{BCFLS}}-j}^{IP}},{\mu _{A_{BCFLS-j}}^{RN}}\!+{\iota }{\mu _{A_{BCFLS-j}}^{IN}}}\right)}\right)$ and ${A_{BCFLS-j}^{\prime }}=\left ({{\dot {\acute {\text {S}}}_{\gamma _{j}}},\left ({{\mu _{A_{BCFLS-j}^{\prime }}^{P}}, {\mu _{A_{BCFLS-j}^{\prime }}^{N}}}\right)}\right) \!=\! \left ({{\dot {\acute {\text {S}}}_{\gamma _{j}}},\left ({{\mu _{A_{BCFLS-j}^{\prime }}^{RP}} \!+\!{\iota }{\mu _{A_{BCFLS-j}^{\prime }}^{IP}},{\mu _{A_{BCFLS-j}^{\prime }}^{RN}} \!+{\iota }{\mu _{A_{BCFLS-j}^{\prime }}^{IN}}}\right)\!}\right)$ , $j=1,2,\ldots,n$ if ${\mu _{A_{BCFLS-j}}^{RP}}\leq {\mu _{A_{BCFLS-j}^{\prime }}^{RP}}$ , ${\mu _{A_{BCFLS-j}}^{IP}} \leq {\mu _{A_{BCFLS-j}^{\prime }}^{IP}}$ , ${\mu _{A_{BCFLS-j}}^{RN}}\leq {\mu _{A_{BCFLS-j}^{\prime }}^{RN}}$ , ${\mu _{A_{BCFLS-j}}^{IN}}\leq {\mu _{A_{BCFLS-j}^{\prime }}^{IN}}{\forall }j$ then

  $$\begin{align*}&\hspace {-2pc}BCFLWA\left ({{A_{BCFLS-1}},{A_{BCFLS-2}},\ldots,{A_{BCFLS-n}}}\right) \\\leq&BCFLWA\left ({{A_{BCFLS-1}^{\prime }},{A_{BCFLS-2}^{\prime }},\ldots,{A_{BCFLS-n}^{\prime }}}\right) \\{}\tag{10}\end{align*}$$

• Boundedness: By considering the family of BCFLNs i.e. ${A_{BCFLS-j}}=\left ({{\dot {\acute {\text {S}}}_{\gamma _{j}}},\left ({{\mu _{A_{BCFLS-j}}^{P}}, {\mu _{A_{BCFLS-j}}^{N}}}\right)}\right)=\left ({{\dot {\acute {\text {S}}}_{\gamma _{j}}},\left ({{\mu _{A_{BCFLS-j}}^{RP}} +{\iota }\,\,{\mu _{{A_{BCFLS}}-j}^{IP}}, {\mu _{A_{BCFLS-j}}^{RN}}}+{{\iota } {\mu _{A_{BCFLS-j}}^{IN}}}\right)}\right), j=1,\,\,2,\ldots,\,\,n$ and suppose that $\begin{aligned} {A_{BCFLS}^{-}} =\left ({\begin{matrix}\mathop {\mathrm {min}}\limits _{j}\left \{{{\mu _{A_{BCFLS-j}}^{RP}}}\right \} +{\iota }\,\,\mathop {\mathrm {min}}\limits _{j}\left \{{{\mu _{A_{BCFLS-j}}^{IP}}}\right \}, \\ \mathop {\mathrm {max}}\limits _{j}\left \{{{\mu _{A_{BCFLS-j}}^{RN}}}\right \}+{\iota }\,\,\mathop {\mathrm {max}}\limits _{j}\left \{{{\mu _{A_{BCFLS-j}}^{IN}}}\right \}\end{matrix}}\right) \end{aligned}$ , and

  $$\begin{align*} {A_{BCFLS}^{+}}=\left ({\begin{matrix}\mathop {\mathrm {max}}\limits _{j}\left \{{{\mu _{A_{BCFLS-j}}^{RP}}}\right \} +{\iota }~\mathop {\mathrm {max}}\limits _{j}\left \{{{\mu _{A_{BCFLS-j}}^{IP}}}\right \}, \\ \mathop {\mathrm {min}}\limits _{j}\left \{{{\mu _{A_{BCFLS-j}}^{RN}}}\right \} +{\iota }~\mathop {\mathrm {min}}\limits _{j}\left \{{{\mu _{A_{BCFLS-j}}^{IN}}}\right \}\end{matrix}}\right),\end{align*}$$ then $$\begin{align*} {A_{BCFLS}^{-}}\leq&BCFLWA \\&\times \,\left ({{A_{BCFLS-1}},{A_{BCFLS-2}},\ldots,{A_{BCFLS-n}}}\right) \\\leq&{A_{BCFLS}^{+}}\tag{11}\end{align*}$$

Definition 10:

