A Border Approximation Area Approach Considering Bipolar Neutrosophic Linguistic Variable for Sustainable Energy Selection

: In the last few decades, the computational methods under Multi-Criteria Decision-Making (MCDM) have experienced signiﬁcant growth in research interests from various scientiﬁc communities. Multi-Attributive Border Approximation area Comparison (MABAC) is one of the MCDM methods where its computation procedures are based on distances and areas, and able to express a complex decision systematically. Previous literature have suggested the combination of MABAC with fuzzy sets, in which this combination is used to solve problems that are characterized by uncertain and incomplete information. Di ﬀ erently from the fuzzy MABAC, which directly used single membership, this paper proposes bipolar neutrosophic MABAC of which the positive and negative of truth, indeterminate and false memberships of bipolar neutrosophic set are introduced to enhance decision in sustainable energy selection. Fourteen criteria and seven alternatives of sustainable energy are the main MCDM structures that need to be solved using the proposed method. A group of experts were invited to provide rating of performance values of criteria and alternatives of sustainable energy problem using a bipolar neutrosophic linguistic scale. The distances of alternatives from the Border Approximation Area of bipolar neutrosophic MABAC are the main output of the proposed method prior to making the ﬁnal decision. The computational results show that ‘Biomass’ is the optimal alternative to sustainable energy selection. Comparable results are also presented to check the consistency of the proposed method.


Introduction
Sustainable energy is one of the global goals in the 2030 Agenda for Sustainable Development, and plays a key role in ensuring accessibility to affordable, reliable, sustainable and modern energy for all [1]. In other words, sustainable energy plays an important role in the economic growth and social development of a country and the living quality of people [2,3]. The sector of sustainable energy has to balance energy production and consumption, and has no or minimal, negative impact on the environment, but at the same time, gives the opportunity for a country to increase the productivity of its social and economic activities [4]. Under a biophysical and ecological view, energy plays an important role in income determination and economic activities in which these activities will be significantly affected by changes in every energy consumption [5]. According to Qi et al. [6], efficient use of energy and sustainable production of energy industry are one of the most important factors for emerging economies. A study on impact of energy on productivity suggests that is not possible without energy use for the traditional growth model, which is treating energy as a secondary factor Before proceeding to discuss various extensions of MABAC based methods, it is good to recall the fuzzy sets and their affiliations in which this set has been successfully combined with many MCDM methods. The fuzzy set theory has been introduced in many MCDM methods to solve problems with uncertain information. Zadeh [22] proposed the concept of a fuzzy set to deal with problems that are characterized by imprecision, vagueness, and uncertainty. The gradation of membership of fuzzy sets is said to be able to imitate thinking and perception using linguistic information. Then, the intuitionistic fuzzy set (IFS) was proposed by Atanassov [23] as a generalization of the fuzzy set, which is considered membership degree and non-membership degree. In the real environment, IFS can deal only with vague information but not incongruous information. Owing to this, Atanassov and Gargov [24], generalized the IFS in the spirit of ordinary interval-valued fuzzy sets (IVFS) and introduced the concept of interval-valued intuitionistic fuzzy sets (IVIFSs). Besides that, Zhang [25] extended a fuzzy set to the bipolar fuzzy set with the grade of membership from [-1, 1]. The membership grade [−1, 0] is fairly fulfils the couched stand-property in the section while the membership grade [0, 1] of a section shows that the section fairly fulfils the matter and the membership grade 0 of a section presents that the section is unrelated to the parallel property.
Having discussed the development of fuzzy sets, this section continues with another kind of set. The neutrosophic set was introduced by Smarandache [26] as extensions of the concept classic set, fuzzy set, intuitionistic fuzzy set and interval-valued intuitionistic set. Neutrosophic set is a part of neutrosophy, which is a branch of philosophy that studies the origin, nature and scope of neutralities as well as their interactions with different ideational spectra. There are three elements of neutrosophic which are truth membership (T), indeterminacy membership (I) and falsity membership (F). Each member value is a real standard or non-standard subset of the unit interval ]0 _ , 0 + [ in the neutrosophic set. Then, Wang et al. [27] proposed single-valued neutrosophic set (SVNS) that is an extension of neutrosophic set which takes the value from the subset of [0, 1]. There are many researchers that have paid particular attention on the T, I, F components which later lead to define a particular case of neutrosophic sets such as rough neutrosophic set [28], interval-valued neutrosophic sets [29], trapezoidal neutrosophic sets [30], multi-valued neutrosophic set [31], simplified neutrosophic sets [32], and bipolar neutrosophic sets [33].
Bipolarity refers to the propensity of the human mind to reason and make decisions on the basis of positive and negative effects [34]. Positive information is all about possible, satisfactory, permitted, desired or considered that being acceptable, while impossible, rejected or forbidden are negative information. In addition, positive preferences correspond to the wishes as they specify which objects are more desirable than others without rejecting those that do not meet the wishes but negative preferences correspond to the constraints as they specify which values or objects have to be rejected. In Chinese medicine, the concept of yin and yang are two sides in which yang is the masculine or positive side of a system and yin is the feminine or negative side of a system. It is believed that most of the human decision-making is based on double-sided or bipolar judgment thinking which is the positive side and negative side. Based on these premises, Deli et al. [33] extended the ideas of bipolar fuzzy sets and neutrosophic sets to bipolar neutrosophic sets in which positive membership degree, negative membership degree, and its operations were studied. Bipolar fuzzy sets are of great value to deal with uncertain real-life problems and prove to help in dealing with the positive, as well as the negative membership values. Thus, in this paper, we combine bipolar neutrosophic with MCDM method to enhance the decision by considering bipolarity judgements. In particular, this paper aims to propose a new bipolar neutrosphic MABAC where positive and negative memberships of T, I, F are combined with the MABAC. Furthermore, the effectiveness of the proposed method is applied to the case of sustainable energy selection.
To sum up, this paper proposes a new bipolar neutrosophic MABAC in which several key features arise from our proposed method. The first feature is the introduction of new linguistic variables using bipolar neutrosophic numbers. The second feature is the use of bipolar neutrosophic numbers to the MABAC method in which the direct relation matrix was aggregated using bipolar neutrosophic weighted aggregation operator. Finally, this paper provides evidence on the feasibility of the proposed method in sustainable energy selection. This paper is organized as follows. In Section 2, we review the related research of MABAC. Section 3 briefly defines and describes some basic concept of neutrosophic set, single-valued neutrosophic set and bipolar neutrosophic set. In Section 4, we propose bipolar neutrosophic set for the MABAC method. A case study of sustainable energy selection is presented in Section 5. The comparative analysis is given in Section 6. Finally, we conclude the paper in Section 7.

