A Systematic Analysis of EDAS Method and Power Aggregation Operators Based on Bipolar Complex Fuzzy Linguistic for Rural Energy Infrastructure Decision

In rural energy infrastructure decisions, EDAS methods and power aggregation operators can be used in combination to help decision makers evaluate and select different infrastructure options. By using these methods, decision makers can consider multiple factors such as cost, efficiency, sustainability, etc., and integrate them into a comprehensive assessment to make more comprehensive and informed decisions. In this article, we compute the power aggregation operators based on bipolar complex fuzzy linguistic (BCFL) values for weight vectors and without weight vectors, called BCFL power averaging (BCFLPOAV), BCFL power weighted averaging (BCFLPOWAV), BCFL power geometric (BCFLPOGE), BCFL power weighted geometric (BCFLPOWGE) operators, and described their valuable properties. Further, we derive the evaluation based on distance from average solution (EDAS) technique for proposed methods based on BCFL information, where the EDAS technique is one of the multi-attribute decision-making (MADM) problems, which evaluates the traditional decision-making process. Finally, to improve or to show the supremacy and validity of the initiated operators, we illustrate rural energy infrastructure example to try to evaluate the comparative analysis between proposed and existing techniques to find the rationality and flexibility of the derived techniques for the development of rural energy infrastructure.