By considering the family of BCFLNs i.e. ${A_{BCFLS-j}}=\left ({{\dot {\acute {\text {S}}}_{\gamma _{j}}},\left ({{\mu _{A_{BCFLS-j}}^{P}}, {\mu _{A_{BCFLS-j}}^{N}}}\right)}\right)=\left ({{\dot {\acute {\text {S}}}_{\gamma _{j}}},\left ({{\mu _{A_{BCFLS-j}}^{RP}} +{\iota }\,\,{\mu _{{A_{BCFLS}}-j}^{IP}},\,\,{\mu _{A_{BCFLS-j}}^{RN}}+{\iota }\,\,{\mu _{A_{BCFLS-j}}^{IN}}}\right)}\right)$ , $j=1,\,\,2,\ldots,\,\,n$ , the BCFLWG operator is scrutinized as

$$\begin{align*}&\hspace {-2pc}BCFLWG\left ({{A_{BCFLS-1}},~{A_{BCFLS-2}},\ldots,~{A_{BCFLS-n}}}\right) \\=&{\begin{array}{c} n\\ {\otimes }\\ j=1 \end{array}} {\left ({{A_{BCFLS-j}}}\right)^{{\mathsf{w}}_{wv-j}}} \\=&{\left ({{A_{BCFLS-1}}}\right)^{{\mathsf{w}}_{wv-1}}} {\otimes }{\left ({{A_{BCFLS-2}}}\right)^{{\mathsf{w}}_{wv-2}}} \\&\otimes \,\cdots {\otimes }{\left ({{A_{BCFLS-n}}}\right)^{{\mathsf{w}}_{wv-n}}}\tag{12}\end{align*}$$

Theorem 3:

By aggregating Eq (12), we achieved the outcome again in the structure of BCFLN and

$$\begin{align*}&\hspace {-2pc}BCFLWG\left ({{A_{BCFLS-1}},{A_{BCFLS-2}},\ldots,{A_{BCFLS-n}}}\right) \\[2pt]=&\left ({\begin{matrix} {\dot {\acute {\text {S}}}_{\underline {z}\mathop {\prod }_{j=1}^{n} {\left ({\frac {\gamma _{j}} {\underline {z}}}\right)^{{\mathsf{w}}_{wv-j}}}}},\\[2pt] \left ({\begin{matrix} \mathop {\prod }\limits _{j=1}^{n} {\left ({{\mu _{A_{BCFLS-j}}^{RP}}}\right)^{{\mathsf{w}}_{wv-j}}}\\[2pt] +{\iota }~\mathop {\prod }\limits _{j=1}^{n} {\left ({{\mu _{A_{BCFLS-j}}^{IP}}}\right)^{{\mathsf{w}}_{wv-j}}},\\[2pt] -1+\mathop {\prod }\limits _{j=1}^{n}{\left ({1 +{\mu _{A_{BCFLS-j}}^{RN}}}\right)^{{\mathsf{w}}_{wv-j}}}\\[2pt] +{\iota }~\left ({-1+\mathop {\prod }\limits _{j=1}^{n}{\left ({1 +{\mu _{A_{BCFLS-j}}^{RN}}}\right)^{{\mathsf{w}}_{wv-j}}}}\right) \end{matrix}}\right) \end{matrix}}\right)\tag{13}\end{align*}$$

Proof:

We would prove this by utilizing the procedure of mathematical induction. Suppose that $n=2$ , then as shown in the equation at the bottom of the next page,

