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10.1371/journal.pone.0309900
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Analysis of coupling in geographic information systems based on WASPAS method for bipolar complex fuzzy linguistic Aczel-Alsina power aggregation operators
Analysis of coupling in geographic information systems based on MADM framework
Ali Zeeshan Methodology Software Validation Writing – original draft Writing – review & editing 1
Hayat Khizar Data curation Methodology Supervision Visualization Writing – original draft Writing – review & editing 2 *
https://orcid.org/0000-0001-8522-1942
Pamucar Dragan Writing – review & editing 3 4 5 *
1 Department of Information Management, National Yunlin University of Science and Technology, Douliou, Yunlin, Taiwan
2 Department of Mathematics, University of Kotli, AJ&K, Pakistan
3 Széchenyi István University, Győr, Hungary
4 Department of Industrial Engineering & Management, Yuan Ze University, Taoyuan City, Taiwan
5 Department of Mechanics and Mathematics, Western Caspian University, Baku, Azerbaijan
Gul Muhammet Editor
Istanbul University: Istanbul Universitesi, TÜRKIYE
Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

* E-mail: khizarhayat@uokajk.edu.pk (KH); dragan.pamucar@fon.bg.ac.rs (DP)
6 9 2024
2024
19 9 e030990016 6 2024
19 8 2024
© 2024 Ali et al
2024
Ali et al
https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

The model of bipolar complex fuzzy linguistic set is a very famous and dominant principle to cope with vague and uncertain information. The bipolar complex fuzzy linguistic set contained the positive membership function, negative membership function, and linguistic variable, where the technique of fuzzy sets to bipolar fuzzy sets are the special cases of the bipolar complex fuzzy linguistic set. In this manuscript, we describe the model of Aczel-Alsina operational laws for bipolar complex fuzzy linguistic values based on Aczel-Alsina t-norm and Aczel-Alsina t-conorm. Additionally, we compute the Aczel-Alsina power aggregation operators based on bipolar complex fuzzy linguistic data, called bipolar complex fuzzy linguistic Aczel-Alsina power averaging operator, bipolar complex fuzzy linguistic Aczel-Alsina power weighted averaging operator, bipolar complex fuzzy linguistic Aczel-Alsina power geometric operator, and bipolar complex fuzzy linguistic Aczel-Alsina power weighted geometric operator with some dominant and fundamental laws such as idempotency, monotonicity, and boundedness. Moreover, we initiate the model of the Weighted Aggregates Sum Product Assessment technique with the help of consequent theory. In the context of geographic information systems and spatial information systems, coupling aims to find out the relationships among different components within a geographic information system, where coupling can occur at many stages, for instance, spatial coupling, data coupling, and functional coupling. To evaluate the above dilemma, we perform the model of multi-attribute decision-making for invented operators to compute the best technique for addressing geographic information systems. In the last, we deliberate some numerical examples for comparing the ranking results of proposed and prevailing techniques.

The author(s) received no specific funding for this work. Data AvailabilityAll the data are included in the manuscript.
Data Availability

All the data are included in the manuscript.
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pmc1. Introduction

The type and degree of coupling in a geographic information system [1] can affect its performance, such as scalability and flexibility. In the meaning of geographic information systems and spatial data systems, coupling main theme is to evaluate the interrelationships between different components within a geographic information system [1]. Further, finding the best optimal based on some attributes among the collection of alternatives is a very complex task, because many people have evaluated such kind of procedure based on classical set theory, where the decision-making procedure, MADM procedure is very famous for depicting vague and unreliable information, but they lost a lot of information due to limited options [2]. To handle such kinds of problems, Zadeh [3] exposed the fuzzy set (FS). FSs have a truth grade defined from universal set to unit interval, such as Ξϖ(x)∈[0,1]. Furthermore, FS theory has only a truth grade, supporting grade, or positive grade, but in many situations, we noticed that the truth grade is not enough for depicting vague and complex information, because in many cases we noticed that the involvement of negative, support against, and negative information. To handle such kind of problems, Zhang [4] introduced the bipolar FS (BFS) theory. BFSs is the modified version of the FSs theory, because FSs contained just a simple truth grade, but the BFSs theory contained the positive truth grade “Ξϖ(x)∈[0,1]” and negative truth grade “Ψϖ(x)∈[0,1], where the idea of FS is the dominant part of BFSs. The involvement of two-dimensional theory has played an important role during decision-making procedures and genuine-life problems, for instance, to buy a car from any company, we will put our opinion in the following shape, name of the car and production data of the car, for managing such kind of problems, the truth grade of FSs is not enough because they just deal with one-dimensional information. For this, Ramot et al. [5] exposed the complex FS (CFS), where the truth grade in CFS is computed in the shape: Ξϖ(x)+iΩϖ(x), where Ξϖ,Ωϖ:X→[0,1]. Further, Mahmood and Rehman [6] exposed the novel technique of bipolar CFSs (BCFSs), where the positive membership grade and negative membership grade are as follows: Ξϖ(x)+iΩϖ(x) and Ψϖ(x)+iΦϖ(x) with a characteristic, such as Ξϖ(x),Ωϖ(x)∈[0,1] and Ψϖ(x),Φϖ(x)∈[−1,0],i=−1, these two grades are the major parts of the BCFSs. BCFSs are very well-known due to their unlimited features and because of their structure, the FSs, CFSs, and BFSs are the special parts of the BCFSs. The linguistic set theory was initiated by Zadeh [7–9], where linguistic variables are used in many situations of life, for instance, if we talked about the weather, we used the following information, such as very cold, cold, normal, hot, very hot are given linguistic terms.

1.1. Literature review

The decision-making technique is a very reliable and dominant technique for addressing the best optimal among the collection of finite information. Because of unlimited ambiguity and uncertainty, the decision-making technique has not worked dominantly because of a crisp set. To address the above problems FS has been proposed. FS theory has a lot of applications in many fields, for instance, fuzzy n-soft sets were invented by Akram et al. [10], fuzzy superior Mandelbrot sets were derived by Mahmood and Ali [11], and hesitant fuzzy n-soft sets were derived by Akram et al. [12]. Further, the model of FS has only a membership function, but in various genuine-life situations, we need the technique of BFS, because it covers the membership function in the form of positive and negative ways. After the utilization of the BFSs, many well-known and dominant ideas have been proposed by different scholars, for instance, bipolar fuzzy metric space was invented by Zararsiz and Riaz [13], and analysis of ideal in BCI-algebra in BFSs was discovered by Abughazalah et al. [14], and bipolar vague soft sets were presented by Sakr et al. [15]. Moreover, CFSs are more superior and effective than FSs, because of their features, and due to this reason, many people have used them in many areas, for instance, distance measures for CFSs were proposed by Liu et al. [16], complex dual type-2 hesitant fuzzy sets were proposed by Mahmood et al. [17], and complex multi-fuzzy hypersoft sets was initiated by Saeed et al. [18]. After the utilization of the BCFSs, many well-known and dominant ideas have been proposed by different scholars, for instance, bipolar complex fuzzy soft sets were derived by Gwak et al. [19] and bipolar complex linear systems were presented by Akram et al. [20]. Furthermore, linguistic sets have been utilized by different scholars in different fields, for instance, fuzzy linguistic sets (FLS) [21], complex fuzzy linguistic sets (CFLS) [22, 23], bipolar fuzzy linguistic sets (BFLSs) [24], and bipolar complex fuzzy linguistic sets (BCFLSs) [25]. Further, Yager [26] evaluated power operators for crisp values. Moreover, power-geometric operators for classical set theory were initiated by Xu and Yager [27]. Zavadskas et al. [28] exposed the WASPAS method for crisp values. Mardani et al. [29] presented the WASPAS theory for FSs and Jaleel [30] exposed it for BCFSs. Bi et al. [31] initiated the arithmetic operators for CFSs and Hu et al. [32] derived the power operators for CFSs. Jana et al. [33] exposed the Dombi operators for BFSs. Mahmood et al. [34] evaluated the aggregation operators for BCFSs. Further, Aczel-Alsina t-norm (AATN) and Aczel-Alsina t-conorm (AATCN) were proposed by Aczel and Alsina [35]. Moreover, Mahmood et al. [36] presented the Aczel-Alsina operators for BCFSs. Further, some fundamental techniques and methods are described in the following, for instance, Schweizer-Sklar operators [37], the VIKOR technique [38], the SWARA-COPRAS technique [39], renewable energy resources [40], a best-worst technique [41], decision-making technique [42], DEMATEL-ISM integration method [43], IFSs and their applications [44], BHARAT decision-making model [45], Hamy mean operators [46], Aczel-Alsina operators [47], analysis of decision accuracy in DEMATEL technique [48], Einstein operators [49], MADM technique based on vague sets [50], Bonferroni mean operators [51], decision-making strategy [52, 53], fuzzy soft code [54], Fermatean fuzzy sets [55], analysis of convexity based on hyper-soft sets [56], Pythagorean fuzzy linear programming [57], analysis of hybrid model based on IFS [58], the parsimonious spherical fuzzy sets [59, 60], the AHP model [61], linear programming [62] and the aggregation operators [63].

1.2. Research gap and major problems

The model of FSs to BCFL set is a very common technique that is used for addressing different kinds of problems in genuine-life dilemmas. Further, various kind of operators, methods, and measures was proposed based on it by different scholars. Additionally, during the analysis and revision of the existing techniques, we observed that every decision-maker has problems with the help of major queries, for instance.

Problem 1: How do we define new operational laws?

Problem 2: How do we aggregate the collection of the finite number of alternatives into a singleton set?

Problem 3: How do we get the excellent optimal among the collection of alternatives?

In the consideration or availability of the above problems, no one can derive accurate results because of ambiguity and limitations. Anyhow, the model of the WASPAS technique and Aczel-Alsina power operators for bipolar complex fuzzy linguistic sets are the best solutions to the above problems. The model of the BCFL set has been proposed, but no one can define any kind of operators or any type of method based on it because the structure of the BCFL set is very complex due to linguistic terms, where the positive and negative membership function is also computed in the shape of complex-number, so it is a quite complex and challenging task for scholars to define any kind of operators or method based on it. The major reason for the construction of the WASPAS method is that with the help of the WASPAS technique, we can easily evaluate the best optimal among the collection of information. But there are still problems, if we have a collection of a finite number of values, then what happens? For evaluating such kind of dilemmas, we propose the technique of Aczel-Alsina power operators, because with the help of power operators, we evaluate the weight vectors, if we use the unknown weight vectors then we may get the wrong result because of ambiguity and complications, but if we have known weight vectors, then we will get accurate results, therefore, by using the Aczel-Alsina power operator, we can easily aggregate the collection of information into a singleton set, which can help in the implementation of the WASPAS model to address the problems of MADM technique.

1.3. Motivation/Advantages/Major contributions of the proposed methods

Fuzzy set theory contains a very wide range of applications in many fields and various scholars have developed different kinds of extensions based on FS theory, where BCFL is one of them. The model of the BCFL set is a very reliable and flexible model for coping with uncertain and vague information, because of complex-valued positive and complex-valued negative membership functions with linguistic term sets. Further, the model of FS, linguistic term set, BFS, CFS, and BCFS are the sub-part of the BCFL sets. Moreover, the technique of Aczel-Alsina aggregation operators is also the modified version of many existing operators, called maximum aggregation operators, minimum aggregation operators, Drastic aggregation operators, and algebraic aggregation operators. Additionally, the power aggregation operators are also a dominant technique for aggregating the collection of information into a singleton set. Motivated by the structure and inspired form their advantages, the major advantages of the presented techniques are listed below:

Maximum, minimum, Drastic, Algebraic, power, Aczel-Alsina aggregation operators, and WASPAS method for fuzzy information.

Maximum, minimum, Drastic, Algebraic, power, Aczel-Alsina aggregation operators, and WASPAS method for bipolar fuzzy information.

Maximum, minimum, Drastic, Algebraic, power, Aczel-Alsina aggregation operators, and WASPAS method for linguistic term information.

Maximum, minimum, Drastic, Algebraic, power, Aczel-Alsina aggregation operators, and WASPAS method for complex fuzzy information.

Maximum, minimum, Drastic, Algebraic, power, Aczel-Alsina aggregation operators, and WASPAS method for bipolar fuzzy information.

Maximum, minimum, Drastic, Algebraic, power, Aczel-Alsina aggregation operators, and WASPAS method for bipolar complex fuzzy information.

Maximum, minimum, Drastic, Algebraic, power, Aczel-Alsina aggregation operators, and WASPAS method for fuzzy linguistic term information.

Maximum, minimum, Drastic, Algebraic, power, Aczel-Alsina aggregation operators, and WASPAS method for bipolar fuzzy linguistic term information.

Maximum, minimum, Drastic, Algebraic, power, Aczel-Alsina aggregation operators, and WASPAS method for complex fuzzy linguistic term information.

