Strategic MARCOS Model for Optimizing Renewable Energy Investments Under Pythagorean Hesitant Fuzzy Assessments

Abstract

For sustainable growth, investments in renewable energy must be maximized. Maximizing investments in renewable energy opens the door to a more successful and environmentally friendly future. Analyzing technological viability, cost-effectiveness, regulatory compliance, and environmental impact are all part of this optimization process. This paper delves into a sophisticated methodology designed to tackle uncertainties in decision-making by leveraging the innovative concept of Pythagorean hesitant fuzzy sets (PHFSs). We defined aggregation operations and distance measures for PHFS. After that, we introduced Measurement of Alternatives and Ranking According to the Compromise Solution (MARCOS), a novel methodology under PHFS, it is a robust tool acknowledged for navigating complex decision scenarios with multiple criteria. Following that, we showcased a case study on enhancing renewable energy investments through an AI-based strategy for sustainable development, utilizing the newly developed MARCOS algorithm. The study highlights the significance of its adaptability and efficiency in practical applications. Furthermore, we compared this methodology with the Technique for Establishing Order Performance by Similarity to the Ideal Solution (TOPSIS), offering insights into their respective strengths. This offers a concrete demonstration of its real-world utility and potential impact in decision-making scenarios. Finally, in the last, we conclude the whole study.

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In the current energy situation, it is critical to maximize investments in renewable to promote sustainable growth. The enhancement of investments in renewable power sources is indispensable to boosting development in the present energy environment. Since the investment in renewable energy sources becomes the decision of high importance that contributes to environmental protection, it becomes the strategic and ethical necessity. There is the ready opportunity for combinations of short-term handling of hasty environmental crises and acquisition of long-term-material revenues by business organizations and governments if only they are willing to invest in renewable energy projects. This formulation of the problem entails an analysis of other parameters, including the technical efficiency of approaching technologies, the influence of the technologies on the environment, technical practicability, and conformity with the law. The optimization process entails a thorough examination of multiple aspects, such as the relative cost effectiveness of various technologies, their influence on the environment, the feasibility of the technology, and obedience to legal requirements. In addition, Vision incorporates other emerging technologies that help make better decisions, thus promoting sustainability and profitability due to the use of technologies like artificial intelligence (AI). Stakeholders may proactively create the environment in which organizational investments in renewable energy are maximized, that is, the intersection of sustainable development and economic welfare. It is argued that the application of MCDM as decision making tools is crucial in the investment toward renewable energy for sustainable growth rates. MCDM helps to apply integration to invest choices and offers logical priority order, giving choices including legal requirements, cost, and environmental aspects. As a result, the process facilitates complete decision-making for sustainable objectives that also take into account the legal requirements and interests of the stakeholders. Implementing AI into the MCDM systems combines the findings from the analysis of big data, thus increasing reliability and productivity. Finally, the employment of MCDM techniques ensures the right direction of the investment on renewable energy with respect to sustainability goals and legal requirements in order to attain sustenance and development. In the end, strategic alignment of renewable energy investments with sustainability objectives and legal criteria is ensured by utilizing MCDM techniques, which promotes sustainability and long-term growth.

1.1. Literature Review

A vital device for managing the complex procedures of choosing and ranking options according to several criteria is MCDM [1, 2]. Its versatility extends to a number of areas, such as the choice of air conditioning systems [3], hiring techniques [4], project management [5], and the creation of universal energy policies [6]. The core elements of MCDM are alternatives, evaluation values, and criteria, which are determined by decision makers through a variety of techniques such as [7]. But there are problems when managers do not have enough time, do not know the subject well, or do not have enough money, which makes providing accurate numerical data difficult [8, 9]. As a result, several approaches have been developed to support subjective assessments in the face of uncertainty, including fuzzy sets (FSs) [10], intuitionistic fuzzy sets (IFSs) [11, 12], hesitant fuzzy sets (HFSs) [13–15], and type-2 fuzzy sets (T2FSs) [16, 17].

By combining membership and nonmembership degrees, Pythagorean fuzzy sets (PFSs), first introduced by Yager [18, 19], present a novel evaluation method. These sets provided capabilities for modeling uncertainty in real-world scenarios involving decision making, while preserving the duality conduct from IFSs. The requirement that the total of squared membership and nonmembership degrees not exceed one is one of PFSs’ unique aspects. With this special feature, PFSs can navigate more complex decision spaces with greater versatility, which can lead to successful applications in domains like candidate selection for institutions like the Asian Infrastructure Investment Bank (AIIB) [20], investment decisions [21, 22], and service quality assessment in industries like the local airline industry [3], and solution of MCDM problem based on picture linguistic information by [23, 24].

Torra’s HFSs [14, 15] are incredibly useful when several values need to be taken into account simultaneously. HFSs communicate various values through a variety of possible values, effectively integrating the perspectives of diverse decision makers. Notably, because HFSs can capture uncertainty and hesitations in a very well manners such as in [25, 26], even a single decision maker can benefit from using them MCDM model based on picture hesitant fuzzy soft set approach using an application of sustainable solar energy management [27]. Making decisions in the real world frequently requires dealing with a concept of PFSs membership and nonmembership degrees that is only partially understood. For example, in determining whether x is a member of A or not, a decision maker may designate 0.7, 0.8 and 0.3, 0.5, respectively. Khan et al. [28] presented Pythagorean hesitant fuzzy sets (PHFSs), an improved version of HFSs, as a solution to this problem.

By using PHFSs, the authors in [29] enhanced the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) technique, hence broadening its scope of use. Many alternative approaches that used the power of PHFS were developed as a result of this integration, including TODIM [30], Vikor [31], WASPAS [32], multicriteria assessment of climate change due to greenhouse effect [33], MOORA [34], and ELECTRE [35]. These approaches improve upon decision making procedures by utilizing the adaptability and resilience provided by PHFSs.