$$\begin{align*}&\hspace {-2pc}{\left ({{A_{BCFLS-1}}}\right)^{{\mathsf{w}}_{wv-1}}} \\=&\left ({{\dot {\acute {\text {S}}}_{\underline {z} {\left ({\frac {\gamma _{1}}{\underline {z}}}\right)^{{\mathsf{w}}_{wv-1}}}}}, \left (\begin{matrix} {\left ({{\mu _{A_{BCFLS-1}}^{RP}}}\right)^{{\mathsf{w}}_{wv-1}}}\\ +\,{\iota }~{\left ({{\mu _{A_{BCFLS-1}}^{IP}}}\right)^{{\mathsf{w}}_{wv-1}}},\\ -1+{\left ({1+{\mu _{A_{BCFLS-1}}^{RN}}}\right)^{{\mathsf{w}}_{wv-1}}}\\ +\,{\iota }~\left ({-1+{\left ({1+{\mu _{A_{BCFLS-1}}^{IN}}}\right)^{{\mathsf{w}}_{wv-1}}}}\right) \end{matrix}\right)}\right)\\&\hspace {-2pc} {\left ({{A_{BCFLS-2}}}\right)^{{\mathsf{w}}_{wv-2}}} \\=&\left ({{\dot {\acute {\text {S}}}_{\underline {z} {\left ({\frac {\gamma _{1}}{\underline {z}}}\right)^{{\mathsf{w}}_{wv-2}}}}},\left (\begin{matrix} {\left ({{\mu _{A_{BCFLS-2}}^{RP}}}\right)^{{\mathsf{w}}_{wv-2}}}\\ +\,{\iota }~{\left ({{\mu _{A_{BCFLS-2}}^{IP}}}\right)^{{\mathsf{w}}_{wv-2}}},\\ -1+{\left ({1+{\mu _{A_{BCFLS-2}}^{RN}}}\right)^{{\mathsf{w}}_{wv-2}}}\\ +\,{\iota }~\left ({-1+{\left ({1+{\mu _{A_{BCFLS-2}}^{IN}}}\right)^{{\mathsf{w}}_{wv-2}}}}\right) \end{matrix}\right)}\right) \end{align*}$$ $$\begin{align*}&\hspace {-2pc}{\left ({{A_{BCFLS-1}}}\right)^{{\mathsf{w}}_{wv-1}}}{\otimes } {\left ({{A_{BCFLS-2}}}\right)^{{\mathsf{w}}_{wv-2}}} =&\left ({{\dot {\acute {\text {S}}}_{\underline {z}{\left ({\frac {\gamma _{1}}{\underline {z}}}\right)^{{\mathsf{w}}_{wv-1}}}}}~,~\left(\begin{matrix} {\left ({{\mu _{A_{BCFLS-1}}^{RP}}}\right)^{{\mathsf{w}}_{wv-1}}}\\ +\,{\iota }~{\left ({{\mu _{A_{BCFLS-1}}^{IP}}}\right)^{{\mathsf{w}}_{wv-1}}},\\ -1+{\left ({1+{\mu _{A_{BCFLS-1}}^{RN}}}\right)^{{\mathsf{w}}_{wv-1}}}\\ +\,{\iota }~\left ({-1+{\left ({1+{\mu _{A_{BCFLS-1}}^{IN}}}\right)^{{\mathsf{w}}_{wv-1}}}}\right) \end{matrix}\right)}\right)\\&\otimes \,\left ({{\dot {\acute {\text {S}}}_{\underline {z}{\left ({\frac {\gamma _{1}} {\underline {z}}}\right)^{{\mathsf{w}}_{wv-2}}}}},\left (\begin{matrix} {\left ({{\mu _{A_{BCFLS-2}}^{RP}}}\right)^{{\mathsf{w}}_{wv-2}}}\\ +\,{\iota }~{\left ({{\mu _{A_{BCFLS-2}}^{IP}}}\right)^{{\mathsf{w}}_{wv-2}}},\\ -1+{\left ({1+{\mu _{A_{BCFLS-2}}^{RN}}}\right)^{{\mathsf{w}}_{wv-2}}}\\ +\,{\iota }~\left ({-1+{\left ({1+{\mu _{A_{BCFLS-2}}^{IN}}}\right)^{{\mathsf{w}}_{wv-2}}}}\right) \end{matrix}\right)}\right) \end{align*}$$ $$\begin{align*} =&\left (\begin{matrix} {\dot {\acute {\text {S}}}_{\underline {z}{\left ({\frac {\gamma _{1}}{\underline {z}}}\right)^{{\mathsf{w}}_{wv-1}}} \underline {z}{\left ({\frac {\gamma _{2}} {\underline {z}}}\right)^{{\mathsf{w}}_{wv-2}}}}},\\ \left (\begin{matrix} \begin{matrix} \begin{matrix} {\left ({{\mu_{A_{BCFLS-1}}^{RP}}}\right)^{{\mathsf{w}}_{wv-1}}} {\left ({{\mu _{A_{BCFLS-2}}^{RP}}}\right)^{{\mathsf{w}}_{wv-2}}}\\ +\,{\iota }~\left ({{\left ({{\mu _{A_{BCFLS-1}}^{IP}}}\right)^{{\mathsf{w}}_{wv-1}}} {\left ({{\mu _{A_{BCFLS-2}}^{IP}}}\right)^{{\mathsf{w}}_{wv-2}}}}\right), \end{matrix}\\ -1+{\left ({1+{\mu _{A_{BCFLS-1}}^{RN}}}\right)^{{\mathsf{w}}_{wv-1}}} {\left ({1+{\mu _{A_{BCFLS-2}}^{RN}}}\right)^{{\mathsf{w}}_{wv-2}}}\end{matrix}\\ +\,{\iota }~\left ({-1+{\left ({1+{\mu _{A_{BCFLS-1}}^{IN}}}\right)^{{\mathsf{w}}_{wv-1}}} {\left ({1+{\mu _{A_{BCFLS-2}}^{IN}}}\right)^{{\mathsf{w}}_{wv-2}}}}\right)\end{matrix}\right) \end{matrix}\right)\end{align*}$$ $$\begin{align*} =&\left (\begin{matrix} {\dot {\acute {\text {S}}}_{\underline {z}\mathop {\prod }_{j=1}^{2} {\left ({\frac {\gamma _{j}}{\underline {z}}}\right)^{{\mathsf{w}}_{wv-j}}}}},\\ \left (\begin{matrix} \mathop {\prod }\limits _{j=1}^{2}{\left ({{\mu _{A_{BCFLS-j}}^{RP}}}\right)^{{\mathsf{w}}_{wv-j}}}\\ +\,{\iota }~\mathop {\prod }\limits _{j=1}^{2} {\left ({{\mu _{A_{BCFLS-j}}^{IP}}}\right)^{{\mathsf{w}}_{wv-j}}},\\ -1+\mathop {\prod }\limits _{j=1}^{2}{\left ({1+{\mu _{A_{BCFLS-j}}^{RN}}}\right)^{{\mathsf{w}}_{wv-j}}}\\ +\,{\iota }~\left ({-1+\mathop {\prod }\limits _{j=1}^{2} {\left ({1+{\mu _{A_{BCFLS-j}}^{RN}}}\right)^{{\mathsf{w}}_{wv-j}}}}\right) \end{matrix}\right)\end{matrix}\right)\end{align*}$$ $\Rightarrow$ Eq. (13) holds for $n=2$ . Next,consider that Eq. (13) holds for $n =\mathcal {E}$ , then as shown in the equation at the top of page 10, $$\begin{align*} BCFLWG\left ({{A_{BCFLS-1}},~{A_{BCFLS-2}},\ldots,~{A_{BCFLS-\mathcal {E}}}}\right)=\left ({\begin{matrix} {\dot {\acute {\text {S}}}_{\underline {z}\mathop {\prod }_{j=1}^{\mathcal {E}}{\left ({\frac {\gamma _{j}} {\underline {z}}}\right)^{{\mathsf{w}}_{wv-j}}}}},\\[2pt] \left ({\begin{matrix} \mathop {\prod }\limits _{j=1}^{\mathcal {E}} {\left ({{\mu _{A_{BCFLS-j}}^{RP}}}\right)^{{\mathsf{w}}_{wv-j}}}\\[2pt] +\,{\iota }~\mathop {\prod }\limits _{j=1}^{\mathcal {E}} {\left ({{\mu _{A_{BCFLS-j}}^{IP}}}\right)^{{\mathsf{w}}_{wv-j}}},\\[2pt] -1+\mathop {\prod }\limits _{j=1}^{\mathcal {E}} {\left ({1+{\mu _{A_{BCFLS-j}}^{RN}}}\right)^{{\mathsf{w}}_{wv-j}}}\\[2pt] +\,{\iota }~\left ({-1+\mathop {\prod }\limits _{j=1}^{\mathcal {E}} {\left ({1+{\mu _{A_{BCFLS-j}}^{RN}}}\right)^{{\mathsf{w}}_{wv-j}}}}\right) \end{matrix}}\right)\end{matrix}}\right)\end{align*}$$ Next to show that Eq. (13) holds for $n =\mathcal {E}+ 1$ , as shown in the equation at the bottom of page 10. $$\begin{align*}&\hspace {-1.4pc}BCFLWG \left ({{A_{BCFLS-1}},{A_{BCFLS-2}},\ldots,{A_{BCFLS-\mathcal {E}}},{A_{BCFLS-\mathcal {E}+1}}\!}\right) \\[2pt]=&\left ({\begin{matrix} {\dot {\acute {\text {S}}}_{\underline {z}\mathop {\prod }_{j=1}^{\mathcal {E}} {\left ({\frac {\gamma _{j}}{\underline {z}}}\right)^{{\mathsf{w}}_{wv-j}}}}},\\[2pt] \left ({\begin{matrix} \mathop {\prod }\limits _{j=1}^{\mathcal {E}} {\left ({{\mu _{A_{BCFLS-j}}^{RP}}}\right)^{{\mathsf{w}}_{wv-j}}}\\[2pt] +\,{\iota }\mathop {\prod }\limits _{j=1}^{\mathcal {E}} {\left ({{\mu _{A_{BCFLS-j}}^{IP}}}\right)^{{\mathsf{w}}_{wv-j}}},\\[2pt] -1+\mathop {\prod }\limits _{j=1}^{\mathcal {E}} {\left ({1+{\mu _{A_{BCFLS-j}}^{RN}}}\right)^{{\mathsf{w}}_{wv-j}}}\\[2pt] +\,{\iota }\left ({-1+\mathop {\prod }\limits _{j=1}^{\mathcal {E}} {\left ({1+{\mu _{A_{BCFLS-j}}^{RN}}}\right)^{{\mathsf{w}}_{wv-j}}}}\right) \end{matrix}}\right)\end{matrix}}\right)\\&\otimes \,\left ({\begin{matrix} {\dot {\acute {\text {S}}}_{\underline {z}{\left ({\frac {\gamma _{1}} {\underline {z}}}\right)^{{\mathsf{w}}_{wv-\mathcal {E}+1}}}}},\\[2pt] \left ({\begin{matrix} {\left ({{\mu _{A_{BCFLS-\mathcal {E}+1}}^{RP}}}\right)^{{\mathsf{w}}_{wv-\mathcal {E}+1}}}\\ +\,{\iota }{\left ({{\mu _{A_{BCFLS-\mathcal {E}+1}}^{IP}}}\right)^{{\mathsf{w}}_{wv-\mathcal {E}+1}}},\\ -1+{\left ({1+{\mu _{A_{BCFLS-\mathcal {E}+1}}^{RN}}}\right)^{{\mathsf{w}}_{wv-\mathcal {E}+1}}}\\ +\,{\iota } \left ({-1+{\left ({1+{\mu _{A_{BCFLS-\mathcal {E}+1}}^{IN}}}\right)^{{\mathsf{w}}_{wv-\mathcal {E}+1}}} }\right)\end{matrix}}\right)\end{matrix}}\right)\\[2pt]=&\left ({\begin{matrix} {\dot {\acute {\text {S}}}_{\underline {z}\mathop {\prod }_{j=1}^{\mathcal {E}+1} {\left ({\frac {\gamma _{j}}{\underline {z}}}\right)^{{\mathsf{w}}_{wv-j}}}}},\\[2pt] \left ({\begin{matrix} \mathop {\prod }\limits _{j=1}^{\mathcal {E}+1} {\left ({{\mu _{A_{BCFLS-j}}^{RP}}}\right)^{{\mathsf{w}}_{wv-j}}}\\[2pt] +\,{\iota }\mathop {\prod }\limits _{j=1}^{\mathcal {E}+1} {\left ({{\mu _{A_{BCFLS-j}}^{IP}}}\right)^{{\mathsf{w}}_{wv-j}}},\\[2pt] -1+\mathop {\prod }\limits _{j=1}^{\mathcal {E}+1}{\left ({1 +{\mu _{A_{BCFLS-j}}^{RN}}}\right)^{{\mathsf{w}}_{wv-j}}}\\[2pt] +\,{\iota }\left ({-1+\mathop {\prod }\limits _{j=1}^{\mathcal {E}+1}{\left ({1 +{\mu _{A_{BCFLS-j}}^{RN}}}\right)^{{\mathsf{w}}_{wv-j}}}}\right) \end{matrix}}\right) \end{matrix}}\right)\end{align*}$$ Consequently, Eq. (13) is valid ${\forall }~n$ .