Maximum, minimum, Drastic, Algebraic, power, Aczel-Alsina aggregation operators, and WASPAS method for bipolar complex fuzzy linguistic term information.

The above information is the special cases of the proposed techniques. It is clear that the proposed techniques and operators are very reliable and superior because of their features, based on the above advantages, the major contribution of this manuscript is listed below:

For evaluating problem 1, we aim to evaluate the novel model of Aczel-Alsina operational laws for BCFL variables.

To address problem 2, we aim to initiate the model of the BCFLAAPOA operator, BCFLAAPOWA operator, BCFLAAPOG operator, and BCFLAAPOWG operator.

For the simplification of the above operators, we aim to derive the major and fundamental properties of the proposed theory, called idempotency, monotonicity, and boundedness.

Using the above operators, which are used for aggregating the collection of information into a singleton set, we compute the WASPAS method based on the initiated operators.

For the justification of the above information, we aim to demonstrate the procedure of the MADM technique based on initiated operators for computing the best technique for addressing geographic information systems.

Finally, we aim to compare the ranking values of initiated techniques with the ranking values of the existing techniques based on illustrated examples to enhance the worth of the proposed theory. The geometrical interpretation of the proposed theory is discussed in Fig 1.

10.1371/journal.pone.0309900.g001 Fig 1 Geometrical abstract of the proposed theory.

1.4. Summary of the proposed manuscript

This manuscript is computed based on BCFL information with some operators, the shape of this manuscript is arranged in the following form:

In Section 2, we discussed the idea of AATN and AATCN. Further, we discussed the idea of a PO averaging (POA) operator and a PO geometric (POG) operator. Moreover, we described the BCFLSs and their basic laws.

In Section 3, we presented the Aczel-Alsina operational laws based on BCFL variables. Further, we evaluated the BCFLAAPOA operator, BCFLAAPOWA operator, BCFLAAPOG operator, and BCFLAAPOWG operator. Some fundamental properties are also presented for the above operators.

In Section 4, we computed the WASPAS method by using the initiated operators.

In Section 5, we demonstrated the procedure of the MADM technique based on initiated operators for computing the best technique for addressing geographic information systems.

In Section 6, we compared the ranking values of initiated techniques with the ranking values of the existing techniques based on illustrated examples to enhance the worth of the proposed theory.

Some concluding remarks are stated in Section 7.

2. Preliminaries

In this section, we revised the technique of AATN, AATCN, POA operator, POG operator, and BCFL set (BCFLS) with some major properties.

Definition 1: [35] Consider μ1, μ2∈[0,1], thus Aπ(μ1,μ2)={Ad(μ1,μ2)ifπ=0min(μ1,μ2)ifπ=∞e−((−log(μ1))π+(−log(μ2))π)1πotherwise

A@π(μ1,μ2)={A@d(μ1,μ2)ifπ=0max(μ1,μ2)ifπ=∞1−e−((−log(1−μ1))π+(−log(1−μ2))π)1πotherwise

Called AATN and AATCN, where π∈[0, ∞]. Further, the model of drastic t-norm Ad(μ1,μ2) and drastic t-conorm A@d(μ1,μ2) are described below: Ad(μ1,μ2)={μ1ifμ2=1μ2ifμ1=10otherwise

A@d(μ1,μ2)={μ1ifμ2=0μ2ifμ1=01otherwise

Where Aπ(μ1,μ2)=μ1*μ2 and A@π(μ1,μ2)=μ1+μ2−μ1*μ2 are called algebraic t-norm and algebraic t-conorm.

Definition 2: [26, 27] Consider any finite family of non-negative integers ϖ1,ϖ2,…,ϖn, thus POA(ϖ1,ϖ2,…,ϖn)=∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))ϖj

POG(ϖ1,ϖ2,…,ϖn)=∏j=1n(ϖj)(1+ω(ϖj))∑j=1n(1+ω(ϖj))

Called POA operator and POG operator, where the term ω(ϖj)=∑j≠k=1nSP(ϖj,ϖk) and SP(ϖj,ϖk)=1−DS(ϖj,ϖk), with some conditions:

SP(ϖj,ϖk)∈[0,1].

SP(ϖj,ϖk)=SP(ϖk,ϖj).

If SP(ϖj,ϖk)<SP(ϖl,ϖm) then DS(ϖj,ϖk)≥DS(ϖl,ϖm).

Where, DS(ϖj,ϖk)=|ϖj−ϖk|

Definition 3: [25] Let X be a universal set. The BCFLS ϖBC based on X is illustrated below: ϖBC={(Ll((x)),Ξϖ(x)+iΩϖ(x),Ψϖ(x)+iΦϖ(x)):x∈X}

Where the positive membership grade and negative membership grade are as follows: Ξϖ(x)+iΩϖ(x) and Ψϖ(x)+iΦϖ(x) with a characteristic, such as Ξϖ(x),Ωϖ(x)∈[0,1] and Ψϖ(x),Φϖ(x)∈[−1,0],i=−1. Further, the representation of linguistic variables is as follows: Ll((x)), where S={Ll((x)):l=1,,2,…,∂}. Finally, we illustrated the simple shape of BCFLN, such as ϖjBC=ϖj=(Llj,Ξϖj+iΩϖj,Ψϖj+iΦϖj),j=1,2,…,n. Moreover, we discussed some operational laws for any two BCFLNs, such as ϖ1BC⊕ϖ2BC=(L∂(l1∂+l2∂−l1∂l2∂),(Ξϖ1+Ξϖ2−Ξϖ1Ξϖ2)+i(Ωϖ1+Ωϖ2−Ωϖ1Ωϖ2),−(Ψϖ1Ψϖ2)+i(−(Φϖ1Φϖ2)))

ϖ1BC⊗ϖ2BC=(L∂(l1∂l2∂),(Ξϖ1Ξϖ2)+i(Ωϖ1Ωϖ2),(Ψϖ1+Ψϖ2+Ψϖ1Ψϖ2)+i(Φϖ1+Φϖ2+Φϖ1Φϖ2))

ρϖ1BC=(L∂(1−(1−l1∂)ρ),1−(1−Ξϖ1)ρ+i(1−(1−Ωϖ1)ρ),−|Ψϖ1|ρ+i(−|Φϖ1|ρ))

(ϖ1BC)ρ=(L∂((l1∂)ρ),(Ξϖ1)ρ+i((Ωϖ1)ρ),−1+(1+Ψϖ1)ρ+i(−1+(1+Φϖ1)ρ))

Moreover, we justify the above information with the help of some examples. For this, we consider two BCFL numbers, such as ϖ1=(L1,0.2+i(0.5),−0.2+i(−0.4)) and ϖ2=(L2,0.5+i(0.7),−0.7+i(−0.4)), where the order of linguistic sets is ∂ = 6 with parameter ρ = 2, then ϖ1BC⊕ϖ2BC=(L6(16+26−16*26),(0.2+0.5−0.2*0.5)+i(0.5+0.7−0.5*0.7),−((−0.2)*(−0.7))+i(−((−0.4)*(−0.4))))

=(L2.666,0.6+i(0.85),−0.14+i(−0.16))

ϖ1BC⊗ϖ2BC=(L6(16*26),(0.2*0.5)+i(0.5*0.7),(−0.2−0.7+(−0.2*−0.7))+i(−0.4−0.4+(−0.4*−0.4)))

=(L0.333,0.1+i(0.35),−0.76+i(−0.64))

2*ϖ1BC=(L6(1−(1−16)2),1−(1−0.2)2+i(1−(1−0.5)2),−|−0.2|2+i(−|−0.4|2))

=(L1.833,0.36+i(0.75),−0.04+i(−0.16))

(ϖ1BC)2=(L6((16)2),(0.2)2+i((0.5)2),−1+(1−0.2)2+i(−1+(1−0.4)2))

=(L0.1666,0.04+i(0.25),−0.36+i(−0.64))

Further, for any BCFL number, the model of score value and accuracy value is described in the following form, such as SC(ϖjBC)=lj∂*(Ξϖj+Ωϖj+Ψϖj+Φϖj)∈[−1,1]

AC(ϖjBC)=lj∂*(Ξϖj+Ωϖj−Ψϖj−Φϖj)∈[0,1]

With the following conditions, such as If SC(ϖ1BC)>SC(ϖ2BC)⇒ϖ1BC>ϖ2BC, if SC(ϖ1BC)<SC(ϖ2BC)⇒ϖ1BC<ϖ2BC, if SC(ϖ1BC)=SC(ϖ2BC)⇒, then AC(ϖ1BC)>AC(ϖ2BC)⇒ϖ1BC>ϖ2BC, if AC(ϖ1BC)<AC(ϖ2BC)⇒ϖ1BC<ϖ2BC.

3. Aczel-Alsina power aggregation operators for BCFLNs

In this section, we describe the model of the BCFLAAPOA operator, BCFLAAPOWA operator, BCFLAAPOG operator, and BCFLAAPOWG operator. Some fundamental properties are also presented for the above operators. For the evaluation of the above operators, we first initiate the model of Aczel-Alsina operational laws based on BCFL information.

Definition 4: For any two BCFLNs ϖjBC=(Llj,Ξϖj+iΩϖj,Ψϖj+iΦϖj),j=1,2,…,n, we describe and define the model of Aczel-Alsina operational laws, such as ϖ1BC⊕ϖ2BC=(L∂(1−e−((−log(1−l1∂))π+(−log(1−l2∂))π)1π),(1−e−((−log(1−Ξϖ1))π+(−log(1−Ξϖ2))π)1π)+i(1−e−((−log(1−Ωϖ1))π+(−log(1−Ωϖ2))π)1π),−(e−((−log(|Ψϖ1|))π+(−log(|Ψϖ2|))π)1π)+i(−(e−((−log(|Φϖ1|))π+(−log(|Φϖ2|))π)1π)))

ϖ1BC⊗ϖ2BC=(L∂(e−((−log(l1∂))π+(−log(l2∂))π)1π),(e−((−log(Ξϖ1))π+(−log(Ξϖ2))π)1π)+i(e−((−log(Ωϖ1))π+(−log(Ωϖ2))π)1π),−1+(e−((−log(1+Ψϖ1))π+(−log(1+Ψϖ2))π)1π)+i(−1+(e−((−log(1+Φϖ1))π+(−log(1+Φϖ2))π)1π)))

ρϖ1BC=(L∂(1−e−(ρ(−log(1−l1∂))π)1π),(1−e−(ρ(−log(1−Ξϖ1))π)1π)+i(1−e−(ρ(−log(1−Ωϖ1))π)1π),−(e−(ρ(−log(|Ψϖ1|))π)1π)+i(−(e−(ρ(−log(|Φϖ1|))π)1π)))

(ϖ1BC)ρ=(L∂(e−(ρ(−log(l1∂))π)1π),(e−(ρ(−log(Ξϖ1))π)1π)+i(e−(ρ(−log(Ωϖ1))π)1π),−1+(e−(ρ(−log(1+Ψϖ1))π)1π)+i(−1+(e−(ρ(−log(1+Φϖ1))π)1π)))

Using the above initiated operational laws, we aim to construct the technique of Aczel-Alsina power aggregation operators for BCFL values.

Definition 5: For any finite family of BCFLNs ϖjBC=(Llj,Ξϖj+iΩϖj,Ψϖj+iΦϖj),j=1,2,…,n, the model of the BCFLAAPOA operator is described and illustrated below: BCFLAAPOA:Zn→Z

by BCFLAAPOA(ϖ1BC,ϖ2BC,…,ϖnBC)=BCFLAAPOA(ϖ1,ϖ2,…,ϖn)=(1+ω(ϖ1))∑j=1n(1+ω(ϖj))ϖ1BC⊕(1+ω(ϖ2))∑j=1n(1+ω(ϖj))ϖ2BC⊕…⊕(1+ω(ϖn))∑j=1n(1+ω(ϖj))ϖnBC=⊕j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))ϖjBC

Where ω(ϖj)=∑j≠k=1nSP(ϖj,ϖk) and SP(ϖj,ϖk)=1−DS(ϖj,ϖk), with some conditions:

SP(ϖj,ϖk)∈[0,1].

SP(ϖj,ϖk)=SP(ϖk,ϖj).

If SP(ϖj,ϖk)<SP(ϖl,ϖm) then DS(ϖj,ϖk)≥DS(ϖl,ϖm).

Further, DS(ϖj,ϖk)=12(|lj−lk|∂+14(|Ξϖj−Ξϖk|+|Ωϖj−Ωϖk|+|Ψϖj−Ψϖk|+|Φϖj−Φϖk|))

Where Zn contained the collection of BCFLNs. Further, by using the information in Def. (4), we aim to calculate the aggregated values of the information in Def. (5).