Literature Review
There are many researchers who proposed various decision-making methods based on MABAC and applied in solving various fields. Pamučar andĆirović [21], for example, proposed the application of new combination Decision Making Trial and Evaluation Laboratory (DEMATEL) with MABAC model to select the optimal transport handling units in the logistics center. Božanić et al. [35] presented the application of MABAC to support in making a decision on using forces in a defensive operation of Land Forces and formulation of a decision strategy. The combination of four tools; Geographic Information Systems, DEMATEL, Analytic Network Process and MABAC was proposed [36] to identify locations for the installation of wind farms, which will provide significant support to planners in the strategy for the development and management of wind energy. Xue et al. [37] introduced the MABAC approach for material selection problems with incomplete weight information under IVIFSs, Pythagorean fuzzy sets and MABAC applied by Peng and Yang [38] to a research and development project selection problem to find the best project. MABAC method, based on trapezoidal interval type-2 fuzzy numbers, proposed to select the most suitable candidate for a software company, which leads to hiring a system analysis engineer based on few attributes [39]. Recently, Xu et al. [40] applied the combination of heterogeneous criteria information and extended MABAC method in green supplier selection model.
The application of a hybrid model using the fuzzyficated Saaty's scale, the analytic hierarchy process method and MABAC method was developed to select the locations for the development of laying-up positions by Božanić et al. [41]. Then, Debnath et al. [42] used several decision-making aspects in the unique mechanism of MABAC method. A novel hybrid method encompassing factor relationship and MABAC proposed by Chatterjee et al. [43] for selection and evaluation of non-traditional machining processes. For assessing and prioritizing medical tourism destinations in an uncertain environment, the analytic hierarchy process (AHP) and MABAC method in a rough number proposed [44]. Sharma et al. [45] modified rough AHP-MABAC model for ranking the Indian railway stations based on the decision maker's performance. The hybrid of fuzzy AHP method and fuzzy MABAC proposed in the selection of the locations for river crossing by tanks with a deep wading technique [46]. An interval type-2 fuzzy numbers similarity based MABAC presented by Hu et al. [47] for selection of the suitable medical treatment under a patient-centered environment.
Ali et al. [48] introduced the concept of neutrosophic cubic set and defined some new notions neutrosophic set to apply in pattern recognition problems. The simplified neutrosophic sets have been proposed by Peng et al. [32] for addressing issues with a set of specific numbers. Radwan et al. [49] developed neutrosophic AHP and applied numerical experiment for learning management system selection. To solve green product design selection problems, Tian et al. [50] simplified neutrosophic linguistic information and developed an innovative multi-criteria group decision-making approach that incorporates power aggregating operators and TOPSIS based QUALIFLEX method. Ye [51] presented the concepts of neutrosophic linear equations and applied the effectiveness of handling the indeterminate traffic flow problem. The neutrosophic set for the DEMATEL method designed by Abdel-Basset et al. [52] to analyze and determine the factors influencing the selection of supply chain management suppliers. Awang et al. [53] proposed Shapley weighting vector based on a single-valued neutrosophic aggregating operator in the DEMATEL method and applied the proposed method to the coastal erosion problem. The two types of single-valued neutrosophic covering rough set models introduced by Wang and Zhang [54].
In recent years, there are many researchers developed and applied bipolar neutrosophic sets to solve MCDM problems. Deli et al. [33] developed a bipolar neutrosophic MCDM approach based on bipolar neutrosophic weighted average and geometric operators including score, uncertainty and accuracy functions. TOPSIS method with bipolar neutrosophic information proposed by Dey et al. [55] to solve MCDM problems. Uluçays et al. [56] introduced some similarity measures for bipolar neutrosophic sets and their application to multiple criteria decision making. The bipolar neutrosophic projection, bidirectional projection and hybrid projection measured for solving MCDM problems [57]. Then, Pramanik et al. [58] also defined the TODIM method in bipolar neutrosophic environment to handle multi-attribute group decision-making problems and showed applicability and effectiveness of the proposed method. Bipolar neutrosophic TOPSIS method and bipolar neutrosophic ELECTRE-I method was proposed by Akram et al. [59] and applied to solve MCDM problems where the best possible alternative was selected.