Moreover, the idempotency, monotonicity, and boundedness are satisfied by the diagnosed BCFLWG operators and are presented as follows

• Idempotency: By considering the family of BCFLNs i.e. ${A_{BCFLS-j}}=\left ({{\dot {\acute {\text {S}}}_{\gamma _{j}}}, \left ({{\mu _{A_{BCFLS-j}}^{P}},\,\,{\mu _{A_{BCFLS-j}}^{N}}}\right)}\right)=\left ({{\dot {\acute {\text {S}}}_{\gamma _{j}}}, \left ({{\mu _{A_{BCFLS-j}}^{RP}}+{\iota }\,\,{\mu _{{A_{BCFLS}}-j}^{IP}},\,\,{\mu _{A_{BCFLS-j}}^{RN}}}+{{\iota }\,\,{\mu _{A_{BCFLS-j}}^{IN}}}\right)}\right),\,\,j=1,\,\,2,\ldots,\,\,n$ , if ${A_{BCFLS-j}}={A_{BCFLS}}$ for all $j$ , then

  $$\begin{align*}&\hspace {-2pc}BCFLWG\left ({{A_{BCFLS-1}},{A_{BCFLS-2}},\ldots,{A_{BCFLS-n}}}\right) \\&\qquad \qquad \qquad \qquad \quad =\,{A_{BCFLS}}\tag{14}\end{align*}$$

• Monotonicity: By considering two families of BCFLNs i.e. ${A_{BCFLS-j}}=\left ({{\dot {\acute {\text {S}}}_{\gamma _{j}}},\left ({{\mu _{A_{BCFLS-j}}^{P}},\,\,{\mu _{A_{BCFLS-j}}^{N}}}\right)}\right) =\left ({{\dot {\acute {\text {S}}}_{\gamma _{j}}},\left ({{\mu _{A_{BCFLS-j}}^{RP}}+{\iota } {\mu _{{A_{BCFLS}}-j}^{IP}}, {\mu _{A_{BCFLS-j}}^{RN}}}+{{\iota }\,\,{\mu _{A_{BCFLS-j}}^{IN}}}\right)}\right)$ and ${A_{BCFLS-j}^{\prime }}=\left ({{\dot {\acute {\text {S}}}_{\gamma _{j}}},\left ({{\mu _{A_{BCFLS-j}^{\prime }}^{P}},} { {\mu _{A_{BCFLS-j}^{\prime }}^{N}}}\right)}\right)=\left ({{\dot {\acute {\text {S}}}_{\gamma _{j}}}, \left ({{\mu _{A_{BCFLS-j}^{\prime }}^{RP}} +{\iota }\,\,{\mu _{A_{BCFLS-j}^{\prime }}^{IP}},} {{\mu _{A_{BCFLS-j}^{\prime }}^{RN}} +{\iota }\,\,{\mu _{A_{BCFLS-j}^{\prime }}^{IN}}}\right)}\right),\,\,j=1,\,\,2,\ldots,\,\,n$ if ${\mu _{A_{BCFLS-j}}^{RP}}\leq {\mu _{A_{BCFLS-j}^{\prime }}^{RP}}$ , ${\mu _{A_{BCFLS-j}}^{IP}}\leq {\mu _{A_{BCFLS-j}^{\prime }}^{IP}}$ , ${\mu _{A_{BCFLS-j}}^{RN}}\leq {\mu _{A_{BCFLS-j}^{\prime }}^{RN}}$ , ${\mu _{A_{BCFLS-j}}^{IN}}\leq {\mu _{A_{BCFLS-j}^{\prime }}^{IN}}\,\,{\forall }\,\,j$ then

  $$\begin{align*}&\hspace {-2pc}BCFLWG\left ({{A_{BCFLS-1}},{A_{BCFLS-2}},\ldots,{A_{BCFLS-n}}}\right) \\\leq&BCFLWG\left ({{A_{BCFLS-1}^{\prime }},{A_{BCFLS-2}^{\prime }},\ldots,{A_{BCFLS-n}^{\prime }}}\right) \\{}\tag{15}\end{align*}$$