Theorem 1: Let ϖjBC=(Llj,Ξϖj+iΩϖj,Ψϖj+iΦϖj),j=1,2,…,n be the collection of BCFLNs. Then, by using the information in Def. (4) and Def. (5), we proved that their aggregated value is also a BCFLN, such as BCFLAAPOA(ϖ1,ϖ2,…,ϖn)=(L∂(1−e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−lj∂))π)1π),(1−e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−Ξϖj))π)1π)+i(1−e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−Ωϖj))π)1π),−(e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(|Ψϖj|))π)1π)+i(−(e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(|Φϖj|))π)1π)))

Proof: For the simplification of the above information, we aim to use the technique of mathematical induction. For this, if n = 2, thus (1+ω(ϖ1))∑j=12(1+ω(ϖj))ϖ1=(L∂(1−e−((1+ω(ϖ1))∑j=12(1+ω(ϖj))(−log(1−l1∂))π)1π),(1−e−((1+ω(ϖ1))∑j=12(1+ω(ϖj))(−log(1−Ξϖ1))π)1π)+i(1−e−((1+ω(ϖ1))∑j=12(1+ω(ϖj))(−log(1−Ωϖ1))π)1π),−(e−((1+ω(ϖ1))∑j=12(1+ω(ϖj))(−log(|Ψϖ1|))π)1π)+i(−(e−((1+ω(ϖ1))∑j=12(1+ω(ϖj))(−log(|Φϖ1|))π)1π)))

(1+ω(ϖ2))∑j=12(1+ω(ϖj))ϖ2BC=(L∂(1−e−((1+ω(ϖ2))∑j=12(1+ω(ϖj))(−log(1−l2∂))π)1π),(1−e−((1+ω(ϖ2))∑j=12(1+ω(ϖj))(−log(1−Ξϖ2))π)1π)+i(1−e−((1+ω(ϖ2))∑j=12(1+ω(ϖj))(−log(1−Ωϖ2))π)1π),−(e−((1+ω(ϖ2))∑j=12(1+ω(ϖj))(−log(|Ψϖ2|))π)1π)+i(−(e−((1+ω(ϖ2))∑j=12(1+ω(ϖj))(−log(|Φϖ2|))π)1π)))

Thus, by combining the above two equations, such as BCFLAAPOA(ϖ1BC,ϖ2BC)=(1+ω(ϖ1))∑j=12(1+ω(ϖj))ϖ1BC⊕(1+ω(ϖ2))∑j=12(1+ω(ϖj))ϖ2BC

=(L∂(1−e−((1+ω(ϖ1))∑j=12(1+ω(ϖj))(−log(1−l1∂))π)1π),(1−e−((1+ω(ϖ1))∑j=12(1+ω(ϖj))(−log(1−Ξϖ1))π)1π)+i(1−e−((1+ω(ϖ1))∑j=12(1+ω(ϖj))(−log(1−Ωϖ1))π)1π),−(e−((1+ω(ϖ1))∑j=12(1+ω(ϖj))(−log(|Ψϖ1|))π)1π)+i(−(e−((1+ω(ϖ1))∑j=12(1+ω(ϖj))(−log(|Φϖ1|))π)1π)))⊕(L∂(1−e−((1+ω(ϖ2))∑j=12(1+ω(ϖj))(−log(1−l2∂))π)1π),(1−e−((1+ω(ϖ2))∑j=12(1+ω(ϖj))(−log(1−Ξϖ2))π)1π)+i(1−e−((1+ω(ϖ2))∑j=12(1+ω(ϖj))(−log(1−Ωϖ2))π)1π),−(e−((1+ω(ϖ2))∑j=12(1+ω(ϖj))(−log(|Ψϖ2|))π)1π)+i(−(e−((1+ω(ϖ2))∑j=12(1+ω(ϖj))(−log(|Φϖ2|))π)1π)))

=(L∂(1−e−(∑j=12(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−lj∂))π)1π),(1−e−(∑j=12(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−Ξϖj))π)1π)+i(1−e−(∑j=12(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−Ωϖj))π)1π),−(e−(∑j=12(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(|Ψϖj|))π)1π)+i(−(e−(∑j=12(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(|Φϖj|))π)1π)))

Hence, the proposed theory holds for n = 2. Further, we assume that the proposed theory also holds for n = n′, such as BCFLAAPOA(ϖ1,ϖ2,…,ϖn′)=(L∂(1−e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(1−lj∂))π)1π),(1−e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(1−Ξϖj))π)1π)+i(1−e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(1−Ωϖj))π)1π),−(e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(|Ψϖj|))π)1π)+i(−(e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(|Φϖj|))π)1π)))

Then, we proved that the proposed theory also holds for n = n′+1, such as BCFLAAPOA(ϖ1BC,ϖ2BC,…,ϖn′+1BC)=BCFLAAPOA(ϖ1,ϖ2,…,ϖn′+1)

=(1+ω(ϖ1))∑j=1n′+1(1+ω(ϖj))ϖ1BC⊕(1+ω(ϖ2))∑j=1n′+1(1+ω(ϖj))ϖ2BC⊕…⊕(1+ω(ϖn′+1))∑j=1n′+1(1+ω(ϖj))ϖn′+1BC

=⊕j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))ϖjBC⊕(1+ω(ϖn′+1))∑j=1n′+1(1+ω(ϖj))ϖn′+1BC

=(L∂(1−e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(1−lj∂))π)1π),(1−e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(1−Ξϖj))π)1π)+i(1−e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(1−Ωϖj))π)1π),−(e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(|Ψϖj|))π)1π)+i(−(e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(|Φϖj|))π)1π)))⊕(1+ω(ϖn′+1))∑j=1n′+1(1+ω(ϖj))ϖn′+1BC

=(L∂(1−e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(1−lj∂))π)1π),(1−e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(1−Ξϖj))π)1π)+i(1−e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(1−Ωϖj))π)1π),−(e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(|Ψϖj|))π)1π)+i(−(e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(|Φϖj|))π)1π)))⊕(L∂(1−e−((1+ω(ϖn′+1))∑j=1n′+1(1+ω(ϖj))(−log(1−ln′+1∂))π)1π),(1−e−((1+ω(ϖn′+1))∑j=1n′+1(1+ω(ϖj))(−log(1−Ξϖn′+1))π)1π)+i(1−e−((1+ω(ϖn′+1))∑j=1n′+1(1+ω(ϖj))(−log(1−Ωϖn′+1))π)1π),−(e−((1+ω(ϖn′+1))∑j=1n′+1(1+ω(ϖj))(−log(|Ψϖn′+1|))π)1π)+i(−(e−((1+ω(ϖn′+1))∑j=1n′+1(1+ω(ϖj))(−log(|Φϖn′+1|))π)1π)))

=(L∂(1−e−(∑j=1n′+1(1+ω(ϖj))∑j=1n′+1(1+ω(ϖj))(−log(1−lj∂))π)1π),(1−e−(∑j=1n′+1(1+ω(ϖj))∑j=1n′+1(1+ω(ϖj))(−log(1−Ξϖj))π)1π)+i(1−e−(∑j=1n′+1(1+ω(ϖj))∑j=1n′+1(1+ω(ϖj))(−log(1−Ωϖj))π)1π),−(e−(∑j=1n′+1(1+ω(ϖj))∑j=1n′+1(1+ω(ϖj))(−log(|Ψϖj|))π)1π)+i(−(e−(∑j=1n′+1(1+ω(ϖj))∑j=1n′+1(1+ω(ϖj))(−log(|Φϖj|))π)1π)))

Hence, the proposed theory holds for n = n′+1, it means that the proposed theory holds for non-negative integers. Moreover, we simplify some basic or major properties for the above operators, called idempotency, monotonicity, and boundedness.

Property 1: Let ϖjBC=(Llj,Ξϖj+iΩϖj,Ψϖj+iΦϖj),j=1,2,…,n be the collection of BCFLNs. Then, Idempotency: When ϖj=ϖ,j=1,2,…,n, thus BCFLAAPOA(ϖ1,ϖ2,…,ϖn)=ϖ

Monotonicity: When ϖj≤ϖj@,j=1,2,…,n, thus BCFLAAPOA(ϖ1,ϖ2,…,ϖn)≤BCFLAAPOA(ϖ1@,ϖ2@,…,ϖn@)

Boundedness: When ϖj−=minj(ϖj) and ϖj+=maxj(ϖj), thus ϖj−≤BCFLAAPOA(ϖ1,ϖ2,…,ϖn)≤ϖj+

Proof: By using the information in Def. (4) and Def. (5), we prove the required results.

When ϖj=ϖ,j=1,2,…,n, thus BCFLAAPOA(ϖ1,ϖ2,…,ϖn)=(L∂(1−e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−lj∂))π)1π),(1−e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−Ξϖj))π)1π)+i(1−e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−Ωϖj))π)1π),−(e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(|Ψϖj|))π)1π)+i(−(e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(|Φϖj|))π)1π)))

=(L∂(1−e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−l∂))π)1π),(1−e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−Ξϖ))π)1π)+i(1−e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−Ωϖ))π)1π),−(e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(|Ψϖ|))π)1π)+i(−(e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(|Φϖ|))π)1π)))

=(L∂(1−e−((−log(1−l∂))π)1π),(1−e−((−log(1−Ξϖ))π)1π)+i(1−e−((−log(1−Ωϖ))π)1π),−(e−((−log(|Ψϖ|))π)1π)+i(−(e−((−log(|Φϖ|))π)1π)))

=(L∂(1−elog(1−l∂)),(1−elog(1−Ξϖ))+i(1−elog(1−Ωϖ)),−(elog(|Ψϖ|))+i(−(elog(|Φϖ|))))=(Ll,Ξϖ+iΩϖ,Ψϖ+iΦϖ)=ϖ.

When ϖj≤ϖj@,j=1,2,…,n, thus lj≤lj@,Ξϖj≤Ξϖj@,Ωϖj≤Ωϖj@ and Ψϖj≤Ψϖj@,Φϖj≤Φϖj@, then lj≤lj@⇒−lj∂≥−lj@∂⇒1−lj∂≥1−lj@∂

⇒−log(1−lj∂)≤−log(1−lj@∂)

⇒(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−lj∂))π≤(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−lj@∂))π

⇒−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−lj∂))π)1π≥−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−lj@∂))π)1π

⇒−e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−lj∂))π)1π≤−e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−lj@∂))π)1π

⇒∂(1−e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−lj∂))π)1π)≤∂(1−e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−lj@∂))π)1π)

Hence, we concluded that.

⇒L∂(1−e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−lj∂))π)1π)≤L∂(1−e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−lj@∂))π)1π)

Further, we simplify the positive membership function, such as: Ξϖj≤Ξϖj@⇒1−Ξϖj≥1−Ξϖj@

⇒(−log(1−Ξϖj))π≤(−log(1−Ξϖj@))π

⇒∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−Ξϖj))π≤∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−Ξϖj@))π

⇒−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−Ξϖj))π)1π≥−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−Ξϖj@))π)1π

⇒1−e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−Ξϖj))π)1π≤1−e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−Ξϖj@))π)1π

Similarly, we evaluate the imaginary part for positive membership function, such as Ωϖj≤Ωϖj@⇒1−e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−Ωϖj))π)1π≤1−e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1−Ωϖj@))π)1π

Further, we determine the negative membership function, such as Ψϖj≤Ψϖj@⇒|Ψϖj|≥|Ψϖj@|⇒(−log(|Ψϖj|))π≥(−log(|Ψϖj@|))π

⇒∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(|Ψϖj|))π≥∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(|Ψϖj@|))π

⇒−(e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(|Ψϖj|))π)1π)≤−(e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(|Ψϖj@|))π)1π)

Similarly, we evaluate the imaginary part for negative membership function, such as Φϖj≤Φϖj@⇒−(e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(|Φϖj|))π)1π)≤−(e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(|Φϖj@|))π)1π)

Finally, by using the technique of score values, we can easily derive the required result, such as BCFLAAPOA(ϖ1,ϖ2,…,ϖn)≤BCFLAAPOA(ϖ1@,ϖ2@,…,ϖn@).

3. When ϖj−=minj(ϖj) and ϖj+=maxj(ϖj), thus ϖj−=minj(ϖj)≤(ϖ1,ϖ2,…,ϖn)

ϖj+=maxj(ϖj)≥(ϖ1,ϖ2,…,ϖn)

thus ϖj−≤BCFLAAPOA(ϖ1,ϖ2,…,ϖn)≤ϖj+.

Definition 6: For any finite family of BCFLNs ϖjBC=(Llj,Ξϖj+iΩϖj,Ψϖj+iΦϖj),j=1,2,…,n, the model of the BCFLAAPOWA operator is described and illustrated below: BCFLAAPOWA:Zn→Z

by BCFLAAPOWA(ϖ1,ϖ2,…,ϖn)=Ψ1(1+ω(ϖ1))∑j=1nΨj(1+ω(ϖj))ϖ1⊕Ψ2(1+ω(ϖ2))∑j=1nΨj(1+ω(ϖj))ϖ2⊕…⊕Ψn(1+ω(ϖn))∑j=1nΨj(1+ω(ϖj))ϖn=⊕j=1nΨj(1+ω(ϖj))∑j=1nΨj(1+ω(ϖj))ϖj

Where ω(ϖj)=∑j≠k=1nSP(ϖj,ϖk) and SP(ϖj,ϖk)=1−DS(ϖj,ϖk), with some conditions:

SP(ϖj,ϖk)∈[0,1].