Preliminaries
This section introduces the definitions and some concepts related to neutrosophic set, single valued neutrosophic set and bipolar neutrosophic set.

Neutrosophic Set
Definition 1 [26] Let X be a space of points (objects) with generic elements in X denoted by x, then, the neutrosophic set (N) in X is defined as,

Single Valued Neutrosophic Set
Definition 2 [27] Let X be universal space of points with a generic element in X denoted by x. A single valued neutrosophic set (S) in X is defined as: There is no restriction on the sum of T S (x),

Bipolar Neutrosophic Set
Definition 3 [33] Let X be a universal space of points. A bipolar neutroshopic set (B) in X is defined as an object of the form: The positive membership degree T + (x), I + (x), F + (x) denotes a truth-membership, an indeterminacy-membership and a falsity-membership of an element x ∈ X corresponding to a bipolar neutrosophic set (B) while the negative membership degree T − (x), I − (x), F − (x) denotes the truth-membership, an indeterminacy-membership and a falsity-membership of an element x ∈ X to some implicit counter property corresponding to B.

The Properties of Bipolar Neutrosophic Set
be two bipolar neutrosophic sets. Then, the inequality property defined as be two bipolar neutrosophic sets. Then, the equality property defined as be two bipolar neutrosophic sets. Then, the union defined for all x ∈ X.
for all x ∈ X.
x ∈ X be a bipolar neutrosophic set in X. Then, the complement of B is denoted by B c and is defined by be a bipolar neutrosophic number. Then, the score function s( b 1 ), accuracy function a( b 1 ) and certainty function c( b 1 ) are defined as follows:

Fundamental of MABAC
Definition 11 [21] Alternative could belong to the border approximation area ( G) upper approximation area The G + presents the area where ideal alternative is located A + while the G − presents the area where the anti-ideal alternative is located A − . Illustration of areas and alternatives are shown in Figure 1.
1 Figure 1. Presentation of G, G + and G − approximation areas.
Definition 12 [21] The belonging of alternative to the approximation area is G, G + or G − determined using the equation: where q are the distance from the approximation border area ( q ij ). According to the principle of MABAC method, we know that if q = 0, the alternative could belong to the G; if q > 0, the alternative belongs to G + and if q < 0, the alternative belongs to the G − . The alternative (A i ) should belong to the G + by as many criteria as possible to be selected the best one from the set.