The use of the Measurement of Alternatives and Ranking According to the Compromise Solution (MARCOS) method comes into play due to some difficulties that may be faced when making a decision when more than one criterion is available but in conflict.

1.2. Motivation and Objectives

The reasons for undertaking this research are the increasing realization on the importance of enhancing investment in renewable energy in a world characterized by unstable energy sources. The development and improvement of the investments in the renewable power sources require enhancement for increased development as well as solving the challenges that relate to the environment. This research acknowledges the fact that persistent investment in renewable energy is both tactically and morally mandatory to halt pollution and destruction of the natural environment, while at the same time, renewable energy also records humongous lucrative value for any company or jurisdiction in the long run.

To research and promote the analysis of investments in renewable energy as an approach of advancing sustainable development and tackling crises in the environment.

For the purpose of creating a more complex matrix that will help assess various aspects of renewable energy sources and investment possibilities with account of technology’s technical, environmental, cost, technical and legal factors’ impact.

To apply new strategies such as AI in decision making and improve the credibility, efficiency as well as effectiveness of investment decisions regarding renewable energy sources.

To balance the decision-making process of investing in renewable energy and to incorporate both the legal requirements and the environmental factors and priorities in a systematic, integrated, and logical manner and use MCDM to control the complexity of the decision process and to arrive at a consistent decision.

To apply PFSs for managing uncertainty and avoiding subjective verdicts in decision-making and enhance the quality of investment decisions.

To tackle the demanding situations related to subjective exams and uncertainty in renewable power investment decisions by using FS theories and their editions.

To make sure that renewable electricity investments are strategically aligned with sustainability targets and legal requirements, fostering lengthy-term boom and environmental stewardship.

1.3. Organization of the Paper

The format of the paper is as follows: the scope of the investigation is summarized in the introduction. In Section 2, basic ideas are covered that are required for the debate that follows. In Section 3, the novel PHFS methodology is presented. After that, a thorough case study on asset management and urban renewal is presented in Section 4. A comprehensive compared examination of the TOPSIS approach and the PHFS-MARCOS technique is carried out in Section 5. The paper’s conclusion is provided in Section 6, which summarizes the main conclusions and discusses how they may affect decision-making processes (Table 1).

Table 1

List of symbols.

Symbols	Description
$\mathrm{\beth }$	Predetermined set
$\gimel$	Elements of predetermined set
$\mu$	Membership degree
$\nu$	Nonmembership degree
$\mathcal{l}$	Pythagorean fuzzy set
$\Xi$	Hesitant fuzzy set
$\wp$	Pythagorean hesitant fuzzy set
$\omega$	Weight vectors
${\vartheta }^{+}$	Maximum element with membership
${\vartheta }^{-}$	Minimum element with membership
${\theta }^{+}$	Maximum element with nonmembership
${\theta }^{-}$	Minimum element with nonmembership
$\mathfrak{I}$	Set of alternatives
$⅁$	Set of attributes
${\mathfrak{A}}^{+}$	Ideal solution
${\mathfrak{A}}^{-}$	Anti-ideal solution
${\mathfrak{U}}_i$	Alternatives’ utility degree
$F{\mathfrak{U}}_i$	Alternatives’ utility function

2. Preliminaries

This section introduces several fundamental definitions and operations that played a crucial role in developing the proposed tasks.

Definition 1 (see [3]).

Suppose $\mathrm{\beth }$ is a predetermined set. A representation of a PFS, denoted as $\mathcal{l}$, on $\mathrm{\beth }$ can be expressed as follows:$$\begin{array}{}\text{(1)}& \mathcal{l}=\gimel ,\mathcal{l}{\mu }_{\mathcal{l}}\gimel ,{\nu }_{\mathcal{l}}\gimel \gimel \in \mathrm{\beth }.\end{array}$$

The provided information pertains to functions ${\mu }_{\mathcal{l}}\gimel$ and ${\nu }_{\mathcal{l}}\gimel$, where $\gimel$ maps to a value between 0 and 1, representing the membership degree and nonmembership degrees of $\gimel$ to $\mathcal{l}$, respectively. For all $\gimel \in \mathrm{\beth }$, the following condition holds: $0\le {{\mu }_{\mathcal{l}}\gimel }^2+{{\nu }_{\mathcal{l}}\gimel }^2\le 1$.

Definition 2 (see [15]).

Let $\mathrm{\beth }$ denote a predetermined set. A HFS $\Xi$ on $\mathrm{\beth }$ can be expressed using a function that maps $\mathrm{\beth }$ to a subset of the interval (0, 1). For clarity, Rodriguez, Martinez, and Herrera [13] introduced the following mathematical notation to represent the HFS:$$\begin{array}{}\text{(2)}& \Xi =\gimel ,{\mu }_{\Xi }\gimel \gimel \in \mathrm{\beth }.\end{array}$$

The data represents ${\mu }_{\Xi }\gimel$ as a collection of values within the range of 0–1, indicating the possible membership degree of the element $\gimel \in \mathrm{\beth }$ to the set $\Xi$.

Definition 3 (see [36]).