SP(ϖj,ϖk)=SP(ϖk,ϖj).

If SP(ϖj,ϖk)<SP(ϖl,ϖm) then DS(ϖj,ϖk)≥DS(ϖl,ϖm).

Further, DS(ϖj,ϖk)=12(|lj−lk|∂+14(|Ξϖj−Ξϖk|+|Ωϖj−Ωϖk|+|Ψϖj−Ψϖk|+|Φϖj−Φϖk|))

Where the weight vector is represented by Ψj∈[0,1] with a condition that is ∑j=1nΨj=1. The term Zn contained the collection of BCFLNs. Further, by using the information in Def. (4), we aim to calculate the aggregated values of the information in Def. (6).

Theorem 2: Let ϖjBC=(Llj,Ξϖj+iΩϖj,Ψϖj+iΦϖj),j=1,2,…,n be the collection of BCFLNs. Then, by using the information in Def. (4) and Def. (6), we proved that their aggregated value is also a BCFLN, such as BCFLAAPOWA(ϖ1,ϖ2,…,ϖn)=(L∂(1−e−(∑j=1nΨj(1+ω(ϖj))∑j=1nΨj(1+ω(ϖj))(−log(1−lj∂))π)1π),(1−e−(∑j=1nΨj(1+ω(ϖj))∑j=1nΨj(1+ω(ϖj))(−log(1−Ξϖj))π)1π)+i(1−e−(∑j=1nΨj(1+ω(ϖj))∑j=1nΨj(1+ω(ϖj))(−log(1−Ωϖj))π)1π),−(e−(∑j=1nΨj(1+ω(ϖj))∑j=1nΨj(1+ω(ϖj))(−log(|Ψϖj|))π)1π)+i(−(e−(∑j=1nΨj(1+ω(ϖj))∑j=1nΨj(1+ω(ϖj))(−log(|Φϖj|))π)1π)))

Proof: The proof of this Theorem is similar to the proof of Theorem 1. Moreover, we simplify some basic or major properties for the above operators, called idempotency, monotonicity, and boundedness.

Property 2: Let ϖjBC=(Llj,Ξϖj+iΩϖj,Ψϖj+iΦϖj),j=1,2,…,n be the collection of BCFLNs. Then,

Idempotency: When ϖj=ϖ,j=1,2,…,n, thus BCFLAAPOWA(ϖ1,ϖ2,…,ϖn)=ϖ

Monotonicity: When ϖj≤ϖj@,j=1,2,…,n, thus BCFLAAPOWA(ϖ1,ϖ2,…,ϖn)≤BCFLAAPOWA(ϖ1@,ϖ2@,…,ϖn@)

Boundedness: When ϖj−=minj(ϖj) and ϖj+=maxj(ϖj), thus ϖj−≤BCFLAAPOWA(ϖ1,ϖ2,…,ϖn)≤ϖj+

Proof: The proof of this Property is similar to the proof of Property 1.

Definition 7: For any finite family of BCFLNs ϖjBC=(Llj,Ξϖj+iΩϖj,Ψϖj+iΦϖj),j=1,2,…,n, the model of the BCFLAAPOG operator is described and illustrated below: BCFLAAPOG:Zn→Z

by BCFLAAPOG(ϖ1,ϖ2,…,ϖn)=ϖ1(1+ω(ϖ1))∑j=1n(1+ω(ϖj))⊗ϖ2(1+ω(ϖ2))∑j=1n(1+ω(ϖj))⊗ϖ3(1+ω(ϖ3))∑j=1n(1+ω(ϖj))⊗…⊗ϖn(1+ω(ϖn))∑j=1n(1+ω(ϖj))=⊗j=1n(ϖj)(1+ω(ϖj))∑j=1n(1+ω(ϖj))

Where ω(ϖj)=∑j≠k=1nSP(ϖj,ϖk) and SP(ϖj,ϖk)=1−DS(ϖj,ϖk), with some conditions:

SP(ϖj,ϖk)∈[0,1].

SP(ϖj,ϖk)=SP(ϖk,ϖj).

If SP(ϖj,ϖk)<SP(ϖl,ϖm) then DS(ϖj,ϖk)≥DS(ϖl,ϖm).

Further, DS(ϖj,ϖk)=12(|lj−lk|∂+14(|Ξϖj−Ξϖk|+|Ωϖj−Ωϖk|+|Ψϖj−Ψϖk|+|Φϖj−Φϖk|))

Where the weight vector is represented by Ψj∈[0,1] with a condition that is ∑j=1nΨj=1. The term Zn contained the collection of BCFLNs. Further, by using the information in Def. (4), we aim to calculate the aggregated values of the information in Def. (7).

Theorem 3: Let ϖjBC=(Llj,Ξϖj+iΩϖj,Ψϖj+iΦϖj),j=1,2,…,n be the collection of BCFLNs. Then, by using the information in Def. (4) and Def. (7), we proved that their aggregated value is also a BCFLN, such as BCFLAAPOG(ϖ1,ϖ2,…,ϖn)=(L∂(e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(lj∂))π)1π),(e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(Ξϖj))π)1π)+i(e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(Ωϖj))π)1π),−1+(e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1+Ψϖj))π)1π)+i(−1+(e−(∑j=1n(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1+Φϖj))π)1π)))

Proof: For the simplification of the above information, we aim to use the technique of mathematical induction. For this, if n = 2, thus (ϖ1)(1+ω(ϖ1))∑j=12(1+ω(ϖj))=(L∂(e−((1+ω(ϖ1))∑j=12(1+ω(ϖj))(−log(l1∂))π)1π),(e−((1+ω(ϖ1))∑j=12(1+ω(ϖj))(−log(Ξϖ1))π)1π)+i(e−((1+ω(ϖ1))∑j=12(1+ω(ϖj))(−log(Ωϖ1))π)1π),−1+(e−((1+ω(ϖ1))∑j=12(1+ω(ϖj))(−log(1+Ψϖ1))π)1π)+i(−1+(e−((1+ω(ϖ1))∑j=12(1+ω(ϖj))(−log(1+Φϖ1))π)1π)))

(ϖ2BC)(1+ω(ϖ2))∑j=12(1+ω(ϖj))=(L∂(e−((1+ω(ϖ2))∑j=12(1+ω(ϖj))(−log(l2∂))π)1π),(e−((1+ω(ϖ2))∑j=12(1+ω(ϖj))(−log(Ξϖ2))π)1π)+i(e−((1+ω(ϖ2))∑j=12(1+ω(ϖj))(−log(Ωϖ2))π)1π),−1+(e−((1+ω(ϖ2))∑j=12(1+ω(ϖj))(−log(1+Ψϖ2))π)1π)+i(−1+(e−((1+ω(ϖ2))∑j=12(1+ω(ϖj))(−log(1+Φϖ2))π)1π)))

Thus, by using the above information, we have BCFLAAPOG(ϖ1BC,ϖ2BC)=(ϖ1BC)(1+ω(ϖ1))∑j=12(1+ω(ϖj))⊗(ϖ2BC)(1+ω(ϖ2))∑j=12(1+ω(ϖj))

=(L∂(e−((1+ω(ϖ1))∑j=12(1+ω(ϖj))(−log(l1∂))π)1π),(e−((1+ω(ϖ1))∑j=12(1+ω(ϖj))(−log(Ξϖ1))π)1π)+i(e−((1+ω(ϖ1))∑j=12(1+ω(ϖj))(−log(Ωϖ1))π)1π),−1+(e−((1+ω(ϖ1))∑j=12(1+ω(ϖj))(−log(1+Ψϖ1))π)1π)+i(−1+(e−((1+ω(ϖ1))∑j=12(1+ω(ϖj))(−log(1+Φϖ1))π)1π)))⊗(L∂(e−((1+ω(ϖ2))∑j=12(1+ω(ϖj))(−log(l2∂))π)1π),(e−((1+ω(ϖ2))∑j=12(1+ω(ϖj))(−log(Ξϖ2))π)1π)+i(e−((1+ω(ϖ2))∑j=12(1+ω(ϖj))(−log(Ωϖ2))π)1π),−1+(e−((1+ω(ϖ2))∑j=12(1+ω(ϖj))(−log(1+Ψϖ2))π)1π)+i(−1+(e−((1+ω(ϖ2))∑j=12(1+ω(ϖj))(−log(1+Φϖ2))π)1π)))

=(L∂(e−(∑j=12(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(lj∂))π)1π),(e−(∑j=12(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(Ξϖj))π)1π)+i(e−(∑j=12(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(Ωϖj))π)1π),−1+(e−(∑j=12(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1+Ψϖj))π)1π)+i(−1+(e−(∑j=12(1+ω(ϖj))∑j=1n(1+ω(ϖj))(−log(1+Φϖj))π)1π)))

Hence, the proposed model is held for n = 2. Further, we assume that the proposed model also holds for n = n′, such as BCFLAAPOG(ϖ1,ϖ2,…,ϖn′)=(L∂(e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(lj∂))π)1π),(e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(Ξϖj))π)1π)+i(e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(Ωϖj))π)1π),−1+(e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(1+Ψϖj))π)1π)+i(−1+(e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(1+Φϖj))π)1π)))

Then, we proved that the proposed model also holds for n = n′+1, such as BCFLAAPOG(ϖ1BC,ϖ2BC,…,ϖn′+1BC)=BCFLAAPOG(ϖ1,ϖ2,…,ϖn′+1)

=⊗j=1n′(ϖjBC)(1+ω(ϖj))∑j=1n′(1+ω(ϖj))⊗(ϖn′+1BC)(1+ω(ϖn′+1))∑j=1n′+1(1+ω(ϖj))

=(L∂(e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(lj∂))π)1π),(e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(Ξϖj))π)1π)+i(e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(Ωϖj))π)1π),−1+(e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(1+Ψϖj))π)1π)+i(−1+(e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(1+Φϖj))π)1π)))⊗(ϖn′+1BC)(1+ω(ϖn′+1))∑j=1n′+1(1+ω(ϖj))

=(L∂(e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(lj∂))π)1π),(e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(Ξϖj))π)1π)+i(e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(Ωϖj))π)1π),−1+(e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(1+Ψϖj))π)1π)+i(−1+(e−(∑j=1n′(1+ω(ϖj))∑j=1n′(1+ω(ϖj))(−log(1+Φϖj))π)1π)))⊕(L∂(e−((1+ω(ϖn′+1))∑j=1n′+1(1+ω(ϖj))(−log(ln′+1∂))π)1π),(e−((1+ω(ϖn′+1))∑j=1n′+1(1+ω(ϖj))(−log(Ξϖn′+1))π)1π)+i(e−((1+ω(ϖn′+1))∑j=1n′+1(1+ω(ϖj))(−log(Ωϖn′+1))π)1π),−1+(e−((1+ω(ϖn′+1))∑j=1n′+1(1+ω(ϖj))(−log(1+Ψϖn′+1))π)1π)+i(−1+(e−((1+ω(ϖn′+1))∑j=1n′+1(1+ω(ϖj))(−log(1+Φϖn′+1))π)1π)))

=(L∂(e−(∑j=1n′+1(1+ω(ϖj))∑j=1n′+1(1+ω(ϖj))(−log(lj∂))π)1π),(e−(∑j=1n′+1(1+ω(ϖj))∑j=1n′+1(1+ω(ϖj))(−log(Ξϖj))π)1π)+i(e−(∑j=1n′+1(1+ω(ϖj))∑j=1n′+1(1+ω(ϖj))(−log(Ωϖj))π)1π),−1+(e−(∑j=1n′+1(1+ω(ϖj))∑j=1n′+1(1+ω(ϖj))(−log(1+Ψϖj))π)1π)+i(−1+(e−(∑j=1n′+1(1+ω(ϖj))∑j=1n′+1(1+ω(ϖj))(−log(1+Φϖj))π)1π)))

Hence, the proposed theory holds for n = n′+1, it means that the proposed theory holds for non-negative integers. Moreover, we simplify some basic or major properties for the above operators, called idempotency, monotonicity, and boundedness.

Property 3: Let ϖjBC=(Llj,Ξϖj+iΩϖj,Ψϖj+iΦϖj),j=1,2,…,n be the collection of BCFLNs. Then,

Idempotency: When ϖj=ϖ,j=1,2,…,n, thus BCFLAAPOG(ϖ1,ϖ2,…,ϖn)=ϖ

Monotonicity: When ϖj≤ϖj@,j=1,2,…,n, thus BCFLAAPOG(ϖ1,ϖ2,…,ϖn)≤BCFLAAPOG(ϖ1@,ϖ2@,…,ϖn@)

Boundedness: When ϖj−=minj(ϖj) and ϖj+=maxj(ϖj), thus

ϖj−≤BCFLAAPOG(ϖ1,ϖ2,…,ϖn)≤ϖj+

Proof: The proof of this Property is similar to the proof of Property 1.