Proposed Bipolar Neutrosophic MABAC
In this section, computational procedures of the proposed method are presented. The proposed method comprises three stages where in the first stage, the new linguistic variable for bipolar neutrosophic set is proposed. In the second stage, we utilize the bipolar neutrosophic weighted aggregation operators to aggregate the direct relation matrix and deneutrosophication for bipolar neutrosophic is introduced to find the crisp number. Finally, in the third stage, we rank alternatives based on distances between alternatives and BAA. Figure 2 presents the overall framework and stages of the proposed method.

Stage 1: Linguistic Variable for Bipolar Neutrosophic
Step 1. Construct bipolar neutrosophic linguistic variables. Linguistic variable is a variable whose values are not numbers but words or sentences in a natural or artificial language [60]. In the early days, most research used to adopt 3-scale linguistic variables and 4-scale linguistic variables. For example, Gabus and Fontela [61] adopted a 4-scale linguistic variable that was used for the DEMATEL method. However, the other linguistic variables such as the 5-scale linguistic variable or even the 8-scale linguistic variable have been used by many researchers. In the case of using single-valued neutrosophic numbers (SVNNs), Biswas et al. [62] used five-scale linguistic variables in a TOPSIS based research. Table 1 shows the linguistic variable and its respective single-valued neutrosophic number.     ii. Complement Property The linguistic variable of bipolar neutrosophic is verified using Definition 5 [33]. Let X = {x 1 , x 2 , x 3 , x 4 , x 5 }. Then, B be a bipolar neutrosophic set in X. Therefore, we have a complement of B denoted by B c verified as: Finally, based on the above properties, we propose the new linguistic variable of bipolar neutrosophic number for MABAC method using the 5-scale linguistic variable of "very poor" until "very good". Table 2 presents the new linguistic variable and its respective bipolar neutrosophic numbers.
The defined linguistic variable is intended to be used in MABAC. Aggregation and deneutrophication of the proposed method is presented as follows:

Stage 2: Aggregation and Deneutrosophication of Bipolar Neutrosophic
Step 2. Aggregated direct relation matrix. In this step, the bipolar neutrosophic weighted average operator is used to aggregate the bipolar . . , n) be a family of bipolar neutrosophic numbers. Bipolar neutrosophic weighted average operator (A w ) is defined as: where w j is weight for neutrosophic information.
Step 3. Deneutrosophication of bipolar neutrosophic. Deneutrosophication is the process to obtain crisp numbers from neutrosophic numbers. Deneutrosophication of bipolar neutrosophic set in X can be defined as a process of mapping B into the neutrosophic set. Therefore, deneutrosophication can be obtained using Equation (2).