• Boundedness: By considering the family of BCFLNs i.e. ${A_{BCFLS-j}}=\left ({{\dot {\acute {\text {S}}}_{\gamma _{j}}},\left ({{\mu _{A_{BCFLS-j}}^{P}}, {\mu _{A_{BCFLS-j}}^{N}}}\right)}\right)=\left ({{\dot {\acute {\text {S}}}_{\gamma _{j}}},} {\left ({{\mu _{A_{BCFLS-j}}^{RP}}+{\iota } {\mu _{{A_{BCFLS}}-j}^{IP}}, {\mu _{A_{BCFLS-j}}^{RN}}+{\iota } {\mu _{A_{BCFLS-j}}^{IN}}}\right)}\right), j=1,\,\,2,\ldots,\,\,n$ and suppose that

  $$\begin{align*} {A_{BCFLS}^{-}}=\left ({\begin{matrix}\mathop {\mathrm {min}}\limits _{j}\left \{{{\mu _{A_{BCFLS-j}}^{RP}}}\right \} +{\iota }~\mathop {\mathrm {min}}\limits _{j}\left \{{{\mu _{A_{BCFLS-j}}^{IP}}}\right \},\\ \mathop {\mathrm {max}}\limits _{j}\left \{{{\mu _{A_{BCFLS-j}}^{RN}}}\right \}+{\iota } \mathop {\mathrm {max}}\limits _{j}\left \{{{\mu _{A_{BCFLS-j}}^{IN}}}\right \}\end{matrix}}\right),\end{align*}$$ and $$\begin{align*}{A_{BCFLS}^{+}}=\left ({\begin{matrix}\mathop {\mathrm {max}}\limits _{j}\left \{{{\mu _{A_{BCFLS-j}}^{RP}}}\right \} +{\iota }~\mathop {\mathrm {max}}\limits _{j}\left \{{{\mu _{A_{BCFLS-j}}^{IP}}}\right \},\\ \mathop {\mathrm {min}}\limits _{j}\left \{{{\mu _{A_{BCFLS-j}}^{RN}}}\right \}+{\iota } \mathop {\mathrm {min}}\limits _{j}\left \{{{\mu _{A_{BCFLS-j}}^{IN}}}\right \}\end{matrix}}\right),\end{align*}$$ then $$\begin{align*} {A_{BCFLS}^{-}}\leq&BCFLWG \\&\times \,\left ({{A_{BCFLS-1}},{A_{BCFLS-2}},\ldots,{A_{BCFLS-n}}}\right) \\\leq&{A_{BCFLS}^{+}}\tag{16}\end{align*}$$

SECTION V. Application (Multi-Attribute Decision-Making)

The progressive beginning of enormous information technology and the consistent development of different work procedure advancements has carried new aspects. The coming of the internet of things and other associated gadgets has investigated every possibility of affecting supply chain usefulness on an essential level. It has prodded the advancement of digital transformation through all parts of supply chain tasks involving warehousing, obtainment, and support. Digitization has addressed all parts of organizations together, on a very basic level varying how you work and convey worth to the clients. Likewise, a social change expects associations to ceaselessly stir things up, and try and become familiar with disappointment. Digital transformation of the supply chain is not just about carrying out innovation, it is tied in with utilizing novel technologies to generally transform how your business works and conveys worth to the clients. Following are the major benefits of the digital supply chain transformation

• Accelerates innovation: The single aim of all digitalization procedures is innovation. This development over the classical method of the supply chain will assist with reinforcing the organization’s plan of action and simultaneously, assist construct associations with providers but also customers.

• Better decision-making: The organizations would take faster and quicker decisions for every particular function if digital technologies are involved in the supply chain. Additionally, the organizations can assess the performance precisely and proficiently by aggregating exchanges and accessible data at the large scale level, consequently, taking the right decisions to prevent contortions made by normal costing.

• End-to-end customer engagement: Digitalization in the supply chain would enhance the clients’ engagement in his excursion. For instance, after giving the order, the provider’s tracking system would assist the client to locate the position of the provider which will guarantee that clients have a good sense of safety and control.

• Organizational flexibility: A significant advantage for an association when it unifies specific explicit capabilities is higher worth through better quality and efficiency.

• Increases automation: Automation decides the most suitable transportation mode transporter, and timetable while thinking about time speed needs and different components.

In any case, numerous associations are as yet reluctant about digital transformation since they don’t have any idea where to begin or what it involves, or how to decide what sort of supply chain digital transformation strategy is the finest one and on which criteria they have to judge. Because of all these issues here, in this article, we going to establish a MADM technique based on the AOs introduced for BCFLS. The MADM technique is described as follows