I. INTRODUCTION
Rural energy infrastructure decision-making involves how to plan, build and manage energy infrastructure in rural areas to meet the energy needs of farmers and rural communities.Such decisions often require a combination of factors, including energy sources, technology options, economic viability, and social impact.When making rural energy infrastructure decisions, decision support methods and tools may be needed to help systematically analyze and evaluate various factors.For example, a multi-criteria decision analysis method (MCDA) can be used to synthesize different stakeholder views and objectives.
The associate editor coordinating the review of this manuscript and approving it for publication was Xiaojie Su .
In various cases, systematic reviews may be followed by meta-analysis [1], where dominant and flexible.Metaanalysis improves the statistical power and precision of the investigations, especially if people study systematic may systematic procedures to ensure that the review is replicable [2], comprehensive, and transparent.Both techniques have a lot of benefits because, with the help of these techniques, we can help individuals improve the research techniques for the research community [3].These methods have been used in many fields based on classical values, where systematic review and meta-analysis are the basic part of the decisionmaking process, with the help of these techniques, we can easily find the best optimal among the collection of information.Further, we noticed that the range of the classical set is very limited because we have just zero and one, but in many real-life cases, we needed a stronger theory, for this, Zadeh [4] exposed the fuzzy set (FS) theory with a strong range, where the range of FS is unit interval.Further, FS theory is very flexible and very dominant because of its structure, where the truth grade in FS is defined based on a universal set to the unit interval, and because of these reasons many applications have been done by different scholars, for instance, fuzzy superior mandelbrot set [5], fuzzy n-soft sets [6], complex fuzzy nsoft sets [7], hesitant fuzzy n-soft sets [8], evolving research agenda based on fuzzy information [9], on a novel view of fuzzy logic and their applications [10], and setback in ranking fuzzy numbers [11].Furthermore, Zhang [12] initiated the bipolar FS (BFS), where the BFS contained the positive and negative truth information based on fixed set-to-unit intervals, such as [1,0] and [-1, 0].The FS is very flexible, but the positive truth grade is not enough for dealing with uncertain and vague information, therefore, the technique of BFS is very dominant to cope with it.Some applications of BFS are discussed in the shape, for instance, (a,b)-bipolar fuzzified rough model [13], pessimistic multi-granulation rough BFSs and their applications [14], BFS theory and their applications [15], and applications on bipolar vague soft information [16].
In FSs and BFSs, the truth grade was computed in the shape of a real number, but in many scenarios, we needed a truth grade in the shape of a complex number, because when we buy any kind of new car, for this, we provide two types of information, such as the name of the car and version of the car and in the presence of the FSs, such kind of information is not enough.For this, Ramot et al. [17] computed the complex FS (CFS), where the truth grade in CFS was derived in the shape of complex numbers whose range is [0, 1] + i [0, 1].Further, many applications have been done by different scholars based on CFSs, for instance, cross-entropy measures for CFSs [18], complex fuzzy large-scale learning problems [19], identifying signals based on CFSs [20], and a new design of Mamdani complex fuzzy inference systems [21].Additionally, the truth grade in CFSs is not enough for managing vague and uncertain information, therefore, Mahmmod and Ur Rehman [22] investigated the novel theory of bipolar CFSs (BCFSs) with two same grades in different directions, called positive truth grade and negative truth grade, such as In 1975, Zadeh [23], [24], [25] initiated the linguistic term set (LTS), which is very flexible and very effective in coping with uncertain and vague information, where to analyze the temperature, we have the following possibilities, such as very low, low, normal, high, very high, and many others, called linguistic information, which played a valuable and dominant role in real-life problems.Furthermore, after the construction of the LTSs, many theories have been developed by different scholars, for instance, the linguistic approach based on FSs was presented by Tong and Bonissone [26], Martinez et al. [27] derived the bipolar linguistic decision-making problems, the linguistic variables based on CFSs was presented by Alkouri and Salleh [28], Mahmood et al. [29] derived the complex linguistic fuzzy sets, Mahmood et al. [30] presented the bipolar complex fuzzy linguistic sets (BCFLSs) and their applications.
The basic theory of the EDAS technique was initiated by Keshavarz et al. [31] by using different measures.Further, Ghorabaee et al. [32] derived the extended EDAS technique based on fuzzy information.Jana and Pal [33] exposed the bipolar fuzzy EDAS technique and its applications.Furthermore, the power operator based on a crisp set was initiated by Yager [34], which is very reliable for aggregating the collection of information.Khan et al. [35] evaluated the complex fuzzy rough aggregation operators and their application.Mardani et al. [36] presented the fuzzy aggregation operators.Jana et al. [37] derived the bipolar fuzzy Dombi operators and their applications.Bi et al. [38] presented the complex fuzzy geometric operators.Hu et al. [39] initiated the complex fuzzy power operators.Bi et al. [40] exposed the complex fuzzy averaging operators.Further, Mahmood et al. [41] exposed the Aczel-Alsina operators for BCFSs.The simple aggregation operators for BCFS were proposed by Mahmood et al. [42].
In the above four paragraphs, we briefly discussed a lot of ideas, operators, and techniques, where these techniques are very reliable, but the major problem is that the idea of BCFLS has been proposed but up to date no one can derive any kind of operators based on it, because the structure of BCFLS is very complicated and unreliable due to their features.At the time, it was very hurdle to cope with linguistic information and truth and falsity grades in different directions.The power operators, EDAS method, and their related properties are very reliable, but no one derives it based on BCFL information.The major contribution of the proposed theory is listed below: 1) To compute the BCFLPOAV, BCFLPOWAV, BCFL-POGE, and BCFLPOWGE operators, and describe their valuable properties.2) To derive the EDAS technique for proposed methods based on BCFL information.3) To utilize the proposed EDAS method in the environment of MADM problems, which evaluates the traditional decision-making process.4) To provide a new paradigm for the planning of rural energy infrastructure by introducing advanced decision support methods such as EDAS as well as innovative energy aggregation algorithms.This approach takes into account a variety of factors, including environmental, social and economic dimensions, leading to more comprehensive, efficient and sustainable decision-making.By cutting across traditional frameworks, the research opens new avenues for energy access and sustainable development in rural areas, with far-reaching implications..This section explained the idea of EDAS techniques [31] based on some classical set theory, where the basic steps of the EDAS technique are listed below: Step 1: Compute the matrix by using the i alternatives and criteria, where Step 2: Evaluate the averaging family of the above information, such as Step 3: Further, we investigate the positive distance from average (PDA) and negative distance from average (NDA) by using the above information, such as and Step 4: Derive the weighted summation based on the PDA and NDA, such as Step 5: Evaluate the normalized values of the SUM (P i ) and SUM (N i ), such as Step 6: Find the appraisal score of the above information, such as After evaluating the above process, we can easily derive our required result.

B. BCFLSS
This section is very basic because it talks about the prevailing concepts, called BCFLSs, and some operational laws based on BCFLNs, such as score, accuracy, and their properties.Definition 1: [34] The mathematical shape of the power operator under the non-negative integers is described below: , where Sup H , H used support information depending on the distance measures with the following conditions: Definition 2: [30] The mathematical shape of BCFLS is described below: Additionally, we know that L ∈S = s : = 1, 2, . . ., be any BCFLNs.Then Signified as a score and accuracy values with few characteristics, such as if