Definition 8: For any finite family of BCFLNs ϖjBC=(Llj,Ξϖj+iΩϖj,Ψϖj+iΦϖj),j=1,2,…,n, the model of the BCFLAAPOWG operator is described and illustrated below: BCFLAAPOWG:Zn→Z

by BCFLAAPOWG(ϖ1,ϖ2,…,ϖn)=ϖ1Ψ1(1+ω(ϖ1))∑j=1nΨj(1+ω(ϖj))⊗ϖ2Ψ2(1+ω(ϖ2))∑j=1nΨj(1+ω(ϖj))⊗ϖ3Ψ3(1+ω(ϖ3))∑j=1nΨj(1+ω(ϖj))⊗…⊗ϖnΨn(1+ω(ϖn))∑j=1nΨj(1+ω(ϖj))=⊗j=1n(ϖj)Ψj(1+ω(ϖj))∑j=1nΨj(1+ω(ϖj))

Where ω(ϖj)=∑j≠k=1nSP(ϖj,ϖk) and SP(ϖj,ϖk)=1−DS(ϖj,ϖk), with some conditions:

SP(ϖj,ϖk)∈[0,1].

SP(ϖj,ϖk)=SP(ϖk,ϖj).

If SP(ϖj,ϖk)<SP(ϖl,ϖm) then DS(ϖj,ϖk)≥DS(ϖl,ϖm).

Further, DS(ϖj,ϖk)=12(|lj−lk|∂+14(|Ξϖj−Ξϖk|+|Ωϖj−Ωϖk|+|Ψϖj−Ψϖk|+|Φϖj−Φϖk|))

Where the weight vector is represented by Ψj∈[0,1] with a condition that is ∑j=1nΨj=1. The term Zn contained the collection of BCFLNs. Further, by using the information in Def. (4), we aim to calculate the aggregated values of the information in Def. (8).

Theorem 4: Let ϖjBC=(Llj,Ξϖj+iΩϖj,Ψϖj+iΦϖj),j=1,2,…,n be the collection of BCFLNs. Then, by using the information in Def. (4) and Def. (8), we proved that their aggregated value is also a BCFLN, such as BCFLAAPOWG(ϖ1,ϖ2,…,ϖn)=(L∂(e−(∑j=1nΨj(1+ω(ϖj))∑j=1nΨj(1+ω(ϖj))(−log(lj∂))π)1π),(e−(∑j=1nΨj(1+ω(ϖj))∑j=1nΨj(1+ω(ϖj))(−log(Ξϖj))π)1π)+i(e−(∑j=1nΨj(1+ω(ϖj))∑j=1nΨj(1+ω(ϖj))(−log(Ωϖj))π)1π),−1+(e−(∑j=1nΨj(1+ω(ϖj))∑j=1nΨj(1+ω(ϖj))(−log(1+Ψϖj))π)1π)+i(−1+(e−(∑j=1nΨj(1+ω(ϖj))∑j=1nΨj(1+ω(ϖj))(−log(1+Φϖj))π)1π)))

Proof: The proof of this Theorem is similar to the proof of the Theorem 3. Moreover, we simplify some basic or major properties for the above operators, called idempotency, monotonicity, and boundedness.

Property 4: Let ϖjBC=(Llj,Ξϖj+iΩϖj,Ψϖj+iΦϖj),j=1,2,…,n be the collection of BCFLNs. Then,

Idempotency: When ϖj=ϖ,j=1,2,…,n, thus BCFLAAPOWG(ϖ1,ϖ2,…,ϖn)=ϖ

Monotonicity: When ϖj≤ϖj@,j=1,2,…,n, thus BCFLAAPOWG(ϖ1,ϖ2,…,ϖn)≤BCFLAAPOWG(ϖ1@,ϖ2@,…,ϖn@)

Boundedness: When ϖj−=minj(ϖj) and ϖj+=maxj(ϖj), thus ϖj−≤BCFLAAPOWG(ϖ1,ϖ2,…,ϖn)≤ϖj+

Proof: The proof of this Property is similar to the proof of Property 1. Further, we simplify the information in Theorem 4 by using some numerical examples. Consider ϖ1=(L0.2087,0.0945+i(0.126),−0.732+i(−0.791)),ϖ2=(L0.2314,0.0974+i(0.1294),−0.739+i(−0.797)),ϖ3=(L0.2545,0.1004+i(0.1329),−0.745+i(−0.801)),ϖ4=(L0.2779,0.1034+i(0.1364),−0.752+i(−0.806)), and ϖ5=(L0.3018,0.1065+i(0.14),−0.758+i(−0.811)), with Ψ=(0.3,0.2,0.1,0.3,0.1)T represents the weight vector and π = 2, thus SP(ϖ1,ϖ2)=1−DS(ϖ1,ϖ2)=1−12(|l1−l2|∂+14(|Ξϖ1−Ξϖ2|+|Ωϖ1−Ωϖ2|+|Ψϖ1−Ψϖ2|+|Φϖ1−Φϖ2|))=0.9958,SP(ϖ1,ϖ3)=0.9916,SP(ϖ1,ϖ4)=0.9874,SP(ϖ1,ϖ5)=0.9833,SP(ϖ2,ϖ3)=0.9958,SP(ϖ2,ϖ4)=0.9916,SP(ϖ2,ϖ5)=0.9874,SP(ϖ3,ϖ4)=0.9958,SP(ϖ3,ϖ5)=0.9916,SP(ϖ4,ϖ5)=0.9958.

Further, ω(ϖ1)=∑j≠k=15SP(ϖ1,ϖk)=SP(ϖ1,ϖ2)+SP(ϖ1,ϖ3)+SP(ϖ1,ϖ4)+SP(ϖ1,ϖ5)=0.9958+0.9916+0.9874+0.9833=3.9581,ω(ϖ2)=3.9707,ω(ϖ3)=3.9749,ω(ϖ4)=3.9707,ω(ϖ5)=3.9581

Ψ1(1+ω(ϖ1))∑j=15Ψj(1+ω(ϖj))=0.3*(1+3.9707)24.833=0.2995,Ψ2(1+ω(ϖ2))∑j=15Ψj(1+ω(ϖj))=0.2002,Ψ3(1+ω(ϖ3))∑j=15Ψj(1+ω(ϖj))0.1002,Ψ4(1+ω(ϖ4))∑j=15Ψj(1+ω(ϖj))0.3003,Ψ5(1+ω(ϖ5))∑j=15Ψj(1+ω(ϖj))0.0998

Thus, BCFLAAPOWG(ϖ1,ϖ2,ϖ3,ϖ4,ϖ5)=(L∂(e−(∑j=1nΨj(1+ω(ϖj))∑j=1nΨj(1+ω(ϖj))(−log(lj∂))π)1π),(e−(∑j=1nΨj(1+ω(ϖj))∑j=1nΨj(1+ω(ϖj))(−log(Ξϖj))π)1π)+i(e−(∑j=1nΨj(1+ω(ϖj))∑j=1nΨj(1+ω(ϖj))(−log(Ωϖj))π)1π),−1+(e−(∑j=1nΨj(1+ω(ϖj))∑j=1nΨj(1+ω(ϖj))(−log(1+Ψϖj))π)1π)+i(−1+(e−(∑j=1nΨj(1+ω(ϖj))∑j=1nΨj(1+ω(ϖj))(−log(1+Φϖj))π)1π)))

=(L1.4962,0.2669+i(0.4147),−0.446+i(−0.503)).

4. WASPAS method for proposed operators

In this section, we compute the technique WASPAS based on the initiated operators by using the information of BCFL values. In the investigation of the WASPAS method, we used the BCFLAAPOA operator, BCFLAAPOWA operator, BCFLAAPOG operator, and BCFLAAPOWG operator to enhance the worth of the proposed theory. The major steps of the WASPAS technique are listed below:

Step 1: Defining a matrix consisting of bipolar complex fuzzy linguistic variables M=[ϖ11ϖ12ϖ13…ϖ1nϖ21ϖ22ϖ23…ϖ2nϖ31ϖ32ϖ33⋯ϖ3n………⋯…ϖm1ϖm2ϖm3⋯ϖmn]

Where, ϖijBC=(Llij,Ξϖij+iΩϖij,Ψϖij+iΦϖij),i,j=1,2,…,m,n. The geometrical representation of the WASPAS technique is illustrated in the shape of Fig 2.

10.1371/journal.pone.0309900.g002 Fig 2 Geometrical representation of the WASPAS technique.

L=lij=Llijmax(lij),ifmaxi(lij)ispreferable

Ξ=ϖij=Ξϖijmax(Ξϖij),Ω=ϖij=Ωϖijmax(Ωϖij),ifmax(Ξϖij),max(Ωϖij)ispreferable

Ψ=ϖij=−|Ψϖij|max(|Ψϖij|),Φ=ϖij=−|Φϖij|max(|Φϖij|),ifmax(|Ψϖij|),max(|Φϖij|)ispreferable

And

L=lij=Lmin(lij)lij,ifmini(lij)ispreferable

Ξ=ϖij=min(Ξϖij)Ξϖij,Ω=ϖij=min(Ωϖij)Ωϖij,ifmin(Ξϖij),min(Ωϖij)ispreferable

Ψ=ϖij=−min(|Ψϖij|)|Ψϖij|,Φ=ϖij=−min(|Φϖij|)|Φϖij|,ifmin(|Ψϖij|),min(|Φϖij|)ispreferable

Step 3: Calculate the bipolar complex fuzzy linguistic weighted normalized matrix for BCFLAAPOA operator and BCFLAAPOG operator, such as Ll=ij=L∂*(1−e−(Ψj(−log(1−lij∂))π)1π),Ξ=ϖij=1−(e−(Ψj(−log(1−Ξϖij))π)1π),Ξ=ϖij=1−(e−(Ψj(−log(1−Ωϖij))π)1π),Ψ=ϖij=−(e−(Ψj(−log(|Ψϖij|))π)1π),Φ=ϖij=−(e−(Ψj(−log(|Φϖij|))π)1π)

And Ll=ij=L∂*(e−(Ψj(−log(lij∂))π)1π),Ξ=ϖij=(e−(Ψj(−log(Ξϖij))π)1π),Ξ=ϖij=(e−(Ψj(−log(Ωϖij))π)1π),Ψ=ϖij=−1+(e−(Ψj(−log(1+Ψϖij))π)1π),Φ=ϖij=−1+(e−(Ψj(−log(1+Φϖij))π)1π)

Step 4: Calculate the optimality function for the BCFLAAPOA operator and BCFLAAPOG operator, such as

Qi=BCFLAAPOA(ϖi1BC,ϖi2BC,…,ϖimBC)

Ri=BCFLAAPOWA(ϖi1BC,ϖi2BC,…,ϖimBC)

Si=BCFLAAPOG(ϖi1BC,ϖi2BC,…,ϖimBC)

Ti=BCFLAAPOWG(ϖi1BC,ϖi2BC,…,ϖimBC)

Step 5: Calculate the score values of all aggregate functions, such as SC(ϖjBC)=lj∂*(Ξϖj+Ωϖj+Ψϖj+Φϖj)

Step 6: Calculate the integrated utility function of the WASPAS method, such as Ki=N*12(SC(Qi)+SC(Ri))+(1−N)*12(SC(Si)+SC(Ti))2

Step 7: Rank all the alternatives and examine the best one.

5. Coupling in geographic information systems for BCFL-WASPAS method

In the context of geography information systems, the technique model of “coupling” is used for the investigation of the degree of integration among different systems or components with a geography information system. Effective and reliable coupling in geography information systems improves the system’s functionality, flexibility, usability, and validity, leading to more efficient spatial information management and systems. This section aims to evaluate the problem of coupling in geographic information systems based on the WASPAS technique for BCFL variables. The WASPAS technique is computed based on initiated techniques, called BCFLAAPOA operator, BCFLAAPOWA operator, BCFLAAPOG operator, and BCFLAAPOWG operator. Coupling in geographic information systems is a very beneficial technique for evaluating or analyzing the relationship between different components within a geographic information system. In this application, we used five levels of geographic information systems and with the help of the WASPAS technique, we found the best one, such as

Data Coupling “ϖ1BC”: The data coupling is used for how spatial and attribute information are linked within the geographic information systems.

Data Capture and Input “ϖ2BC”: Data capture and input I geographic information systems are used to processes by which spatial and criteria information are arranged and entered into geographic information systems.

Modeling and Simulation “ϖ3BC”: Modeling and simulation are very important techniques used in geographic information systems and many other fields to understand, predict, and analyze complicated systems and procedures.

