Fuzzy Decision Support Modeling for Hydrogen Power Plant Selection Based on Single Valued Neutrosophic Sine Trigonometric Aggregation Operators

In recent decades, there has been a massive growth towards the prime interest of the hydrogen energy industry in automobile transportation fuel. Hydrogen is the most plentiful component and a perfect carrier of energy. Generally, evaluating a suitable hydrogen power plant site is a complex selection of multi-criteria decision-making (MCDM) problem concerning proper location assessment based on numerous essential criteria, the decision-makers expert opinion, and other qualitative/quantitative aspects. This paper presents the novel single-valued neutrosophic (SVN) multi-attribute decision-making method to help decision-makers choose the optimal hydrogen power plant site. At first, novel operating laws based on sine trigonometric function for single-valued neutrosophic sets (SVNSs) are introduced. The well-known sine trigonometry function preserves the periodicity and symmetric in nature about the origin, and therefore it satisfies the decision-maker preferences over the multi-time phase parameters. In conjunction with these properties and laws, we define several new aggregation operators (AOs), called SVN weighted averaging and geometric operators, to aggregate SVNSs. Subsequently, on the basis of the proposed AOs, we introduce decision-making technique for addressing multi-attribute decision-making (MADM) problems and provide a numerical illustration of the hydrogen power plant selection problem for validation. A detailed comparative analysis, including a sensitivity analysis, was carried out to improve the understanding and clarity of the proposed methodologies in view of the existing literature on MADM problems.


Introduction
Fossil fuels and renewable energy are the most important natural resources for the social and economic growth of a country. Invariably and exceptionally, it is clearly observed that energy demand is increasing significantly over time throughout the world. The major dependence on fossil fuels leads directly to carbon dioxide emissions that harm the environment and also rapidly exhaust the natural stock. The implementation of the electrification technique decreased the emission factor dramatically, but could not ultimately be considered a viable solution. However, hydrogen energy, wind power, ‫ג‬ ∂ : ℵ → Θ, respectively, of the elementh to the neutrosophic set ∂, where Θ = [0, 1] is the unit interval. Furthermore, it is required that 0 ≤ ∂ (h) + ∂ (h) + ‫ג‬ ∂ (h) ≤ 3, for eachh ∈ ℵ.
Then, the value of the operator sin (∂) is SVNN.
be two STSVNNs. Then the operational laws are as follows To compare the STSVNNs, we have mentioned the following definitions.
Then, the score and accuracy of ∂ is denoted and defined as Next we discussed some basic properties of STSVNNs based on proposed STOLs.
Proof. Straightforward from the Definition 12, so we omit the proofs of them.
Proof. Straightforward from the Definition 12, so we omit the proofs of them.

Novel Sine Trigonometric Aggregation Operators for SFNs
In this section, we present some novel aggregation operators based on the proposed STOLs of SVNNs. we define the following weighted averaging and geometric aggregation operators (AOs).
.., n) and the weight vector of Proof. We prove Theorem 6, by employing mathematical induction on n. As for each g, Then the following steps of the mathematical induction have been executed.
Therefore, Equation (1) holds for any n. The proof is completed. 16 First, we find the l g = sin 2 π 2 ∂ g we get Thus, we have Similarly, if n g = sin 2 π Thus, we have Therefore, Next, we give the some properties of the proposed ST − SV NWA aggregation operator. As these aggregation operators are based on the sine trigonometric function, they preserve the idempotency, boundedness, monotonically, and symmetry.
Proof. As ∂ g = ∂ (g = 1, 2, 3, ..., n) . Then, by Theorem 6, we get Proof. As, for any g, min g Then, based on the monotonicity of sine function, we have Similarly, Based on the sore function, we get . Now, we discuss the three cases: , then the result holds.
∂ g , and therefore ac sin ∂ g = ac sin ∂ − g . Therefore, we finally obtain Proved.
Proof. Follows from Theorem 8, so we omit here.
Proof of above theorems are follows form Theorems 7-10 similarly.
that is, when n = z + 1, Equation (3) also holds. Therefore, Equation (3) holds for any n. The proof is completed.
Proof of above theorems follows from Theorems 7-10 similarly.