If $\mathrm{\beth }$ is a fixed set, then a PHFS $\wp$ on $\mathrm{\beth }$ can be defined as follows:$$\begin{array}{}\text{(3)}& \wp =\gimel ,\wp {\mu }_{\wp }\gimel ,{\nu }_{\wp }\gimel \gimel \in \mathrm{\beth },\end{array}$$in which ${\mu }_{\wp }\gimel$ and ${\nu }_{\wp }\gimel$ are two sets of some values in (0, 1), denoting the possible Pythagorean membership degrees and the Pythagorean nonmembership degrees of the element $\gimel$ to the set $\wp$, respectively, with the following conditions:$$\begin{array}{}\text{(4)}& 0\le \vartheta ,\theta \le 1,0\le {{\vartheta }^{+}}^2+{{\theta }^{+}}^2\le 1,\end{array}$$where $\vartheta \in {\mu }_{\wp }\gimel$, $\theta \in {\nu }_{\wp }\gimel$, ${\vartheta }^{+}={\max }_{\vartheta \in {\mu }_{\wp }\gimel }\vartheta$, and ${\theta }^{+}={\max }_{\theta \in {\nu }_{\wp }\gimel }\theta$ for all $\gimel \in \mathrm{\beth }$.

Example 1.

A hesitant PFS is exemplified as $\wp =0.2,0.4,0.5,0.7,0.7,0.7$. In this representation, every element is expressed through a tuple comprising the membership degree and nonmembership degree; the squares sum up to be less than or equal to 1. This particular example puts forward how PHFS easily supports the uncertainty and hesitation associated with a decision-making process by admitting multiple membership values for every element presented. This consequently offers a flexible and comprehensive approach to handle complex fuzzy data.

Definition 4 (see [36]).

Given a PHFE​ $\wp =\wp \mu ,\nu$, the minimum and the maximum of each element are calculated as follows:$$\begin{array}{}\text{(5)}& {\vartheta }^{-}=\underset{\vartheta \in \mu }{\min }\vartheta ,{\theta }^{-}=\underset{\theta \in \mu }{\min }\theta ,\text{(6)}& {\vartheta }^{+}=\underset{\vartheta \in \mu }{\max }\vartheta ,{\theta }^{+}=\underset{\theta \in \mu }{\max }\theta ,\end{array}$$where the minimum and the maximum of the element $\mu$ are denoted by ${\vartheta }^{-}$ and ${\vartheta }^{+}$, respectively. In a similar manner, the minimum and the maximum of the element $\nu$ are ${\theta }^{-}$ and ${\theta }^{+}$.

Definition 5 (see [36]).

Assuming that $\wp =\wp \mu ,\nu ,{\wp }_1=\wp {\mu }_1,{\nu }_1$ and ${\wp }_2=\wp {\mu }_2,{\nu }_2$ represent the three Pythagorean hesitant fuzzy elements (PHFEs), the following operations among them are described:$$\begin{array}{}\text{(7)}& {\wp }_1\oplus {\wp }_2={\displaystyle \bigcup }_{\begin{array}{c}{\vartheta }_1\in {\mu }_1,\\ {\theta }_1\in {\nu }_1,\\ {\vartheta }_2\in {\mu }_2,\\ {\theta }_2\in {\nu }_2\end{array}}\sqrt{{{\vartheta }_1}^2+{{\vartheta }_2}^2-{{\vartheta }_1}^2{{\vartheta }_2}^2},{\theta }_1{\theta }_2.\end{array}$$

Definition 6 (see [36]).

Assuming an PHFE $\wp =\wp \mu ,\nu$, the scoring function of $\wp$ is $s\wp =1/\mathrm{\ast }\mu {{\displaystyle \sum }}_{\vartheta \in \mu }{\vartheta }^2-1/\mathrm{\ast }\nu {\sum }_{\theta \in \nu }{\theta }^2$ and the function of accuracy of $\wp$ is $p\wp =1/\mathrm{\ast }\mu {\sum }_{\vartheta \in \mu }{\vartheta }^2+1/\mathrm{\ast }\nu {\sum }_{\theta \in \nu }{\theta }^2$, where $\mathrm{\ast }\mu$ and $\mathrm{\ast }\nu$ are the numbers of the elements in $\mu$ and $\nu$, respectively.

Definition 7 (see [36]).

Assume that ${\wp }_1=\wp {\mu }_1,{\nu }_1$ and ${\wp }_2=\wp {\mu }_2,{\nu }_2$. Given two PHFEs, ${\wp }_1$ and ${\wp }_2$, we define the Pythagorean hesitant fuzzy hamming distance between them as follows:$$\begin{array}{}\text{(8)}& d_1{\wp }_1,{\wp }_2=\frac{1}{2}\frac{1}{\mathrm{\ast }\mu }{\displaystyle \sum }_{\kappa =1}^{\mathrm{\ast }\mu }\mid {{\vartheta }_1^{\kappa }}^2-{{\vartheta }_2^{\kappa }}^2\mid +\frac{1}{\mathrm{\ast }\nu }{\displaystyle \sum }_{\kappa =1}^{\mathrm{\ast }\nu }\mid {{\theta }_1^{\kappa }}^2-{{\theta }_2^{\kappa }}^2\mid ,\end{array}$$where $\mathrm{\ast }\mu =\mathrm{\ast }{\mu }_1=\mathrm{\ast }{\mu }_2\text{\hspace{0.17em}and}\mathrm{\ast }\nu =\mathrm{\ast }{\nu }_1=\mathrm{\ast }{\nu }_2$. ${\vartheta }_1^{\kappa },{\vartheta }_2^{\kappa },{\theta }_1^{\kappa }$ and ${\theta }_2^{\kappa }$ are the kth greatest values of ${\wp }_1$ and ${\wp }_2$, respectively, representing the membership and nonmembership degrees.