Functional Coupling “ϖ4BC”: Functional coupling in geographic information systems is used for the integration and interaction among different functional modules or components within a geographic information system.

Spatial Coupling “ϖ5BC”: Spatial coupling in geographic information systems is used for the way different spatial datasets or layers interact and relate to each other within a geographic information system.

Further, for the selection of the above alternative we will use the following attributes, such as growth analysis, social impact, political impact, environmental impact, and internet resources with weight vectors (0.3,0.2,0.1,0.3,0.1)T. The major steps of the WASPAS technique are listed below:

Step 1: Defining a matrix consisting of bipolar complex fuzzy linguistic variables, see Table 1.

10.1371/journal.pone.0309900.t001 Table 1 BCFL decision matrix.

	ϖ1AT	ϖ2AT	ϖ3AT	ϖ4AT	ϖ5AT	
ϖ1BC	(L1,0.4+i(0.5),−0.2+i(−0.3))	(L1.1,0.41+i(0.51),−0.21+i(−0.31))	(L1.2,0.42+i(0.52),−0.22+i(−0.32))	(L1.3,0.43+i(0.53),−0.23+i(−0.33))	(L1.4,0.44+i(0.54),−0.24+i(−0.34))	
ϖ2BC	(L2,0.7+i(0.8),−0.4+i(−0.2))	(L2.1,0.71+i(0.81),−0.41+i(−0.21))	(L2.2,0.72+i(0.82),−0.42+i(−0.22))	(L2.3,0.73+i(0.83),−0.43+i(−0.23))	(L2.4,0.74+i(0.84),−0.44+i(−0.24))	
ϖ3BC	(L3,0.8+i(0.9),−0.3+i(−0.4))	(L3.1,0.81+i(0.91),−0.31+i(−0.41))	(L3.2,0.82+i(0.92),−0.32+i(−0.42))	(L3.3,0.83+i(0.93),−0.33+i(−0.43))	(L3.4,0.84+i(0.94),−0.34+i(−0.44))	
ϖ4BC	(L4,0.5+i(0.6),−0.2+i(−0.1))	(L4.1,0.51+i(0.61),−0.21+i(−0.11))	(L4.2,0.52+i(0.62),−0.22+i(−0.12))	(L4.3,0.53+i(0.63),−0.23+i(−0.13))	(L4.4,0.54+i(0.64),−0.24+i(−0.14))	
ϖ5BC	(L5,0.6+i(0.7),−0.5+i(−0.4))	(L5.1,0.61+i(0.71),−0.51+i(−0.41))	(L5.2,0.62+i(0.72),−0.52+i(−0.42))	(L5.3,0.63+i(0.73),−0.53+i(−0.43))	(L5.4,0.64+i(0.74),−0.54+i(−0.44))	

Where, ϖijBC=(LIij,Ξϖij+iΩϖij,Ψϖij+iΦϖij),i,j=1,2,…,m,n.

Step 2: Normalize the bipolar fuzzy linguistic decision matrix, see Table 2.

10.1371/journal.pone.0309900.t002 Table 2 Normalized BCFL decision matrix.

	ϖ1AT	ϖ2AT	ϖ3AT	ϖ4AT	ϖ5AT	
ϖ1BC	(L07142.,0.9090+i(0.9259),−0.8333+i(−0.8823))	(L0.7857,0.9318+i(0.9444),−0.875+i(−0.9117))	(L0.8571,0.9545+i(0.9629),−0.9166+i(−0.9411))	(L0.9285,0.9772+i(0.9814),−0.9583+i(−0.9705))	(L1,1+i(1),−1+i(−1))	
ϖ2BC	(L0.8333,0.9459+i(0.9523),−0.9090+i(−0.8333))	(L0.875,0.9594+i(0.9642),−0.9218+i(−0.875))	(L0.9166,0.9729+i(0.9761),−0.9545+i(−0.9166))	(L0.9583,0.9864+i(0.9881),−0.9772+i(−0.9583))	(L1,1+i(1),−1+i(−1))	
ϖ3BC	(L0.8823,0.9523+i(0.9574),−0.8823+i(−0.9090))	(L0.9117,0.9642+i(0.9680),−0.9117+i(−0.9318))	(L0.9411,0.9761+i(0.9787),−0.9411+i(−0.9545))	(L0.9705,0.9881+i(0.9893),−0.9705+i(−0.9772))	(L1,1+i(1),−1+i(−1))	
ϖ4BC	(L0.9090,0.9259+i(0.9375),−0.8333+i(−0.7142))	(L0.9318,0.9444+i(0.9531),−0.875+i(−0.7857))	(L0.9545,0.9629+i(0.9687),−0.9166+i(−0.8571))	(L0.9772,0.9814+i(0.9843),−0.9583+i(−0.9285))	(L1,1+i(1),−1+i(−1))	
ϖ5BC	(L0.9259,0.9375+i(0.9459),−0.9259+i(−0.9090))	(L0.9444,0.9531+i(0.9594),−0.9444+i(−0.9318))	(L0.9629,0.9687+i(0.9729),−0.9629+i(−0.9545))	(L0.9814,0.9843+i(0.9864),−0.9814+i(−0.9772))	(L1,1+i(1),−1+i(−1))	

Step 3: Calculate the bipolar complex fuzzy linguistic weighted normalized matrix for the BCFLAAPOA operator and BCFLAAPOG operator, see Table 3.

10.1371/journal.pone.0309900.t003 Table 3 Weighted normalized BCFL decision matrix.

	ϖ1AT	ϖ2AT	ϖ3AT	ϖ4AT	ϖ5AT	
ϖ1BC	(L0.2087.,0.0944+i(0.1259),−0.7315+i(−0.7914))	(L0.2314.,0.0974+i(0.1293),−0.7385+i(−0.7965))	(L0.2544.,0.1003+i(0.1328),−0.7452+i(−0.8014))	(L0.2779.,0.1034+i(0.1364),−0.7516+i(−0.8062))	(L0.3017.,0.1065+i(0.14),−0.7579+i(−0.8109))	
ϖ2BC	(L0.4543.,0.2085+i(0.2684),−0.8369+i(−0.7315))	(L0.4815.,0.2137+i(0.2757),−0.841+i(−0.7385))	(L0.5093.,0.2190+i(0.2832),−0.8449+i(−0.7452))	(L0.5377.,0.2245+i(0.2911),−0.8488+i(−0.7516))	(L0.5667.,0.2302+i(0.2994),−0.8526+i(−0.7579))	
ϖ3BC	(L0..7557,0.2684+i(0.3605),−0.7914+i(−0.8369))	(L0.7901.,0.2757+i(0.3735),−0.7965+i(−0.841))	(L0.8255.,0.2832+i(0.3877),−0.8014+i(−0.8449))	(L0.8619.,0.2911+i(0.4033),−0.8062+i(−0.8488))	(L0.8994.,0.2994+i(0.4209),−0.8109+i(−0.8526))	
ϖ4BC	(L1.1528.,0.1259+i(0.1630),−0.7315+i(−0.6394))	(L1.2009,0.1293+i(0.1671),−0.7385+i(−0.6513))	(L1.2510,0.1328+i(0.1713),−0.7452+i(−0.6624))	(L1.3034,0.1364+i(0.1756),−0.7516+i(−0.6728))	(L1.3584,0.14+i(0.1799),−0.7579+i(−0.6825))	
ϖ5BC	(L1.7634,0.1630+i(0.2085),−0.8740+i(−0.8369))	(L1.8492,0.1671+i(0.2137),−0.8774+i(−0.841))	(L1.9431,0.1713+i(0.2190),−0.8807+i(−0.8449))	(L2.0469,0.1756+i(0.2245),−0.8839+i(−0.8488))	(L2.1635,0.1799+i(0.2302),−0.8872+i(−0.8526))	

Step 4: Calculate the optimality function for the BCFLAAPOA operator, BCFLAAPOWA operator, BCFLAAPOG operator, and BCFLAAPOWG operator, see Table 4.

10.1371/journal.pone.0309900.t004 Table 4 Aggregated BCFL matrix.

	BCFLAAPOA	BCFLAAPOWA	BCFLAAPOG	BCFLAAPOWG	
ϖ1BC	(L0.1130,0.0449+i(0.0601),−0.8798+i(−0.9082))	(L0.1099,0.0445+i(0.0596),−0.8788+i(−0.9075))	(L1.5145,0.3683+i(0.4160),−0.4478+i(−0.5045))	(L1.4962,0.3669+i(0.4146),−0.4459+i(−0.5029))	
ϖ2BC	(L0.2278,0.1019+i(0.1350),−0.9293+i(−0.8798))	(L0.2240,0.1011+i(0.1338),−0.9287+i(−0.8788))	(L2.0529,0.5170+i(0.5781),−0.5550+i(−0.4478))	(L2.0381,0.5154+i(0.5761),−0.5535+i(−0.4459))	
ϖ3BC	(L0.3749,0.1350+i(0.1934),−0.9082+i(−0.9293))	(L0.3699,0.1338+i(0.1908),−0.9075+i(−0.9287))	(2.5336,0.5781+i(0.6629),−0.5045+i(−0.5550))	(L2.5193,0.5761+i(0.6596),−0.5029+i(−0.5535))	
ϖ4BC	(L0.582,0.0601+i(0.0784),−0.8798+i(−0.8355))	(L0.5744,0.0596+i(0.0778),−0.8788+i(−0.8337))	(L3.0358,0.4160+i(0.4647),−0.4478+i(−0.3759))	(L3.0197,0.4146+i(0.4632),−0.4459+i(−0.3733))	
ϖ5BC	(L0.9479,0.0784+i(0.1019),−0.9462+i(−0.9293))	(L0.9318,0.0778+i(0.1011),−0.9462+i(−0.9293))	(L3.6774,0.4646+i(0.5170),−0.6029+i(−0.5550))	(L3.6534,0.4632+i(0.5154),−0.6015+i(−0.5535))	

Step 5: Calculate the score values of all aggregate functions, see Table 5.

10.1371/journal.pone.0309900.t005 Table 5 Score values of the aggregated values.

	ϖ1AT	ϖ2AT	ϖ3AT	ϖ4AT	
ϖ1BC	−0.03171	−0.03084	−0.0424	−0.04174	
ϖ2BC	−0.0597	−0.5872	0.03161	0.03125	
ϖ3BC	−0.09431	−0.0932	0.07665	0.07525	
ϖ4BC	−0.15295	−0.15079	0.02882	0.02944	
ϖ5BC	−0.26783	−0.26334	−0.10801	−0.10747	

Step 6: Calculate the integrated utility function of the WASPAS method, such as K1=−0.01942,K2=0.00212,K3=0.01252,K4=−0.01259,K5=−0.07755

Step 7: Rank all the alternatives and examine the best one, such as K3>K2>K4>K1>K5

The most valuable and dominant decision is Modeling and Simulation “ϖ3BC” between the above fives. Further, we calculate the ranking values by using the proposed operators without following the WASPAS technique, then the ranking result is given in the shape of Table 6 (see Table 5).

10.1371/journal.pone.0309900.t006 Table 6 Ranking values of the aggregated values.

Methods	Ranking values	Best decision	
BCFLAAPOA Operator	K1>K2>K3>K4>K5	K1	
BCFLAAPOWA Operator	K1>K3>K4>K5>K2	K1	
BCFLAAPOG Operator	K3>K2>K4>K1>K5	K3	
BCFLAAPOWG Operator	K3>K2>K4>K1>K5	K3	

The most valuable and dominant decision is Modeling and Simulation “ϖ3BC” according to the theory of the BCFLAAPOG operator and BCFLAAPOWG operator, but in the consideration of the BCFLAAPOA operator and BCFLAAPOWA operator, the best decision is Data Coupling “ϖ1BC”. Additionally, we described the significant of the parameters by using the technique of proposed BCFLAAPOWG operator are described in Table 7.

10.1371/journal.pone.0309900.t007 Table 7 Stability of the parameter using the BCFLAAPOWG operator.

Parameter	Score values	Ranking values	
π = 2	-0.0194,0.0021,0.0125,-0.0126,-0.0775	K3>K2>K4>K1>K5	
π = 4	0.0077,0.048,0.0708,0.0491,-0.0211	K3>K4>K2>K1>K5	
π = 6	0.0195,0.0674,0.0949,0.0749,0.0032	K3>K4>K2>K1>K5	
π = 8	0.0258,0.0777,0.1077,0.0866,0.0161	K3>K4>K2>K1>K5	
π = 10	0.0297,0.0841,0.1154,0.0969,0.024	K3>K4>K2>K1>K5	

For different values of parameter, the proposed BCFLAAPOWG operator is given the same best optimal which states the stability of the initiated operators. Further, we will evaluate the supremacy of the proposed theory by comparing our results with some existing techniques.