III. BCFL POWER AGGREGATION OPERATORS
In this section, we exposed the power operators based on BCFL information, called BCFLPOAV operator, BCFLPOWAV operator, BCFLPOGE operator, and BCFLPOWGE operator, and also investigated their reliable properties based on BCFLNs, such as Proof: Using the technique of mathematical induction, we have When = 2, thus Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
For = 2, we corrected.Further, we assume it for = ′ , then Hence, the proposed theory is held for all positive integers.Further, the mathematical shape of the weight vector is of the shape: BCFLNs.Then we present that the aggregated value of BCFLPOWAV H 1 , H 2 , .., H is again a BCFLN, such as Proof: Using the technique of mathematical induction, we have Case 1: When = 2, thus Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
For = 2, we corrected.Further, we assume it for = ′ , then BCFLPOGE H1 , H2 , .., Then, we evaluate it for = ′ + 1, such as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Hence, the proposed theory is held for all positive integers.
Signifies the theory of the BCFLPOWGE operator.Further, we explained the value of F H = = 1 ̸ = Sup H , H , where Sup H , H = 1− Dis H , H , with the below three rules, such as Further, the mathematical shape of the weight vector is of the shape:

., be any
BCFLNs.Then we present that the aggregated value of BCFLPOWGE H 1 , H 2 , .., H is again a BCFLN, such as x 3m . . .
where each term will be computed in the shape of BCFLNs.
Step 2: Evaluate the family of the above information, such as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Step 3: Further, we investigate the positive distance from average (PDA) and negative distance from average (NDA) by using the above information, such as Using the above formula, we find the value of the linguistic term, truth, and falsity grades separately, and this process will continuously be applied to the remaining formulas.Further, we used the below technique for evaluating the linguistic term, truth grade, and falsity grades, such as The same formula will be used based on BCFLPOWAV, BCFLPOGE, and BCFLPOWGE operators.
Step 4: Derive the weighted summation based on the PDA and NDA, such as Step 5: Evaluate the normalized values of the SUM (P i ) and SUM (N i ), such as Step 6: Find the appraisal score of the above information, such as Step 7: Finally, we evaluate the score values of the appraisal values to find the best optimal, such as Step 8: Evaluate the ranking values to examine the best one.After evaluating the above process, we can easily derive our required result.

V. A SYSTEMATIC REVIEW AND META-ANALYSIS: THEORY AND APPLICATIONS
At present, ZD village needs to build biomass energy centralized power generation facilities.The project's environmental protection design concept is advanced, meeting the ultra-low emission environmental protection standards.After the operation of the project, more than 1 million tons of straw biomass fuel will be efficiently used every year, and zero waste water discharge can be achieved, providing green electricity of about 500 million KWH, effectively alleviating the harm caused by straw incineration to the ecological environment.For China's rural areas to achieve carbon peak, carbon neutral ''30•60'' goal contribution.At present, there are five construction options for the project, which need to be screened, and the following five criteria need to be considered.

1) ''H AT
1 '': Sustainability and environmental impacts: Policy makers should assess the potential environmental impacts of energy infrastructure to ensure that options selected are consistent with sustainable development principles and reduce negative impacts on natural resources.
2) ''H AT 2 '': Social participation and community needs: Ensure that the needs and views of local communities are taken into account in decision-making processes and promote social participation.This can be achieved by holding civic engagement events and listening to feedback from residents.3) ''H AT 3 '': Economic feasibility and cost-effectiveness: Conduct a comprehensive economic feasibility analysis, taking into account the costs of infrastructure construction, operation and maintenance.Ensure that the selected option is economically viable to maximize cost effectiveness.4) ''H AT 4 '': Technical adaptability and reliability: Select reliable and adaptable energy technologies suitable for rural areas.Factors such as local resources and climatic conditions are considered to ensure the long-term reliability of the technology.5) ''H AT 5 '': Policy and Regulatory compliance: Ensuring that decisions are made in accordance with national and local energy policies and regulations, and that relevant standards and norms are followed to reduce legal risks and facilitate the smooth implementation of projects.To evaluate the best one, we have used the following five attributes, such as growth analysis, social analysis, political analysis, environmental analysis, and research strategy.Then, we used our computed procedure for evaluating the considered problems, such as Step 1: Compute the matrix by using the i alternatives and criteria, where H  1. Where each term will be computed in the shape of BCFLNs.
Step 2:Evaluate the averaging family of the above information by using the idea of BCFLPOAV operator, see Table 2.
Step 3:Further, we investigate the positive distance from average (PDA) and negative distance from average (NDA) by using the above information, see Table 3.
Step 4: Derive the weighted summation based on the PDA and NDA, see Table 4.
Step 5: Evaluate the normalized values of the SUM (P i ) and SUM (N i ), see Table 5.
Step 6: Find the appraisal score of the above information, see Table 6.
Step 7: Finally, we evaluate the score values of the appraisal values to find the best optimal, see Table 7. Step 8: Evaluate ranking values to examine the best one, see Table 8.
The most preferable and most valuable decision is H 4 , represented the quantitative analysis which a major part of the meta-analysis.Further, we follow the same procedure for evaluating the best decision based on BCFLPOWAV operator, BCFLPOGE operator, and BCFLPOWGE operator, see Table 9 using the following weight vectors, such as (0.3, 0.2, 0.1, 0.3, 0.1) T .
Thus, using the information in Table 9, the ranking values are stated in Table 10.The most preferable and most valuable decision is H 4 .This program is the ZD village needs to build biomass energy centralized power generation facilities need to focus on.