Let us consider ${A_{BCFLS}}=\left \{{{A_{BCFLS-1}},{A_{BCFLS-2}}}, {{A_{BCFLS-3}},\ldots,{A_{BCFLS-n}}}\right \}{}$ as $n$ alternatives and ${\mathcal {B} _{\mathcal {A}}}=\left \{{{\mathcal {B} _{\mathcal {A-} 1}},{\mathcal {B} _{\mathcal {A-} 2}},{ \mathcal {B} _{\mathcal {A-} 3}},\ldots,{\mathcal {B} _{\mathcal {A-} m}}}\right \}{}$ as $m$ attributes along with weight vector ${\mathsf{w}}_{wv}=\big ({{\mathsf{w}}_{wv-1}},{\mathsf{w}}_{wv-2},{\mathsf{w}}_{wv-3},\ldots, {{\mathsf{w}}_{wv-m}}\big)$ with the properties that ${\mathsf{w}}_{wv-j} \in \left [{0,1}\right]$ and $\mathop {\sum }_{j=1}^{n}{{\mathsf{w}} _{wv-j}}=1$ . The decision analyst would give information about each alternative based on the attributes in the environment of BCFLS, i.e. ${A_{BCFLS}}=\left ({{\dot {\acute {\text {S}}}_{\gamma \left ({\mathsf{x}} \right)}},\left ({{\mu _{A_{BCFLS}}^{P}}\left ({\mathsf{x}} \right),{\mu _{A_{BCFLS}}^{N}}\left ({\mathsf{x}} \right)}\right)}\right) =\left ({{\dot {\acute {\text {S}}}_{\gamma \left ({\mathsf{x}} \right)}},\left ({{\mu _{A_{BCFLS}}^{RP}}\left ({\mathsf{x}} \right)} {+{\iota }{\mu _{A_{BCFLS}}^{IP}}\left ({\mathsf{x}} \right),{\mu _{A_{BCFLS}}^{RN}}\left ({\mathsf{x}} \right)+{\iota }{\mu _{A_{BCFLS}}^{IN}}\left ({\mathsf{x}} \right)}\right)}\right)$ to construct a DM matrix. Afterward, the following steps would be followed for making the decision.

• Step 1:

  The information provided by the decision analyst can be of two types i.e. benefit and cost type. If it is benefit type so there is no such requirement for normalizing the information but if it is cost type then one has to normalize the information by employing the following formula to get a normalized DM matrix.

  $$\begin{align*} {N_{BCFLS}}= \begin{cases} \left (\dot {\acute {\text {S}}}_{\gamma \left ({\mathsf{x}}\right)},\left ({\mu _{A_{BCFLS}}^{P}} \left ({\mathsf{x}}\right), {\mu _{A_{BCFLS}}^{N}}\left ({\mathsf{x}}\right)\right)\right)\\[0.2pc] \qquad for~benefit~type\\ \left (\dot {\acute {\text {S}}}_{\gamma \left ({\mathsf{x}}\right)},\left ({\mu _{A_{BCFLS}}^{P}} \left ({\mathsf{x}}\right), {\mu _{A_{BCFLS}}^{N}}\left ({\mathsf{x}}\right)\right)^{C}\right)\\[0.2pc] \qquad for~cost~type\end{cases}\tag{17}\end{align*}$$ where, $$\begin{align*} {\left ({{\mu _{A_{BCFLS}}^{P}}\left ({\mathsf{x}} \right),~{\mu _{A_{BCFLS}}^{N}}\left ({\mathsf{x}} \right)}\right)^{C}} =\left ({\begin{matrix} 1-{\mu _{A_{BCFLS}}^{RP}}\left ({\mathsf{x}} \right)\\[0.3pc] +{\iota }\left ({1-{\mu _{A_{BCFLS}}^{IP}}\left ({\mathsf{x}} \right)}\right),\\[0.4pc] -1-{\mu _{A_{BCFLS}}^{RN}}\left ({\mathsf{x}} \right)\\[0.2pc] +{\iota }\left ({-1-{\mu _{A_{BCFLS}}^{IN}}\left ({\mathsf{x}} \right)}\right) \end{matrix}}\right).\end{align*}$$

• Step 2:

  If the information is benefit type then employ introduced AOs for BCFLNs such as BCFLWA and BCFLWG operators to achieve the aggregated values of the DM matrix and if the information is cost type then employ introduced AOs for BCFLNs to achieve the aggregated values of normalized DM matrix.

• Step 3:

  To achieve SV of the aggregated values employing Def (7). In case the SV of any two aggregated values is similar, then find the accuracy value (AV) by employing Def (8).

• Step 4:

  Rank the alternatives by noticing their SVs or AVs.

To display the practical use of this MADM technique and its usefulness in the prioritization of strategies of digital transformation in the supply chain we present the following numerical example.

A. Numerical Example

Suppose an organization wants to transform its supply chain from conventional to digital. A technology expert provided 4 strategies for the supply chain digital transformation to the organization i.e. ${A_{BCFLS-1}}=Blockchain$ , ${A_{BCFLS-2}}=Artificial\,\,intelligence$ , ${A_{BCFLS-3}}=Immersive\,\,technologies$ , ${A_{BCFLS-4}}=Supply\,\,chain\,\,as\,\,a\,\,service$ , which the organization has to prioritize and choose the finest one. For this, the decision analyst of the organization considers 4 attributes regarding the digital transformation of the supply chain which are ${ \mathcal {B} _{\mathcal {A-} 1}}=Agility$ , ${\mathcal {B} _{\mathcal {A-} 2}}=Connectivity$ , ${\mathcal {B} _{\mathcal {A-} 3}}=Standardisation$ , ${\mathcal {B} _{\mathcal {A-} 4}}=Collaboration$ , along with weight vector ${\mathsf{w}}_{wv}=\left ({0.16,\,\,0.27,0.38,\,\,0.19}\right)$ . The decision analyst of the organization provides his data or information in the environment of BCFLS which is portrayed in Table 1.