3. PHFS-MARCOS Methodology for Handling MCDM Problems

The purpose of this part is to develop a novel PHFS-MARCOS decision method to handle the MCDM problem, which is characterized by decision data, that is, the PHFS format. Specifically, the suggested method focuses on using the MARCOS strategy to determine the scheme’s order of priority. These models are integrated by the proposed approach with PHFS to improve the accuracy and rationality of decision analysis in uncertain environments. Decision algorithm depicts the steps involved in the recently proposed PHFS-MARCOS method, which describes the methodical approach used for more accurate and logical decision analysis in an uncertain situation.

However, there is a quality weakness of using the MARCOS method that should be noted. A significant strength of the MARCOS method is that it highlighting some of the limitations of this technique. In the framework of the considered limitation, we can distinguish the dependence on the input data accuracy and existence. However, if the input data are imprecise or incomplete, it will slightly affect the values of the MARCOS method. Also, the method involves the computation of weights for the criteria, which may bring the issue of subjectivity, and hence bias, into the decision. It also addresses the interaction between the normalization process in MARCOS and the relative ranking of the alternatives and indicates that the choice of a specific technique of normalization may define the results. Also, as with any optimization model, it is noteworthy that, although MARCOS’s computational requirements are low, the use of this model may entail difficulties in complexity and excessive time when using a vast number of equivalents and criteria, respectively. Finally, similar to other MCDM techniques, MARCOS supposes that the decision-maker utility matrix is stable, not subjected to modification while decision-making processes are often oriented in unceasingly changing conditions.

3.1. Developing a Complete Decision Matrix With PHFS Integrated for Decision-Making

In order to address fundamental ambiguity immediately, we explore here the incorporation of PHFS into a MCDM approach. Choosing the best solutions requires careful consideration in order to achieve the best possible outcomes. Algorithms are important elements in this scenario; they are tools for systematic, well-designed decision making. According to this conceptualization, an algorithm is a planned and organized series of operations evolved to provide the best preferable solution to a given problem. Supposing that we have a set of attribute described in symbol $\mathfrak{I}={\mathfrak{I}}_1,{\mathfrak{I}}_2,{\mathfrak{I}}_3,\dots {\mathfrak{I}}_n$ as well as the set of alternatives in $⅁={⅁}_1,{⅁}_2,{⅁}_3,\dots {⅁}_n$. All of these attributes have their respective weight vectors $\omega ={\omega }_1,{\omega }_2,\dots .{\omega }_n$. According to the needs of the weighting scheme, the weight vectors must satisfy conditions that the weight must be within the closed interval (0, 1), and the sum of the weights is equal to 1 (Algorithm 1).

Algorithm 1: MARCOS technique.

1. Expert decision matrix.

2. If the data are of the cost type, normalize the matrix by converting membership to nonmembership.

3. Analyze the positive ideal solution (PIS) and negative ideal solution (NIS) to assess an expanded initial Decision Making (DM).

   $\text{PIS}=\max {\epsilon }_{ij}$

   $\text{NIS}=\min {\epsilon }_{ij}$

4. Determine the PIS and NIS distances using Definition 7.

5. Closeness coefficient (CC): the CC may be found using ${\xi }_{ij}^{+}$ and ${\xi }_{ij}^{-}$ as follows:

   ${\mathcal{C}}_{ij}={\xi }_{ij}^{-}/{\xi }_{ij}^{-}+{\xi }_{ij}^{+}.$

6. Extended decision matrix (EDM): make an EDM by combining $\mathfrak{I}ij$, along with the anti-ideal ${\mathfrak{A}}^{-}=\mathfrak{I}i{1}^{-},\mathfrak{I}i{2}^{-},\mathfrak{\dots },\mathfrak{I}i{n}^{-}$ and ideal ${\mathfrak{A}}^{+}={\mathfrak{I}}_{ij}^{+};j=1,2,\dots ,n$ solutions.

   $\mathfrak{A}=\begin{array}{cccc}{\mathfrak{I}}_{i1}^{-}& {\mathfrak{I}}_{i2}^{-}& \dots & {\mathfrak{I}}_{in}^{-}\\ {\mathfrak{I}}_{11}& {\mathfrak{I}}_{12}& \dots & {\mathfrak{I}}_{1n}\\ {\mathfrak{I}}_{21}& {\mathfrak{I}}_{22}& \dots & {\mathfrak{I}}_{2n}\\ ⋮& ⋮& \dots & ⋮\\ {\mathfrak{I}}_{m1}& {\mathfrak{I}}_{m2}& \dots & {\mathfrak{I}}_{mn}\\ {\mathfrak{I}}_{i1}^{+}& {\mathfrak{I}}_{i2}^{+}& \dots & {\mathfrak{I}}_{in}^{+}\end{array}.$

  Here,

   ${\mathfrak{I}}_{ij}^{-}=\min {\mathfrak{I}}_{ij}$

  and

   ${\mathfrak{I}}_{ij}^{-}=\max {\mathfrak{I}}_{ij}$

7. Normalization: transform the extended decision matrix E into its normalized form, denoted as $E={n_{ij}}_{m+2\times n}$, utilizing the provided equation.

   $n_{ij}={\mathfrak{I}}_{ij}/{\mathfrak{I}}_{ij}^{+},$

  where ${\mathfrak{I}}_{ij}$ and ${\mathfrak{I}}_{ij}^{+}$ are the elements in the E matrix.

8. A matrix of weighted decisions: build the definitive weighted decision matrix $F$, as depicted by the following equation:

   $f_{ij}=n_{ij}\times {\nabla }_j.$

  In this instance, an element in the matrix $E^{\prime }$ is indicated by $n_{ij}$, and the weight assigned to the jth criterion is indicated by ${\nabla }_j$.