VI. COMPARATIVE ANALYSIS
This section aims to collect some prevailing techniques based on fuzzy sets and their extensions and try to evaluate our considered data in Table 1 with the help of it, then we compare their results with our proposed ranking results to show the validity and effectiveness of the initiated techniques.Comparative analysis is a major part of all manuscripts because, with the help of comparative analysis, we can easily evaluate the supremacy and validity of the proposed operators.In this manuscript, we used the following existing techniques for compassion, such as Mahmood et al. [30] presented the aggregation operators based on BCFLSs and their applications.Further, Khan et al. [35] evaluated the complex fuzzy rough aggregation operators and their application.Mardani et al. [36] presented the fuzzy aggregation operators.Jana et al. [37] derived the bipolar fuzzy Dombi operators and their applications.Bi et al. [38] presented the complex fuzzy geometric operators.Hu et al. [39] initiated the complex fuzzy power operators.Bi et al. [40] exposed the complex fuzzy averaging operators.Further, Mahmood et al. [41] exposed the Aczel-Alsina operators for BCFSs.The simple aggregation operators for BCFS were proposed by Mahmood et al. [42].Thus, using the data in Table 1, the comparison is available in Table 11.
The most preferable and most valuable decision is H 4 , represented the quantitative analysis, which is a major part of the meta-analysis, the best optimal is obtained with the help of four types of proposed operators and the existing technique of Mahmood et al. [30], which is the special case of the proposed operators.Further, we noticed that some existing techniques have failed, because of many limitations and problems, where up to date no one can derive any kind of operators based on BCFLSs accepted averaging and geometric operators which are the special cases of the proposed operators.Hence, the proposed operators are very flexible, and dominant compared to existing techniques.

VII. CONCLUSION
In this manuscript, we computed the following ideas, for instance, we presented the BCFLPOAV, BCFLPOWAV, BCFLPOGE, and BCFLPOWGE operators, and described their valuable properties.Further, we derived the EDAS technique for the proposed methods based on BCFL information.Moreover, we utilized the proposed EDAS method in the environment of MADM problems, which evaluates the traditional decision-making process.Finally, we improved the supremacy and validity of the initiated operators; we illustrate rural energy infrastructure example to try to evaluate the comparative analysis between proposed and existing techniques to find the rationality and flexibility of the derived techniques for the development of rural energy infrastructure.
The proposed theory is very suitable and dominant for depicting vague and unreliable information in genuine life problems, but in many cases, they are not working, for instance, if someone provides information in the shape of truth grade and falsity grade with 2-tuple linguistic variables, then the proposed theory has been failed, for this, we needed to propose the technique of bipolar complex intuitionistic 2-tuple linguistic sets and their extensions.
In the future, we will describe some Einstein operators, Dombi operators, Hamacher operators, Frank operators, the TOPSIS method, the WASPAS method, and many others based on BCFLSs.Further, we will utilize it in artificial intelligence, machine learning, neural networks, and game theory to enhance the worth of the derived operators to promote clean energy in rural areas.

,
IN −H ∈ [−1, 0].The BCFSs are much better than FSs, BFSs, and CFSs to cope with vague and uncertain information in genuine-life problems.

TABLE 3 .
Representation of PDA and NDA information.

TABLE 4 .
Representation of the weighted summation.

TABLE 5 .
Representation of the normalized values.

TABLE 6 .
Representation of the appraisal score.

TABLE 7 .
Representation of the score values.

TABLE 8 .
Representation of the ranking values.

TABLE 9 .
Representation of the score values for different aggregated values.

TABLE 10 .
Representation of the ranking values for data in Table8.

TABLE 11 .
Representation of the ranking values for data in Table8.