6. Comparative analysis

This section aims to compare the proposed ranking values with the obtained ranking values to enhance or show the supremacy and validity of the proposed theory. For comparison, we will use the information in Table 1, and then have the following existing techniques, such as Mahmood et al. [25] proposed aggregation operators for BCFLSs. Further, Yager [26] evaluated power operators for crisp values. Mardani et al. [29] presented the WASPAS theory for FSs and Jaleel [30] exposed it for BCFSs. Bi et al. [31] initiated the arithmetic operators for CFSs and Hu et al. [32] derived the power operators for CFSs. Jana et al. [33] exposed the Dombi operators for BFSs. Mahmood et al. [34] evaluated the aggregation operators for BCFSs. Further, Mahmood et al. [36] presented the Aczel-Alsina operators for BCFSs. Thus, based on the data in Table 1, the comparative analysis is listed in Table 8.

10.1371/journal.pone.0309900.t008 Table 8 Comparative analysis between proposed and prevailing information.

Methods	Ranking values	Best decision	
Mahmood et al. [25]	K3>K2>K4>K1>K5	K3	
Yager [26]	Failed to evaluate the data in Table 1.	no	
Mardani et al. [29]	Failed to evaluate the data in Table 1.	no	
Jaleel [30]	Failed to evaluate the data in Table 1.	no	
Bi et al. [31]	Failed to evaluate the data in Table 1.	no	
Hu et al. [32]	Failed to evaluate the data in Table 1.	no	
Jana et al. [33]	Failed to evaluate the data in Table 1.	no	
Mahmood et al. [34]	Failed to evaluate the data in Table 1.	no	
Mahmood et al. [36]	Failed to evaluate the data in Table 1.	no	
WASPAS Method	K3>K2>K4>K1>K5	K3	
BCFLAAPOA Operator	K1>K2>K3>K4>K5	K1	
BCFLAAPOWA Operator	K1>K3>K4>K5>K2	K1	
BCFLAAPOG Operator	K3>K2>K4>K1>K5	K3	
BCFLAAPOWG Operator	K3>K2>K4>K1>K5	K3	

The most valuable and dominant decision is Modeling and Simulation “ϖ3BC” according to the theory of the BCFLAAPOG operator, BCFLAAPOWG operator, Mahmood et al. [25], and WASPAS method, but in the consideration of the BCFLAAPOA operator and BCFLAAPOWA operator, the best decision is Data Coupling “ϖ1BC”. Further, the existing techniques and methods failed to evaluate the data in Table 1, because these are the special cases of the proposed operators. The major advantages of the presented techniques are listed below:

Maximum, minimum, Drastic, Algebraic, power, Aczel-Alsina aggregation operators, and WASPAS method for fuzzy information.

Maximum, minimum, Drastic, Algebraic, power, Aczel-Alsina aggregation operators, and WASPAS method for bipolar fuzzy information.

Maximum, minimum, Drastic, Algebraic, power, Aczel-Alsina aggregation operators, and WASPAS method for linguistic term information.

Maximum, minimum, Drastic, Algebraic, power, Aczel-Alsina aggregation operators, and WASPAS method for complex fuzzy information.

Maximum, minimum, Drastic, Algebraic, power, Aczel-Alsina aggregation operators, and WASPAS method for bipolar fuzzy information.

Maximum, minimum, Drastic, Algebraic, power, Aczel-Alsina aggregation operators, and WASPAS method for bipolar complex fuzzy information.

Maximum, minimum, Drastic, Algebraic, power, Aczel-Alsina aggregation operators, and WASPAS method for fuzzy linguistic term information.

Maximum, minimum, Drastic, Algebraic, power, Aczel-Alsina aggregation operators, and WASPAS method for bipolar fuzzy linguistic term information.

Maximum, minimum, Drastic, Algebraic, power, Aczel-Alsina aggregation operators, and WASPAS method for complex fuzzy linguistic term information.

Maximum, minimum, Drastic, Algebraic, power, Aczel-Alsina aggregation operators, and WASPAS method for bipolar complex fuzzy linguistic term information.

Form the above analysis, we clear that the proposed theory is very powerful and dominant because of its features, but the existing techniques are contained a lot of problems. The limitations of the existing techniques are briefly discussed below:

Mahmood et al. [25] proposed aggregation operators for BCFLSs, where the ranking values of the proposed theory of Mahmood et al. [25] follow, such as K3>K2>K4>K1>K5, but the proposed theory is the modified version of the existing techniques of Mahmood et al. [25].

Yager [26] evaluated power operators for crisp values has failed to cope with it because of unlimited complications and problems, where the technique of Yager [26] is a special part of the proposed theory.

Mardani et al. [29] presented that the WASPAS theory for FSs has failed to cope with it because of unlimited complications and problems, where the technique of Mardani et al. [29] is the special part of the proposed theory.

Jaleel [30] exposed it for BCFSs. Bi et al. [31] initiated the arithmetic operators for CFSs and Hu et al. [32] derived the power operators for CFSs. Jana et al. [33] exposed the Dombi operators for BFSs. Mahmood et al. [34] evaluated the aggregation operators for BCFSs. Further, Mahmood et al. [36] presented the Aczel-Alsina operators for BCFSs. These techniques are the modified version of the proposed theory.

Hence, the existing techniques have failed to cope with it, but the proposed theory is more general and more modified than the existing information to cope with it.

7. Conclusion

The model of BCFL information is very strong and reliable to cope with vague and uncertain information in genuine-life problems. The techniques of Aczel-Alsina aggregation operators, power aggregation operators, and the WASPAS technique have various advantages. Based on their features, we have described the following ideas, such as

We diagnosed the model of Aczel-Alsina operational laws for BCFL variables.

We evaluated the BCFLAAPOA operator, BCFLAAPOWA operator, BCFLAAPOG operator, and BCFLAAPOWG operator.

We presented some fundamental properties for the above operators.

We computed the WASPAS method based on initiated operators.

We demonstrated the procedure of the MADM technique based on initiated operators for computing the best technique for addressing geographic information systems.

We compared the ranking values of initiated techniques with the ranking values of the existing techniques based on illustrated examples to enhance the worth of the proposed theory.

7.1. Limitations of the proposed model

The model of the BCFL technique is very proficient because of the positive membership function, negative membership function, and linguistic variable, but in the presence of complex information the technique of the BCFL set is not working, for instance, when we provide the positive and negative information in the form of membership and non-membership, then the technique of BCFL set has been failed. For this, we need to provide the model of bipolar complex intuitionistic fuzzy linguistic sets and their extensions.

7.2. Future directions

In the future, we will discuss the Hamacher operators, Einstein operators, Dombi operators, and Frank operators based on bipolar complex fuzzy linguistic sets and also try to propose some new techniques, called TOPSIS method, MARCOS methods, VIKOR method, AHP method, and CODAS method to enhance the worth of the proposed theory.

This paper is supported by the NRPU-HEC Pakistan Project Number 14662 and the joint project PSF(PSF-NSFC/JSEP/ENG/AJKUKAJK/01)-NSFC (12211540710).

10.1371/journal.pone.0309900.r001
Decision Letter 0
Gul Muhammet Academic Editor
© 2024 Muhammet Gul
2024
Muhammet Gul
https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Submission Version0
8 Jul 2024

PONE-D-24-24446Analysis of Coupling in Geographic Information Systems Based on WASPAS Method for Bipolar Complex Fuzzy Linguistic Aczel-Alsina Power Aggregation OperatorsPLOS ONE

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Reviewer #2: Yes

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Reviewer #1: I have read the manuscript very carefully, the manuscript is very interesting and results

are corrected. After accepting the paper, I have some minor revision, such as:

1) Avoid abbreviations in abstract.

2) In the introduction section, add advantages, clearly present research gap.

3) Extended the literature review section by presenting relevant methods, e.g. (parsimonious spherical fuzzy AHP and the integrated IMF SWARA and Fuzzy Bonferroni operator)

4) present results in figures

5) In Section 2, please include some descriptive analysis for each definition.

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8) In the application, add relatedgeometrical diagrams.

Reviewer #2: This manuscript presents an approach applying Aczel-Alsina aggregation operators to bipolar complex fuzzy linguistic sets for analyzing coupling in geographic information systems. However, the paper requires major revisions before it is suitable for publication. I have the following concerns:

- The abstract does not clearly summarize the key contributions and objectives of the paper.

- The integration of Aczel-Alsina operators with bipolar complex fuzzy linguistic sets appears novel and could provide valuable tools for decision making under uncertainty. However, the novelty claim is not fully justified. - a more extensive literature review is needed to clearly situate this work relative to previous studies applying fuzzy approaches to GIS coupling analysis.

-The literature review appears to be incomplete and lacks critical analysis of existing work in the field. The authors should: Expand the review to include more recent publications (within the last 3-5 years). Provide a more in-depth analysis of how their work relates to and builds upon previous research. Clearly identify the research gap their study aims to fill. Add the recent work such as MADM models[10.3389/frai.2024.1347626, 10.22105/jfea.2023.426042.1331, 10.61356/j.mawa.2024.4241, 10.56578/josa020203, 10.1007/s00500-023-09459-0, 10.22105/jarie.2022.339957.1467, 10.31181/sa2120246, 10.22105/jfea.2023.425520.1326, 10.61356/j.mawa.2024.26761]; Bipolar/ linguistic sets[10.1007/s40314-023-02254-5, ]; aggregation operators [10.56578/josa020103, 10.22105/jfea.2023.422582.1318, 10.1007/s40314-021-01651-y, 10.3934/math.2023577, 10.1038/s41598-023-37497-z, 10.22105/bdcv.2021.142090 ]; various fuzzy set extensions[10.1111/exsy.12542, 10.22105/jfea.2024.424633.1328, 10.3934/math.2022954, 10.22105/bdcv.2023.192676, 10.3233/JIFS-210655, 10.22111/ijfs.2024.45118.7968, ].

- The methodology could be explained more clearly, especially the definitions of the various operators which rely on multiple mathematical terms defined earlier. More intuitive explanations may help non-expert readers. Provide a more detailed explanation of the BCFLAAPOA and BCFLAAPOG operators. Justify the choice of these specific operators over other potential alternatives. Include a step-by-step explanation of how these operators are applied in the context of their research.

-The properties and lemmas stated require formal proofs to demonstrate their validity.

- There are several issues with the formulas and mathematical notation presented:Some formulas appear to be incomplete or incorrectly formatted (e.g., Eq. 35 and 36). The use of symbols is inconsistent throughout the paper (e.g., π and ϖ are used interchangeably). The authors should provide more context and explanation for each formula presented.

-In equation (1)-(4), how are the parameters π and δ defined and how do they relate to the problem context? What is the recommended range of values for π?

-Equations (7)-(8) define power aggregation operators but do not specify the function used to calculate the similarity measure ω(πj). Provide more details on the choice of similarity function?

-Regarding the bipolar complex fuzzy linguistic operational laws in (10)-(13), could you explain intuitively what each component of the tuples represents (e.g. the positive/negative membership grades)? Some illustrative examples may help interpretation.

-In the score and accuracy functions (14)-(15), what is the justification for using the specific formulae? Have other definitions been considered/evaluated?

- In Eq. 36, what is the significance of the parameter π in the exponents?

-How does the BCFLAAPOWG operator differ from the BCFLAAPOG operator in terms of its mathematical properties and practical applications?

-Can the authors provide a numerical example to illustrate the application of Eq. 35 and 36?

-There are numerous formatting issues, particularly with equations and tables.

-Figures and tables should be properly labeled and referenced in the text.

- No empirical data or case study is presented to demonstrate the practical application of the proposed approach. Examples are needed to validate the methodology works as designed.

- The authors should provide a more comprehensive analysis of their findings. Comparisons with existing methods or techniques should be included to demonstrate the advantages of their proposed approach. The practical implications and limitations of the study should be discussed in detail.

- The conclusion is weak and does not effectively summarize the key contributions of the study: The authors should clearly state the main findings and their significance. Future research directions should be outlined more specifically.

- The paper contains numerous grammatical errors and awkward phrasings that need to be addressed to improve readability.

**********

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Reviewer #2: No

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10.1371/journal.pone.0309900.r002
Author response to Decision Letter 0
Submission Version1
31 Jul 2024

Response to the Reviewers/Editor in Chief/Associate Editor in Chief of

PLOS ONE

Subject: PLOS ONE Decision: Revision required

[PONE-D-24-24446] - [EMID:0c89c56569d8837a]

“Analysis of Coupling in Geographic Information Systems Based on WASPAS Method for Bipolar Complex Fuzzy Linguistic Aczel-Alsina Power Aggregation Operators”

Dear Editors and Reviewers:

First, the authors would like to thank the Editor in Chief, Associate Editor, and anonymous referees for spending their time on the manuscript carefully. The comments of the editors and reviewers are valuable. We have taken all the suggestions/comments positively and did our best to incorporate all these suggestions in the revised version. Our pointwise responses to the reviewer’s comments/suggestions are given below.