Fundamental Properties of the Proposed AOs
In this section, we investigated the several relations between the proposed AOs and study their some fundamental properties as follows. = 1, 2) . Then, we have Proof. Since, ∂ g ∈ SV NN (ℵ) (g = 1, 2) . Then, by using Definition 13, we have As for any two non-negative real numbers l and m, their arithmetic mean is greater than or equal to their geometric mean, l+m Similarly, we have Proved.

Lemma 1.
For l g ≥ 0 and m g ≥ 0, then we have ∏ n g=1 l g m g ≤ ∑ n g=1 m g l g and if l 1 = l 2 = ... = l n then equality holds.

Decision-Making Technique
This part presents a decision-making methodology, followed by an illustrative example, to solve decision-making problems (DMPs) under SVNS setting. Multi-attribute decision-making issues can be demonstrated in the form of a decision matrix, in which the columns reflect the set of attributes and the rows are alternatives [59][60][61][62][63]. Thus, for decision matrix D n×m , consider a set of n alternatives {ℵ 1 , ℵ 2 , ℵ 3 , ..., ℵ n } and m criteria/attributes {t 1 , t 2 , t 3 , ..., t m }. The unknown weight vector of m criteria/attributes is denoted by W = {κ 1 , κ 2 , κ 3 , ..., κ m } with subject to g ∈ [0, 1] such that Suppose that the single-valued neutrosophic decision matrix is denoted by D = ∂ ij n×m = ij , ij , ‫ג‬ ij n×m , where ij represents the truth degree of the alternative gratifies the criteria t j considered by decision-maker (DM), ij represents the degree of the alternative is indeterminacy for the criteria t j considered by decision maker (DM), and ‫ג‬ ij represents the degree of the alternative does not gratify the criteria t j considered by decision-maker (DM). The algorithm consists of the following steps.
Step-1 Summarize the values of each alternative in term of decision matrix D (k) = ∂ (k) ij n×m with SVNS information.
Step-2 Construct the normalized decision matrix P = p ij from D = ∂ ij , where p ij is calculated as If criteria are cost type (5) Step-3 Calculate the aggregate information of the decision-makers information either SFWA/SFWG operator.
Step-4 If the attribute weights are known as a prior then utilize them. Otherwise, we compute them by utilizing the concept of the entropy measure. For it, the information of criteria t j based on entropy measure is computed as is a constant for assuring 0 ≤ E j (∂) ≤ 1.
Step-6 Evaluate the scores values sc (∂) of collective single-valued neutrosophic numbers and rank according the maximum score values. If the score values of two ∂ 1 and ∂ 2 are same, then find the accuracy degrees ac (∂ 1 ) and ac (∂ 2 ) , respectively, then we rank the ∂ 1 and ∂ 2 according the maximum degree.
Step-7 Select the optimal alternative according the maximum score value or accuracy degree.

Application of Proposed Decision-Making Technique
In this section, a numerical application about hydrogen power plant selection problem is firstly used to illustrate the designed decision-making method. Then, a comparison between the presented sine trigonometric aggregation operators and the existing aggregation operators of SVNNs are carried out to demonstrate the characteristic and benefit of the presented AOs.