9. Alternatives’ utility degree: utilizing the following formulas, determine the alternatives’ utility degree ${\mathfrak{U}}_i$:

   ${\mathfrak{U}}_i^{-}={\mathfrak{S}}_i/{\mathfrak{S}}^{-},$

   ${\mathfrak{U}}_i^{+}={\mathfrak{S}}_i/{\mathfrak{S}}^{+},$

  where ${\mathfrak{S}}_i={\sum }_{j=1}^n{f}_{i+1j}i=1,2,\dots ,m$,

   ${\mathfrak{S}}^{-}={\sum }_{j=1}^n{f}_{1j}$ and ${\mathfrak{S}}^{+}={\sum }_{j=1}^n{f}_{m+2j}$.

10. Utility function: compute the alternatives’ utility function $F{\mathfrak{U}}_i$ utilizing the given equation.

   $F{\mathfrak{U}}_i={\mathfrak{U}}_i^{+}+{\mathfrak{U}}_i^{-}/1+1-F{\mathfrak{U}}_i^{+}/F{\mathfrak{U}}_i^{+}+1-F{\mathfrak{U}}_i^{-}/F{\mathfrak{U}}_i^{-},$

  In instances where the utility function is delineated by the ideal $F\mathfrak{U}{i}^{+}$ and anti-ideal $F\mathfrak{U}{i}^{-}$, the formulations for each are presented in the following expressions:

   $F{\mathfrak{U}}_i^{+}={\mathfrak{U}}_i^{-}/{\mathfrak{U}}_i^{+}+{\mathfrak{U}}_i^{-},$

   $F{\mathfrak{U}}_i^{-}={\mathfrak{U}}_i^{+}/{\mathfrak{U}}_i^{+}+{\mathfrak{U}}_i^{-}.$

11. Ranking: sort the alternatives according to the utility function’ values, giving priority to those with the highest possible value.

4. Case Study: Optimizing Renewable Energy Investments an AI-Driven Approach for Sustainable Growth

In the global pursuit of sustainable development, the optimization of renewable energy investments stands as a paramount imperative. The 21st century has witnessed an unprecedented convergence of environmental consciousness and technological innovation, propelling the renewable energy sector into the spotlight as a linchpin of a greener, more sustainable future. As the world grapples with the urgent need to mitigate climate change and transition toward cleaner energy sources, the role of renewable energy projects becomes increasingly significant. This case study embarks on an exploration of how an innovative AI-driven approach can revolutionize decision-making processes within the renewable energy sector, ultimately aiming to foster sustainable growth on both environmental and economic fronts.

Traditional methods of investment decision-making within the renewable energy domain often grapple with multifaceted complexities. Balancing economic viability with environmental considerations, navigating regulatory landscapes, and assessing technological feasibility are just a few of the challenges that decision-makers face. In this context, the integration of AI into the decision-making framework emerges as a promising avenue for transformative change. By harnessing the power of AI-driven analytics, organizations can potentially unlock new avenues for maximizing both environmental and economic benefits.

This study delves into the intricacies of leveraging AI to navigate the multifaceted landscape of renewable energy investments, presenting a paradigm shift toward more informed, data-driven, and forward-thinking strategies. At its core, AI offers the capacity to process vast volumes of data, identify patterns, and generate insights at speeds and scales that far surpass human capabilities. By analyzing historical data, market trends, and predictive analytics, AI algorithms can discern complex relationships and extract actionable intelligence to inform decision-making processes. Furthermore, AI holds the promise of enhancing decision-making accuracy and efficiency while reducing inherent biases that may influence traditional approaches. By augmenting human expertise with AI-driven insights, organizations can make more informed and strategic investment decisions that are grounded in empirical evidence rather than subjective judgment.

The convergence of technology and sustainability represents a pivotal juncture in the evolution of the renewable energy sector. With the advent of AI-driven approaches, organizations have the opportunity to transcend conventional limitations and unlock new dimensions of possibility. By leveraging AI to optimize renewable energy investments, stakeholders can align financial objectives with environmental imperatives, thereby driving tangible progress toward a low-carbon future. Against this backdrop, this case study sets out to examine how AI-driven approaches can catalyze sustainable growth within the renewable energy sector. By dissecting the complexities of investment decision-making and showcasing real-world applications of AI in action, this study aims to elucidate the transformative potential of AI in shaping the future of renewable energy.

Through a comprehensive analysis of case studies, theoretical frameworks, and industry best practices, this study seeks to provide actionable insights and practical recommendations for stakeholders across the renewable energy value chain. By distilling key learning and illustrating successful implementations of AI-driven decision-making processes, this study endeavors to empower decision-makers with the knowledge and tools necessary to navigate the complexities of renewable energy investments in an increasingly dynamic and interconnected world. In summary, the intersection of AI and renewable energy represents a nexus of innovation and sustainability. By embracing AI-driven approaches, organizations can unlock new pathways for maximizing the potential of renewable energy investments while advancing the global transition toward a more sustainable and resilient energy future. As we stand on the cusp of a new era in renewable energy, the integration of AI promises to be a transformative force, propelling us toward a world where economic prosperity and environmental stewardship go hand in hand. The criteria for the case study are as follows:

1. Cost-effectiveness ${\mathfrak{I}}_1$: this criterion evaluates the financial aspects of each investment alternative. It includes factors such as the initial investment cost, operational expenses (e.g., maintenance and fuel costs), and the return on investment (ROI) timeline. Essentially, it assesses how efficiently the resources are utilized to generate revenue or achieve the desired outcomes.

2. Environmental impact ${\mathfrak{I}}_2$: this criterion focuses on the ecological consequences of each investment alternative. It considers factors such as carbon footprint reduction, ecosystem disruption, and sustainable resource usage. The goal is to prioritize investments that contribute positively to environmental sustainability, minimize harm to ecosystems, and promote the responsible use of natural resources.