Reviewer 1 Comments:

Reviewer #1:

I have read the manuscript very carefully, the manuscript is very interesting and results

are corrected. After accepting the paper, I have some minor revision, such as:

Response to Reviewer 1#

Dear Sir/Mam, we appreciate your time in handling our paper and providing suggestions for improvement. We believe the quality of the revised version has considerably improved and hope that you find the revised manuscript satisfactory this time.

Comment 1: Avoid abbreviations in abstract.

Response: Dear Sir/Mam, thank you very much for taking an interest and pointing out these deficiencies. Dear Sir/Mam, we have avoided abbreviations from the abstract section as per your suggestion and highlighted them in the revised manuscript, I hope this time you will be satisfied.

Comment 2: In the introduction section, add advantages, clearly present research gap.

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, we have revised the introduction section by including the advantages and research gap as per your suggestion and highlighted in the revised manuscript, I hope this time you will be satisfied.

Comment 3: Extended the literature review section by presenting relevant methods, e.g. (parsimonious spherical fuzzy AHP and the integrated IMF SWARA and Fuzzy Bonferroni operator).

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, we have extended the literature review section (see Ref. [59-62]) as per your suggestion and highlighted in the revised manuscript, I hope this time you will be satisfied.

Comment 4: Present results in figures.

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, we have done the needful, see Figure 1, I hope this time you will be satisfied.

Comment 5: In Section 2, please include some descriptive analysis for each definition.

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, we have improved section 2 by including the descriptive analysis for each definition as per your suggestion and highlighted in the revised manuscript, I hope this time you will be satisfied.

Comment 6: The authors suggested checking the grammar carefully.

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, we have improved the quality of the manuscript especially grammatical mistakes and typos error as per your suggestion and highlighted in the revised manuscript, I hope this time you will be satisfied.

Comment 7: The order of the citation in your manuscript is not the proper way, please correct it.

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, we have very carefully revised the order of the citation as per your suggestion and highlighted it in the revised manuscript, I hope this time you will be satisfied.

Comment 8: In the application, add related geometrical diagrams.

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, we have added the geometrical diagram (see Figure 2) in the application section as per your suggestion and highlighted in the revised manuscript, I hope this time you will be satisfied.

Reviewer 2 Comments:

Reviewer #2: This manuscript presents an approach applying Aczel-Alsina aggregation operators to bipolar complex fuzzy linguistic sets for analyzing coupling in geographic information systems.

However, the paper requires major revisions before it is suitable for publication. I have the following concerns:

Response to Reviewer 2#

Dear Sir/Mam, we appreciate your time in handling our paper and providing suggestions for improvement. We believe the quality of the revised version has considerably improved and hope that you find the revised manuscript satisfactory this time.

Comment 1: The abstract does not clearly summarize the key contributions and objectives of the paper.

Response: Dear Sir/Mam, thank you very much for taking an interest and pointing out these deficiencies. Dear Sir/Mam, we have very carefully revised the abstract section by including the novelty, motivation, objectives, and major contribution of the proposed work in the abstract section as per your suggestion and highlighted in the revised manuscript, I hope this time you will be satisfied.

Comment 2: The integration of Aczel-Alsina operators with bipolar complex fuzzy linguistic sets appears novel and could provide valuable tools for decision making under uncertainty. However, the novelty claim is not fully justified. - a more extensive literature review is needed to clearly situate this work relative to previous studies applying fuzzy approaches to GIS coupling analysis.

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, we have very carefully improved the introduction section by including the novelty, motivation, research gap, and problem description, and also improved the literature review section by including some information based on fuzzy approaches to GIS coupling analysis as per your suggestion and highlighted in the revised manuscript, I hope this time you will be satisfied.

Comment 3: The literature review appears to be incomplete and lacks critical analysis of existing work in the field. The authors should: Expand the review to include more recent publications (within the last 3-5 years). Provide a more in-depth analysis of how their work relates to and builds upon previous research. Clearly identify the research gap their study aims to fill. Add the recent work such as MADM models[10.3389/frai.2024.1347626, 10.22105/jfea.2023.426042.1331, 10.61356/j.mawa.2024.4241, 10.56578/josa020203, 10.1007/s00500-023-09459-0, 10.22105/jarie.2022.339957.1467, 10.31181/sa2120246, 10.22105/jfea.2023.425520.1326, 10.61356/j.mawa.2024.26761]; Bipolar/ linguistic sets[10.1007/s40314-023-02254-5, ]; aggregation operators [10.56578/josa020103, 10.22105/jfea.2023.422582.1318, 10.1007/s40314-021-01651-y, 10.3934/math.2023577, 10.1038/s41598-023-37497-z, 10.22105/bdcv.2021.142090 ]; various fuzzy set extensions[10.1111/exsy.12542, 10.22105/jfea.2024.424633.1328, 10.3934/math.2022954, 10.22105/bdcv.2023.192676, 10.3233/JIFS-210655, 10.22111/ijfs.2024.45118.7968].

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, we have very carefully revised the literature review section (see Ref. [37-62]) as per your suggestion and highlighted in the revised manuscript, I hope this time you will be satisfied.

Comment 4: The methodology could be explained more clearly, especially the definitions of the various operators which rely on multiple mathematical terms defined earlier. More intuitive explanations may help non-expert readers. Provide a more detailed explanation of the BCFLAAPOA and BCFLAAPOG operators. Justify the choice of these specific operators over other potential alternatives. Include a step-by-step explanation of how these operators are applied in the context of their research.

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, we have explained the methodology of the proposed work more clearly, especially the definition of the various operators and their properties. We have provided more explanations for the BCFLAAPOA and BCFLAAPOG operators. We have also justified the proposed operators with the help of potential alternatives explained them step-by-step as per your suggestion and highlighted them in the revised manuscript, I hope this time you will be satisfied.

Comment 5: The properties and lemmas stated require formal proofs to demonstrate their validity.

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, we have included the proof of the proposed Theorems and Properties as per your suggestion and highlighted in the revised manuscript, I hope this time you will be satisfied.

Comment 6: There are several issues with the formulas and mathematical notation presented: Some formulas appear to be incomplete or incorrectly formatted (e.g., Eq. 35 and 36). The use of symbols is inconsistent throughout the paper (e.g., π and ϖ are used interchangeably). The authors should provide more context and explanation for each formula presented.

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, we have very carefully revised the formulas and mathematical notions, especially the information in Eq. (35), Eq. (36), and their related problems. Further, we have also explained all the symbols that are used in the proposed manuscript as per your suggestion and highlighted them in the revised manuscript, I hope this time you will be satisfied.

Comment 7: In equation (1)-(4), how are the parameters π and δ defined and how do they relate to the problem context? What is the recommended range of values for π?

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, the parameters used in Eq. (1) to Eq. (4) play an important role, because, with the help of these parameters, we can easily derive some existing techniques, for instance, if we used the value of π=0, then we will get the model of the drastic t-norm A^d (μ_1,μ_2 ) and drastic t-conorm 〖A^@〗^d (μ_1,μ_2 ), such as

A^d (μ_1,μ_2 )={■(μ_1&if μ_2=1@μ_2&if μ_1=1@0&otherwise)┤

〖A^@〗^d (μ_1,μ_2 )={■(μ_1&if μ_2=0@μ_2&if μ_1=0@1&otherwise)┤

if we used the value of π=π=∞, then we will get the model of the maximum t-norm and minimum t-norm, such as max⁡(μ_1,μ_2 ) and min⁡(μ_1,μ_2 ). But if we use the value of π∈(0,∞), then we get the idea of the Aczel-Alsina t-norm e^(-((-log(μ_1 ))^π+(-log(μ_2 ))^π )^(1/π) ) and Aczel-Alsina t-conorm 1-e^(-((-log(1-μ_1 ))^π+(-log(1-μ_2 ))^π )^(1/π) ), which is the modified version of the algebraic t-norm and t-conorm Where A^π (μ_1,μ_2 )=μ_1*μ_2 and 〖A^@〗^π (μ_1,μ_2 )=μ_1+μ_2-μ_1*μ_2, where the value of δ≥1, I hope this time you will be satisfied.

Comment 8: Equations (7)-(8) define power aggregation operators but do not specify the function used to calculate the similarity measure ω(πj). Provide more details on the choice of similarity function?

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, we have defined the power aggregation operators in Eq. (7) and Eq. (8), but these definitions are defined for fuzzy values, anyhow we have defined the value of distance measures in Def. (2) and also defined it for BCFL values in Def. (5) as per your suggestion and highlighted in the revised manuscript, I hope this time you will be satisfied.

Comment 9: Regarding the bipolar complex fuzzy linguistic operational laws in (10)-(13), could you explain intuitively what each component of the tuples represents (e.g. the positive/negative membership grades)? Some illustrative examples may help interpretation.

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, we have explained the information in Eq. (10) to Eq. (13) with the help of some satiable examples as per your suggestion and highlighted in the revised manuscript, I hope this time you will be satisfied.

Comment 10: In the score and accuracy functions (14)-(15), what is the justification for using the specific formulae? Have other definitions been considered/evaluated?

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, we have easily evaluated the order between any two real numbers, but in the case of complex numbers or in the case of bipolar complex fuzzy linguistic numbers, it is quite complex to describe which one is greater and which is weaker. For evaluating the order between any two BCFL values, we have defined the idea of score values, the concept of score value can easily convert the BCFL number to a simple real number, which can help to order them. But if we obtain the score value of two BCFL numbers that are equal, then we will be using the accuracy values, I hope this time you will be satisfied.

Comment 11: In Eq. 36, what is the significance of the parameter π in the exponents?

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, we have briefly described the significance of the parameter π in the form of Table 7 before the comparison section and highlighted in the revised manuscript, I hope this time you will be satisfied.

Comment 12: How does the BCFLAAPOWG operator differ from the BCFLAAPOG operator in terms of its mathematical properties and practical applications?

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, the technique of the BCFLAAPOWG operator and BCFLAAPOG operator are computed based on power aggregation operators, for the construction of the BCFLAAPOG operator, we have used the power operators without weight vector, such as ((1+ω(ϖ_j )))/(∑_(j=1)^n▒(1+ω(ϖ_j )) ), but in the construction of the BCFLAAPOWG operators, we have used the power operators with weight vectors, such as (Ψ_j (1+ω(ϖ_j )))/(∑_(j=1)^n▒〖Ψ_j (1+ω(ϖ_j )) 〗), which can affect the final results, see the ranking values in Table 6 and Table 8, it means that the weight vector plays an important role in the construction of the above operators, but in mathematical properties, both operators are the same, I hope this time you will be satisfied.

Comment 13: Can the authors provide a numerical example to illustrate the application of Eq. 35 and 36?

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, we have illustrated numerical examples for the aggregation operators in Eq. (35) and Eq. (36) as per your suggestion and highlighted in the revised manuscript, I hope this time you will be satisfied.

Comment 14: There are numerous formatting issues, particularly with equations and tables.

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, we have very carefully revised the formatting problems of the equations, Tables, and Figures as per your suggestion and highlighted in the revised manuscript, I hope this time you will be satisfied.

Comment 15: Figures and tables should be properly labeled and referenced in the text.

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, we have properly labeled all the Figures and Tables in the revised manuscript, I hope this time you will be satisfied.

Comment 16: No empirical data or case study is presented to demonstrate the practical application of the proposed approach. Examples are needed to validate the methodology works as designed.

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear Sir/Mam, we have briefly discussed the case study and also demonstrated the practical application of the proposed work. Further, we have also given some examples for showing the validation of the proposed methodology as per your suggestion and highlighted in the revised manuscript, I hope this time you will be satisfied.

Comment 17: The authors should provide a more comprehensive analysis of their findings. Comparisons with existing methods or techniques should be included to demonstrate the advantages of their proposed approach. The practical implications and limitations of the study should be discussed in detail.

Response: Dear Sir/Mam, thank you very much for pointing out these deficiencies. Dear

Attachment Submitted filename: Response to the reviewers_R#1.docx

10.1371/journal.pone.0309900.r003
Decision Letter 1
Gul Muhammet Academic Editor
© 2024 Muhammet Gul
2024
Muhammet Gul
https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Submission Version1
21 Aug 2024

Analysis of Coupling in Geographic Information Systems Based on WASPAS Method for Bipolar Complex Fuzzy Linguistic Aczel-Alsina Power Aggregation Operators

PONE-D-24-24446R1

Dear Dr. Pamucar,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

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Academic Editor

PLOS ONE

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Reviewer #1: (No Response)

Reviewer #2: All comments have been addressed

**********

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Reviewer #2: Yes

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Reviewer #2: Yes

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10.1371/journal.pone.0309900.r004
Acceptance letter
Gul Muhammet Academic Editor
© 2024 Muhammet Gul
2024
Muhammet Gul
https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
23 Aug 2024

PONE-D-24-24446R1

PLOS ONE

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==== Refs
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