Stage 3: Obtain the Ranking of Alternatives
Step 4. Construction of the initial decision matrix. The first step of the MABAC method is to evaluate alternatives with respect to criteria. We show the alternative in the form of vectors A i = (x i1 , x i2 , . . . , x in ), where x ij is the value of the i-th alternative according to the j-th criterion (i = 1, 2, . . . , m; j = 1, 2, . . . , n). It is arranged as Equation (3): where m indicates the total number of the alternative and n indicates the total number of criteria.
For benefit type criteria which is higher value of the criterion, ii. For cost type criteria which is higher value of the criterion, where x ij , x + i and x − i are the elements from the initial decision matrix for which x + i and x − i are defined as: x + i = max( x 1 , x 2 , . . . , x m ) is the maximum value of the observed criterion according to the alternatives and x − i = min( x 1 , x 2 , . . . , x m ) is the minimum value of the observed criterion with respect to alternatives.
Step 6. Calculation of elements of weighted matrix. The elements from the weighted matrix ( V) are calculated using Equation (7). v ij w i . n ij + 1 (7) where n ij are the elements of the normalized matrix, w i is the weight coefficients of the criteria. We can obtain the weighted matrix using Equation (8).
Step 7. Determining the border approximation area (BAA) matrix. The BAA for each criterion is determined using Equation (9), where v ij are the elements of the weighted matrix. Then, a BAA matrix is given in matrix n × 1 (n columns and one row): G = g 1 g 2 · · · g n (10) Step 8. Calculation of the distance of alternatives from the BAA for each element of the matrix: q 11 q 12 · · · q 1n q 21 q 22 · · · q 2n . . . . . . . . . . . .
The distance from the approximation border area ( q ij ) is determined as the difference of weighted matrix elements ( V) and the values of border approximation area ( G).
where g ij is the BAA for criterion.
Step 9. Ranking the alternatives. The values of the criterion functions for the alternative is obtained as the sum of the distance of the alternative from the BAA. We obtain the final values of the criterion functions for the alternative by calculating the sum of element of matrix by rows: Application of the proposed method is made to the problem of sustainable energy selection. In this problem, the alternatives of sustainable energy are evaluated with respect to the criteria that normally used in sustainable energy. Detailed implementation of the problem is presented in the following section.

Sustainable Energy Selection
The best three experts in the field of sustainable energy were invited to provide linguistic evaluation over the sustainable energy criteria. Two experts are academicians from the Department of Electrical Engineering and Department of Environment Science at a public university in Malaysia. The third expert is an engineer from the Department of Environment at a government ministry in Malaysia. The experts were asked to make evaluation on alternatives of sustainable energy with respect to fourteen criteria using the predefined linguistic variables (see Table 2). Table 3 shows the fourteen criteria and seven alternatives that were used in the evaluation.
Data was collected from the experts, then used as the input for the proposed method. In this section, the detailed computation procedures of bipolar neutrosophic MABAC method are implemented according to the following steps: Step 1. Construct bipolar neutrosophic linguistic scale. The group of experts need to evaluate by rating the alternatives with respect to criteria using the proposed five linguistic scales varying from 'very poor' to 'very good' and converted into bipolar neutrosophic linguistic scales accordingly.
The aggregated direct-relation matrix can be obtained using bipolar neutrosophic weighted average operator (see Equation (1)). Table 4 presents the aggregated direct relation matrix. Table 3. The criteria and alternatives for sustainable energy selection [25].
Criteria Alternative C 1 Efficiency A 1 Biomass energy C 2 Exergy efficiency A 2 Biogas energy C 3 Safety A 3 Geothermal energy C 4 NO X emission A 4 Hydro energy C 5 CO 2 emission A 5 Solar energy C 6 CO emission A 6 Tidal energy C 7 Land use A 7 Wind energy C 8 Noise C 9 Social acceptability C 10 Job creation C 11 Investment cost C 12 Operation and maintenance cost C 13 Fuel cost C 14 Electric cost Step 3. Deneutrosophication of bipolar neutrosophic. Deneutrosophication can be obtained using Equation (2). It is shown in Table 5.  Step 4. Construct the initial decision matrix. The MABAC method is used to evaluate alternatives with respect to criteria. The alternatives are arranged in the form A i = (x i1 , x i2 , . . . , x in ) where x ij is the value of the i-th alternative with respect to the j-th criterion(i = 1, 2, . . . , 7; j = 1, 2, . . . , 14) (see Equation (3)).
Step 5. Normalization of the elements from the initial decision matrix. In this step, C 1 until C 10 are the benefit type criteria and C 11 until C 14 are the cost type criteria by Equation (6). Elements of the normalized matrix of benefit type criteria and cost type criteria are determined using Equations (5) and (6) respectively. Elements of the normalized matrix are shown in Table 6.  Step 6. Obtain the elements of weighted normalized decision matrix. The elements of weighted matrix ( V) are calculated using Equation (7). Table 7 shows the weighted normalized decision matrix.  Step 7. Determining the BAA matrix. Table 8 shows the BAA for each criterion. It is obtained using Equation (9): 0.0000000533 C 10 0.0000000344 C 11 0.0000000088 C 12 0.0000000122 C 13 0.0000000086 C 14 0.0000000140 Step 8. Calculate distance of alternatives. The distances of the alternatives from the BAA for the matrix elements are obtained using Equation (12). The results are presented in Table 9.  Step 9. Ranking the alternative.
The final values for each alternative can be obtained by calculating the sum of the distance of alternatives from the BAA using Equation (14). The final ranking of the alternative is given in Table 10. Based on the sum of distance of alternative from the BAA in Table 10, ranking of the alternatives are accomplished as A 1 > A 6 > A 5 > A 7 > A 2 > A 4 > A 3 . Therefore, A 1 is the best alternative of the sustainable energy.