3. Technological viability ${\mathfrak{I}}_3$: this criterion examines the feasibility and reliability of the technology associated with each investment alternative. It assesses factors such as the reliability of the technology, scalability potential (ability to expand or adapt over time), and the level of innovation and research advancement. The aim is to select investments that leverage proven, reliable technologies with potential for future advancements, and scalability.

4. Regulatory compliance ${\mathfrak{I}}_4$: this criterion evaluates the legal and regulatory aspects of each investment alternative. It considers factors such as permitting requirements, government incentives (e.g., subsidies and tax credits), and compliance with legal constraints and industry standards. The objective is to ensure that investments align with regulatory requirements, minimize legal risks, and capitalize on available incentives to enhance financial returns.

Alternatives:

1. Solar photovoltaic (PV) farm installation ${⅁}_1$: This alternative involves setting up solar PV panels to convert sunlight into electricity. Solar PV farms are known for their relatively low operational costs once installed, minimal environmental impact compared with fossil fuels, and potential for scalability. They typically require large initial investments but offer long-term cost-effectiveness and environmental benefits.

2. Wind turbine farm expansion ${⅁}_2$: this alternative focuses on expanding existing wind turbine farms or setting up new ones to harness wind energy for electricity generation. Wind energy is renewable, abundant, and emits no greenhouse gases during operation. However, it may face challenges related to initial investment costs, variable wind patterns affecting energy output, and regulatory hurdles due to land use and environmental concerns.

3. Hydroelectric power plant development ${⅁}_3$: this alternative involves constructing hydroelectric power plants to generate electricity using flowing water, typically from rivers or dams. Hydroelectric power is reliable, sustainable, and has low operating costs once infrastructure is in place. However, it often requires significant upfront investment, environmental impact assessments, and compliance with water rights and regulatory frameworks.

4. Biomass energy facility construction ${⅁}_4$: this alternative entails building facilities to produce energy from organic materials such as wood, agricultural residues, or municipal solid waste. Biomass energy can provide a renewable source of electricity and heat, utilizing waste materials that would, otherwise, be disposed of in landfills. However, it may face challenges related to technological viability, environmental concerns (e.g., emissions and land use), and regulatory requirements governing waste management and emissions standards.

In the following, the sequential computational process outlined for the specified MCDM problem is presented.

Step 1: expert’s decision matrix in Table 2.

Table 2

Expert’s decision matrix.

⅁	${\mathfrak{I}}_1$	${\mathfrak{I}}_2$
${⅁}_1$	$0.2,0.4,0.6,0.5$	$0.6,0.3,0.2,0.2$
${⅁}_2$	$0.5,0.6,0.5,0.7$	$0.9,0.9,0.3,0.4$
${⅁}_3$	$0.5,0.4,0.2,0.2$	$0.2,0.5,0.6,0.6$
${⅁}_4$	$0.5,0.6,0.6,0.5$	$0.3,0.5,0.6,0.7$
⅁	${\mathfrak{I}}_{\mathbf{3}}$	${\mathfrak{I}}_{\mathbf{4}}$
${⅁}_1$	$0.1,0.1,0.5,0.7$	$0.2,0.4,0.5,0.7,0.7,0.7$
${⅁}_2$	$0.3,0.4,0.8,0.8$	$0.6,0.6,0.7,0.3,0.4,0.5$
${⅁}_3$	$0.6,0.9,0.1,0.2$	$0.1,0.1,0.2,0.3,0.6,0.9$
${⅁}_4$	$0.5,0.5,0.4,0.3$	$0.1,0.3,0.5,0.1,0.5,0.6$

Step 2: no need to normalize the decision matrix because of benefit type data.

Step 3: evaluate an expanded initial DM by measuring the PIS and NIS in Tables 3 and 4.

Table 3

PIS.

${\mathfrak{I}}_1$	${\mathfrak{I}}_2$
$0.5,0.6,0.2,0.2$	$0.9,0.9,0.2,0.2$
${\mathfrak{I}}_{\mathbf{3}}$	${\mathfrak{I}}_{\mathbf{4}}$
$0.6,0.9,0.1,0.2$	$0.6,0.6,0.7,0.1,0.4,0.5$

Table 4

NIS.

${\mathfrak{I}}_1$	${\mathfrak{I}}_2$
$0.2,0.4,0.6,0.7$	$0.2,0.3,0.6,0.7$
${\mathfrak{I}}_{\mathbf{3}}$	${\mathfrak{I}}_{\mathbf{4}}$
$0.1,0.1,0.8,0.8$	$0.1,0.1,0.2,0.7,0.7,0.9$

Step 4: obtain the distance for PIS and NIS by using Definition 7 presented in Tables 5 and 6, respectively.

Table 5

Distance for PIS.

${\mathfrak{I}}_1$	${\mathfrak{I}}_2$	${\mathfrak{I}}_3$	${\mathfrak{I}}_4$
0.2350	0.2925	0.4600	0.3017
0.1650	0.0425	0.5375	0.0133
0.0500	0.4925	0	0.3317
0.1325	0.5125	0.2175	0.1767

Table 6

Deviations from anti-ideal solution.

${\mathfrak{I}}_1$	${\mathfrak{I}}_2$	${\mathfrak{I}}_3$	${\mathfrak{I}}_4$
0.0600	0.2725	0.1350	0.1183
0.1300	0.5225	0.0575	0.4067
0.2450	0.0725	0.5950	0.0883
0.1625	0.0525	0.3775	0.2433

Step 5: the closeness coefficient is determined in Table 7.

Table 7

Closeness coefficient.