TABLE 1 The Information is Provided by the Decision Analysis of the Organization

Now to obtain the finest strategy fordigital transformation of the supply chain, we utilize the introduced MADM as follows

• Step 1:

  The information provided by the decision analyst is benefit type so there is no requirement for normalization.

• Step 2:

  Employed the introduced BCFLWA and BCFLWG operators to achieve the aggregated values portrayed in Table 2.

• Step 3:

  The achieved SV of the aggregated values is depicted in Table 3.

• Step 4:

  Rank the alternatives by noticing their SVs or AVs are depicted in Table 4.

TABLE 2 The Aggregated Values of the Data are Depicted in Table 1

TABLE 3 The SVs of the Aggregated Values are Described in Table 2

TABLE 4 The Ranking Order of the Alternatives by Noticing Their SVs is Presented in Table 3

According to the ranking depicted in Table 4, ${A_{BCFLS-1}}$ i.e. blockchain is the finest strategy for the digital transformation of the supply chain. COMPARISON

For displaying the dominance and significance of the propounded work we do a comparison of the propounded work with certain prevailing notions such as Wang et al. [36], Lu et al. [38], and Mahmood et al. [32]. Let us consider the information portrayed in Table 1 and try to solve this data by propounded technique and operators and by MADM technique and operators established by Wang et al. [26], Lu et al. [38], and Mahmood et al. [32]. Thus, Table 5 contains the outcomes achieved by employing these approaches and operators and Figure 2 is the graphical interpretation of Table 5. Table 6 contains the ranking.

TABLE 5 The SVs are Achieved by Propounded Work and Certain Prevailing Works

TABLE 6 The Ranking Order is Based on the SVs Displayed in Table 5

Fig. 2.

The graphical description of Table 5.

The prevailing notions Wang et al. [36], Lu et al. [38], and Mahmood et al. [32] are unsuccessful in providing any sort of outcome as portrayed in Tables 5 and 6. Wang et al. [36] are unsuccessful due to lack of $2^{nd}$ dimension i.e. can’t contain extra information and lack negative opinions i.e. the negative satisfactory degree. Lu et al. [38] are unsuccessful due to the lack of the $2^{nd}$ dimension as Lu et al. [38] carry the negative satisfactory degree i.e. negative aspect but can’t carry the extra information. Mahmood et al. [32] contain both the $2^{nd}$ dimension and negative opinions but can’t handle the linguistic terms and that’s the reason for the failure of Mahmood et al. [32]. Besides these, no other prevailing theory in the literature can handle the information presented in Table 1. Merely, the propounded MADM and operators can solve such information which one can observe from Tables 5 and 6 we got a result and ${A_{BCFLS-1}}$ i.e. blockchain is the finest strategy for the digital transformation of the supply chain. The proposed operators and technique also generalized FS, fuzzy linguistic set (FLS), BFS, bipolar FLS, CFS, and complex FLS.

SECTION VI. Conclusion

In this manuscript, we fused two different notions that is BCFS and linguistic set (LS) to establish the notion of BCFLS. We also established fundamental operational laws, score, and accuracy functions in the environment of BCFLS. The notion of BCFLS is the modification of various notions such as FS, fuzzy LS (FLS), BFS, bipolar FLS, CFS, complex FLS, and BCFS. Further, in this manuscript, we established average and geometric AOs in the setting of BCFLS such as BCFLWA and BCFLWG operators along with their idempotency, boundedness, and monotonicity properties. Based on these scrutinized AOs, we established a MADM approach in the environment of our interpreted BCFLS and presented a numerical example of the prioritization of the digital transformation of the supply chain to show the utilization and benefits of the scrutinized AOs and MADM in real-life dilemmas. Digitization has addressed all parts of organizations together with supply chain management. A lot of organizations want to get the advantages of digitalization, particularly in the supply chain. But the biggest problem, the organizations are facing is the ambiguities, i.e. how to select and evaluate the best strategy of the SCDT, on which criteria or attributes they can assess the strategies of SCDT, etc. MADM procedure is the finest technique to evaluate and find out the finest strategy of the SCDT. Thus, through the propounded MADM technique and operators we solved a numerical example of SCDT and found the finest strategy of the organization. Furthermore, to display the advantages and supremacy of the established work, we compared our work with certain current work. The established AOs and MADM mechanism has certain limitations that as the proposed work can’t deal with the information in the setting of the bipolar complex fuzzy uncertain linguistic set, bipolar complex fuzzy 2-tuple linguistic set, bipolar complex fuzzy soft set (BCFSS), etc.

A. Future Direction

In the future, we are thinking to expand this work to various notions where the proposed work is not applicable such as bipolar complex fuzzy uncertain linguistic set, bipolar complex fuzzy 2-tuple linguistic set, BCFSS [34], complex bipolar intuitionistic FS [39], complex bipolar picture FS [40] complex hesitant FS [41], [42], picture FS [43], and picture fuzzy soft set [44].