Practical Case Study
Maximizing the reach of technologies and the efficient use of renewable resources has always been a key task for developing sustainable and environmentally friendly energy with a view to future prospects. Invariably, in dealing with all renewable energy projects, the problem of site selection is always a very important one, where experts and decision-makers take all possible qualitative and quantitative factors into account. In particular, selecting the right location for the hydrogen power plant project is an important task that is consistently addressed through a multi-criteria decision-making process. Hydrogen energy is one of the most efficient and cleanest energy sources that contribute significantly to the share of energy in the world.
The sites under consideration must have been chosen through professional communication by the competent experts. All the attributes affecting the site selection have been determined on the basis of the expert's/decision-maker's opinion and the available literature. For the sake of selecting the best site/location, the decision-makers must take the social aspects, environment aspects, technology aspects, financial implications, and also some major characteristic aspects. We take a case study for this selection problem in a conventional frame where there are five available sites, say, S 1 , S 2 , S 3 , S 4 and S 5 , which are under consideration in solving the problem. These sites have been systematically examined with respect to the five main attributes, say, f 1 (Social Aspect) , f 2 (Environment Aspect) , f 3 (Technology Aspect) , f 4 (Economical Aspect), and f 5 (Site Characteristics). Naturally, a better solution is expected if the number of attributes are increased. The problem of selecting the best possible hydrogen power plant site from the available set of alternatives is being mathematically and critically solved under the expert's/decision-maker's opinion and criteria weights taking the single-valued neutrosophic environment. Due to the fuzziness and uncertainty of the experts' cognition, they cannot provide the complete decision information, and the evaluation information is shown in the following Table 1. In this evaluation, the expert was asked to use SVN information and attributes weights are (0.15, 0.28, 0.20, 0.22, 0.15) T .
Step-1 Information result of the expert is listed in Table 1; Table 1. SVN Information (D). Step-2 According to the expert, attributes t 1 , t 3 , and t 5 are benefits type, t 2 and t 4 are cost attributes. Normalized matrix computed as given formula 5, and results are shown in Table 2; Step-3 In this practical case study, only one expert (decision-maker) is involved, so here we do not need to compute the aggregated decision matrix.
Step-4 Known criteria weight vector is: Step-5 Based on the weight vector and utilizing the proposed sine trigonometric AOs, the aggregated single-valued neutrosophic information of each alternatives are obtained in Table 3: Step-6 Compute the score value of the each aggregated single-valued neutrosophic information of each alternative as follows in Table 4. Step-7 Select the optimal alternative according the maximum score value given in Table 5.

Score Ranking
Best Alternative In our case study, we aim to select the the right location for the hydrogen power plant according to five attributes: Social Aspect, Environment Aspect, Technology Aspect, Economical Aspect, and Site Characteristics. After implementing the designed algorithm steps to the collective data in the form of a single-valued neutrosophic set based on the novel sine trigonometric operational rules. Based on the above computational process, we can conclude that the alternative S 2 is the best among the others and therefore it is highly recommended to select for the task/plan that is required.

Verification and the Comparison Analysis
In the following, we provides some suitable examples to show the feasibility as well as effectiveness of the proposed novel decision-making method and make a comparison with the existing studies.
To using existing methods and different aggregation operators to computed aggregated single-valued information information are shown in Tables 6 and 7. Now, we analysis the ranking of the alternative according to their aggregated informations in Tables 8 and 9. Table 8. Overall ranking of the alternatives.

Proposed Operators Ranking Best Alternative
The bast alternative is S 2 . The results achieved using novel single sine trigonometric-valued neutrosophic-weighted aggregation operators were the same as the results demonstrate existing techniques. Therefore, this study proposed the list of novel sine trigonometric aggregation operators to aggregate the single-valued neutrosophic information more effectively and efficiently. Using the proposed sine trigonometric aggregation operators, we sound the best alternative out of a collection of alternatives given by the decision-maker. Therefore, the proposed decision-making methodology based on sine trigonometric operational rules, helps us to find the best solution in decision-support systems as applications.

Conclusions
The process of industrialization has significantly increased energy consumption throughout the world. The objective of the proposed research is to present a novel decision-making approach for the selection of hydrogen power plant sites. To accomplish this task, novel sine trigonometric function-based operational laws are introduced under SVNNs. Utilizing these STOLs proposed some aggregation operators, namely, sine trigonometric SVN weighted averaging/geometric aggregation operators and sine trigonometric SVN-ordered weighted averaging/geometric aggregation operators. The various fundamental relations between the developed AOs are studied and presented in details.
To implement the proposed laws on to the DMPs, we designed a new MADM algorithm with decision-making problems where the preferences are assessed in terms of SVNNs. The utilized single-valued neutrosophic information measures have been found to be significantly efficient to handle the uncertainty in decision-making problems. The functionality of the developed method are tested over the illustrated example of hydrogen power plant site selection and superiority as well as feasibility of the method are examined in details. A comparative analysis with several existing works are also done to check its performance.
In the future research, the method proposed in this paper will be applied to other uncertain fields, such as probabilistic linguistic term sets, interval-value SVNSs, and so on. Besides, the proposed method can be applied to other areas, such as medical health diagnosis, green supplier selection, and so on.