Comparative Analysis
The results obtained from the proposed method are subjected to comparative analysis. Using the original MABAC method as a benchmark method, the comparative analysis is implemented. The linguistic variables used in the proposed method are substituted with crisp numbers of MABAC, then the computational procedures of MABAC are iterated. In the original MABAC method, crisp numbers are not converted into bipolar neutrosophic linguistic scales. Apart from that, no aggregation and deneutrosophication computational steps are needed. The initial decision matrix of the original MABAC is shown in Table 11. This initial decision matrix is indeed the average of expert scores in which these scores are computed further using a normalization equation (see Step 5). The normalized matrix obtained from the initial decision matrix is shown in Table 12.  Computation is continued further with step 6 until step 9, and the sum of the distance of alternatives from the BAA( S) are obtained using Equation (14). Table 13 presents the distance and ranking of alternatives as the result of implementation MABAC method. The final ranking results of MABAC and the propose method are put side by side to observe the consistency. Table 14 presents the comparative results of the two methods. As shown in Table 14, it can be seen that the two methods offer inconsistency results. It is mainly attributed to the different rating numbers, which subsequently affect the distances. In this study, a group of experts applied the bipolar neutrosophic numbers in linguistic evaluation, while in computational procedures of MABAC method, crisp numbers based linguistic variables are used. Furthermore, the proposed bipolar neutrosophic MABAC method needs to reduce the bipolar neutrosophic numbers to crisp numbers through the deneutrosophication. Contrarily, in MABAC method, the process of deneutrosophication are excluded during computation. The result of bipolar neutrosophic MABAC shows that A 1 is an optimal alternative while A 3 is the optimal alternative for the MABAC method. The comparative result showed the significance of the proposed method in decision enhancement.

Conclusions
This paper proposed a combination of bipolar neutrosophic set and the MABAC method to solve the MCDM problem. Specifically, the proposed method was applied to sustainable energy selection in which the optimal solution is suggested. The set of fourteen criteria and seven alternatives in sustainable energy were used in this empirical study. Bipolar neutrosophic judgment based on positive and negative sides were judiciously used in this decision. The MABAC method is a particularly pragmatic and reliable tool for rational decision making because it has simple computation and stable solutions.
The proposed method suggests that A 1 is the optimal alternative. This paper has proceeded with a comparative analysis where the rank of alternatives, using the proposed method, is slightly different from the benchmark method. In short, this study provides several significant contributions and modifications to the fundamental MABAC method. Firstly, the new linguistic variable for bipolar neutrosophic was successfully introduced in the evaluation process. Secondly, the direct relation matrix was aggregated using bipolar neutrosophic weighted aggregation operator and introduced deneutrosophication bipolar neutrosophic. Finally, the bipolar neutrosophic MABAC was applied to the sustainable energy selection where 'biomass energy' was selected as the optimal alternative. The proposed method provided evidence on the feasibility of bipolar judgment coupled with straightforward computation in offering the optimal solution of sustainable energy selection. Nonetheless, this study has some recommendations for future research. The weighted average operator in aggregating direct-relation matrix could be further enhanced by considering new aggregation methods such as generalized Schweizer-Sklar prioritized aggregation operators [63], normalized weighted Bonferroni mean aggregation operator [64], and picture Dombi aggregation operators [65]. Moreover, the sensitivity of the experts' weights towards the evaluation results is not investigated in this paper. Therefore, any new sensitivity analysis particularly on the experts' weights could be explored as a future research direction. The study is not only limited by the computational steps of the proposed method that could be explored for future research, but the bipolar neutrosophic sets also could be combined with other MCDM methods.