${\mathfrak{I}}_1$	${\mathfrak{I}}_2$	${\mathfrak{I}}_3$	${\mathfrak{I}}_4$
0.2034	0.4823	0.2269	0.2817
0.4407	0.9248	0.0966	0.9683
0.8305	0.1283	1	0.2103
0.5508	0.0929	0.6345	0.5794

Step 6: utilize the insertion to create the EDM of $C_{ij}$ as presented in Table 8.

Table 8

EDM.

	${\mathfrak{I}}_1$	${\mathfrak{I}}_2$	${\mathfrak{I}}_3$	${\mathfrak{I}}_4$
${\mathfrak{I}}^{-}$	0.2034	0.0929	0.0966	0.2103
${\mathfrak{I}}_1$	0.2034	0.4823	0.2269	0.2817
${\mathfrak{I}}_2$	0.4407	0.9248	0.0966	0.9683
${\mathfrak{I}}_3$	0.8305	0.1283	1	0.2103
${\mathfrak{I}}_4$	0.5508	0.0929	0.6345	0.5794
${\mathfrak{I}}^{+}$	0.8305	0.9248	1	0.9683

Step 7: normalized EDM is presented in Table 9.

Table 9

Normalized EDM.

	${\mathfrak{I}}_1$	${\mathfrak{I}}_2$	${\mathfrak{I}}_3$	${\mathfrak{I}}_4$
${\mathfrak{I}}^{-}$	0.2449	0.1005	0.0966	0.2172
${\mathfrak{I}}_1$	0.2449	0.5215	0.2269	0.2910
${\mathfrak{I}}_2$	0.5306	1	0.0966	1
${\mathfrak{I}}_3$	1	0.1388	1	0.2172
${\mathfrak{I}}_4$	0.6633	0.1005	0.6345	0.5984
${\mathfrak{I}}^{+}$	1	1	1	1

Step 8: the final weighted DM is presented in Table 10, where weights are W = (0.1750, 0.2580, 0.3646, 0.2024).

Table 10

Weighted DM.

${\mathfrak{I}}^{-}$	0.0429	0.0259	0.0352	0.0440
${\mathfrak{I}}_1$	0.0429	0.1345	0.0827	0.0589
${\mathfrak{I}}_2$	0.0929	0.2580	0.0352	0.2024
${\mathfrak{I}}_3$	0.1750	0.0358	0.3646	0.0440
${\mathfrak{I}}_4$	0.1161	0.0259	0.2313	0.1211
${\mathfrak{I}}^{+}$	0.1750	0.2580	0.3646	0.2024

Step 9: the alternatives’ utility degree ${\mathfrak{U}}_i$ is presented in Table 11.

Table 11

Alternatives’ utility degree.

${\mathfrak{U}}^{-}$	${\mathfrak{U}}^{+}$
2.1558	0.3190
3.9767	0.5885
4.1854	0.6194
3.3412	0.4944

Step 10: determine the alternatives’ utility function, $F{\mathfrak{U}}_i$, as presented in Table 12.

Table 12

Utility function.

$F{\mathfrak{U}}_i$

0.3130

0.5775

0.6078

0.4852

Step 11: the process of ranking the alternatives involves evaluating and arranging them according to the values of their utility functions as presented in Table 13.

Table 13

Utility function.

Ranking

${⅁}_3>{⅁}_2>{⅁}_4>{⅁}_1$

The ranking results are shown in Figure 1.

[figure(s) omitted; refer to PDF]

5. Comparison Analysis

TOPSIS serves as a robust DM method. It not only enables the comparison of alternatives but also aids in identifying the best choice according to predetermined criteria. By taking into account both the relative significance of criteria and the performance of alternatives, TOPSIS empowers decision-makers across a wide array of fields, including business, engineering, and healthcare [37] (Algorithm 2).

5.1. TOPSIS Approach for PHFS

Algorithm 2: TOPSIS technique.

1. Expert’ decision matrix.

2. The matrix normalization by interchanging the membership degree with nonmembership degree.

3. Assess the extended initial decision matrix by examining both PIS and NIS.

   $\text{PIS}=\max {\epsilon }_{ij}$

   $\text{NIS}=\min {\epsilon }_{ij}$

4. Compute the distances of each alternative from both the PIS and NIS individually as follows:

   $d{A}_i,\text{PIS}={{\displaystyle \sum }}_{j=1}^m·d{A}_{ij},\text{PI}{\mathrm{S}}_j$

   $d{A}_i,\text{NIS}={{\displaystyle \sum }}_{j=1}^m·d{A}_{ij},\text{NI}{\mathrm{S}}_j$

  where $d{A}_{ij},\text{PI}{\mathrm{S}}_j$ and $d{A}_{ij},\text{NI}{\mathrm{S}}_j$ are evaluated by using define 2.7.

5. Determine the relative closeness degree (RCD) for every alternative with respect to PIS.

   $\text{RCD}=d{A}_i,\text{NIS}/d{A}_i,\text{PIS}+d{A}_i,\text{NIS}$

6. By assessing the RCD for every alternative, we choose the best optimal solution.

5.2. Numerical Illustration

Step 1: expert’s decision matrix given in Table 2.

Step 2: no need to normalize the decision matrix because of benefit type data.

Step 3: identified the PIS and NIS in Tables 14 and 15, respectively.

Table 14

PIS.

${\mathfrak{I}}_1$	${\mathfrak{I}}_2$
$0.5,0.6,0.2,0.2$	$0.9,0.9,0.2,0.2$
${\mathfrak{I}}_{\mathbf{3}}$	${\mathfrak{I}}_{\mathbf{4}}$
$0.6,0.9,0.1,0.2$	$0.6,0.6,0.7,0.1,0.4,0.5$

Table 15

NIS.

${\mathfrak{I}}_1$	${\mathfrak{I}}_2$
$0.2,0.4,0.6,0.7$	$0.2,0.3,0.6,0.7$
${\mathfrak{I}}_{\mathbf{3}}$	${\mathfrak{I}}_{\mathbf{4}}$
$0.1,0.1,0.8,0.8$	$0.1,0.1,0.2,0.7,0.7,0.9$

Step 4: assess the distances of each alternative with weights W = (0.1750, 0.2580, 0.3646, 0.2024), represented as $d{A}_i,\text{PIS}$ and $d{A}_i,\text{NIS}$, as shown in Table 16.

Table 16

$d{A}_i,\text{PIS}$ and $d{A}_i,\text{NIS}$.

Alternatives	$d{A}_i,\text{PIS}$	$d{A}_i,\text{NIS}$
${⅁}_1$	0.3454	0.1540
${⅁}_2$	0.2385	0.2608
${⅁}_3$	0.2029	0.2964
${⅁}_4$	0.2705	0.2289

Step 5: calculating the RCD of each alternative as shown in Table 17.

Table 17

Relative closeness degree (RCD) of each alternative from PIS.

Alternatives	${⅁}_1$	${⅁}_2$	${⅁}_3$	${⅁}_4$
$\text{RCD}{A}_i,\text{PIS}$	0.3083	0.5223	0.5936	0.4584

Step 6: Based on $\text{RCD}{A}_i,\text{PIS}$ (where $i=1,2,\dots ,4$), we can establish a ranking, as shown in Table 18.

Table 18

Ranking of all possibilities.

Method	Ranking
TOPSIS method	${⅁}_3\ge {⅁}_2\ge {⅁}_4\ge {⅁}_1$

6. Discussion

Although both methods make use of the concepts of ideal and anti-ideal solutions to rank the alternatives, they make quite different use of these reference points. MARCOS relies more heavily on utility ratios, the direct comparison of how much each alternative performs relative to both the ideal and anti-ideal solutions. The ratio-based methodology then gives a more profound view on how much better or worse the alternative is in comparison with these reference points and provides a more delicate evaluation of trade-offs among the alternatives. Meanwhile, TOPSIS uses a methodology of geometric distance such that the alternatives are ranked according to how close they are to the ideal solution and how far they are from the anti-ideal solution. In this regard, MARCOS tends to provide more granular insights into the relative performance of alternatives by bringing out the degree of compromise between the ideal and anti-ideal scenarios. The utility-driven model gives a view not of which alternative is better but by what amount. In contrast, TOPSIS provides an easier, distance-based approach, hence being simpler to apply, but potentially less sensitive to the intricacies of the performance gaps between the alternatives. We conducted a comprehensive review to determine the efficacy of the algorithms we introduced to the PHFS framework. Though the order of the rating sequences may vary significantly, all approaches result in the same optimum option. Figure 2 offers graphical representations, while Table 19 offers a thorough examination of rankings and graphic illustrations for the MARCOS method and the TOPSIS approach.

[figure(s) omitted; refer to PDF]

Table 19

Ranking of comparison between the MARCOS method and TOPSIS method.

Sr.	Methods	Ranking
1	MARCOS	${⅁}_3>{⅁}_2>{⅁}_4>{⅁}_1$
2	TOPSIS	${⅁}_3>{⅁}_2>{⅁}_4>{⅁}_1$

When making judgments, it is important to understand that different conclusions regarding which option is better may arise depending on the method chosen, particularly when rating options. In this instance, two distinct methods yield their own unique optimal outcomes. As a consequence of the MARCOS method’s initial focus on maximizing the rankings of alternatives, ${⅁}_3$ is determined to be the best solution because it achieves the highest rank of all available possibilities. Alternatively, in an attempt to maximize alternative ranks as well, the TOPSIS technique aligns the optimal solution with the alternative that has the highest rank; in this case, ${⅁}_3$. Figure 2 provides a graphic representation of this comparison. The graphic illustrates the differences between the purposed operators technique and the TOPSIS method for finding the best solutions.

7. Conclusion

This study article suggests integrating the MARCOS technique with PHFS, which represents a significant progress in the field. When these approaches function together, a robust and comprehensive framework for making decisions in complex settings is established. Through the combination of PHFS’s uncertainty handling capabilities and the MARCOS methods accuracy, we have enhanced the efficacy of MCDM. Optimizing Renewable Energy Investments for Sustainable Growth is a case study that demonstrates the real-world implementation of the MARCOS method on PHFS. This innovative method offers a more thorough and flexible response, making it essential for dealing with difficult decision-making situations. A comparison study is conducted using the TOPSIS. This work establishes the foundation for further investigation and use of the MARCOS technique on PHFS across numerous industries. The data acquired for this study can be applied to real-world issue solving and serves to reinforce the theoretical foundations of decision science. This work promotes additional investigation into novel approaches to complex system decision-making, which may lead to discoveries with broad applications across multiple industries. As for the further research and application of the MARCOS method, one of the potential areas for its developments lies in combining the explorations of FSs’ extensions. When the problem has a certain amount of uncertainty or imprecision, the theories of FSs and their generalizations are useful. Future lines of work about PHFS include new aggregation operators that can be developed, a combination of PHFS with other fuzzy extensions, and its use in the group decision-making process and real-time applications. Similarly, MARCOS can use hybrid approaches that combine it with other existing models of decision-making and incorporate fuzzy logic to handle uncertainty and further optimize the model to make efficient decisions in real time in dynamic environments. Both methods have excellent prospects for use in AI, big data, and many other multiobjective optimization problems.

Author Contributions

The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.

Funding

No funding is received to support this research.

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