Site Selection for Solar Photovoltaic Power Plant Using MCDM Method with New De-i-Fuzzification Technique

Abstract

Choosing sites for solar photovoltaic (PV) power plants in developing countries like India is a crucial task while considering multiple conflicting factors and sub-factors simultaneously. Multi-criteria decision-making (MCDM) is an optimisation method that provides a framework for handling such situations in an intuitionistic fuzzy environment. The complexity and uncertainty associated with the site selection model are dealt with professionally. The Criteria Importance Through Intercriteria Correlation (CRITIC) method is applied to determine the relative importance of the criteria, identifying airflow speed as the most influential factor, followed by humidity ratio, level of dust haze, availability of labour and resources, and ecological effects. This shows that airflow speed plays an important role in the power plant’s efficiency and performance. The Vlse Kriterijumska Optimizacija I Kompromisno Rešenje (VIKOR) method is then used to prioritise the alternatives as potential locations for setting up a solar PV power plant in India. A new de-i-fuzzification method based on the relative difference between two real numbers is also proposed. Sensitivity analyses and comparative studies are conducted to assess the robustness and effectiveness of the framework. Overall, the results demonstrate that the proposed framework is useful and effective for optimising site selection for solar power plants in India.

1. Introduction

Nowadays, photovoltaic (PV) systems are seen as a potential replacement for fossil fuels as a clean energy source, and solar photovoltaic (PV) technology is one of the fastest-growing renewable energy sources worldwide [1,2]. Large-scale solar power systems, or utility-scale photovoltaic (PV) technology, are emerging in vast, open areas with abundant sunlight, making these locations ideal for large-scale solar projects [3]. Site selection is an important issue for solar photovoltaic (PV) systems, and it is very important to consider various factors in selecting a suitable site for a solar PV system, including solar potential, sunshine duration, PV surface slope, temperature gradient, airflow speed, level of dust haze, humidity ratio, government assistance and policies, climate, soil texture, proximity to the grid network, roads, and residential areas, as well as suitable water resources [4]. It is also advisable to carefully assess various social and environmental impacts, including wildlife habitat, the presence of protected areas, and local community concerns, and in regions where land availability is limited, it is essential to choose a site that balances the potential for energy production with the need to minimize environmental disturbance. Additionally, during the site selection process, it is important to account for the possibility of future expansion and ease of maintenance access. Therefore, the selection of suitable sites is an important step in the development of solar energy projects, which lays the foundation for efficient, cost-effective, and sustainable energy production, increases the long-term performance and economic feasibility of the enterprise, and maximizes energy production while minimizing environmental impact.

Mukherjee, A. K. et al. [5] conducted a comparative study on various renewable energy sources and evaluated the optimal renewable energy sources considering multiple criteria. Based on the evaluation, solar energy was the second most optimal renewable energy source for sustainable development. That study motivated us to study the solar energy domain. Further, Dinçer, S. [6] studied the energy crisis worldwide and the development of a sustainable environment. Kishore, T. S. et al. [7] provided a review of the sustainability of solar energy, focusing on its importance and impact in the global context.

This study targets India, which has strong potential for solar energy generation due to its diverse geography and favourable solar PV conditions [8]. Since selecting a suitable location for a solar project is a complex process and the geographical structure of different regions in India varies, the decision-making process must also be region-based. The study aims to propose an effective decision-making process for setting up PV plants in India that balances power generation, environmental conservation, and economic development to support renewable energy targets and sustainable growth.

Solar PV site selection problems can be solved using various MCDM methods that evaluate multiple criteria, including solar potential, terrain, climate, environmental impact, and proximity to infrastructure [9]. Integration with GIS allows these methods to provide more accurate suitability assessments [10,11]. Overall, MCDM techniques make the PV site selection process systematic, unbiased, and highly reliable. This study uses the CRITIC and VIKOR methods to select solar power plant sites across different regions of India, evaluating 12 criteria and considering Rajasthan, Uttar Pradesh, Andhra Pradesh, Tamil Nadu, Karnataka, Maharashtra, Gujarat, and West Bengal as alternatives. MCDM methods help prioritise conflicting criteria during site selection. The selection of a suitable location for installing a solar PV power plant depends on various criteria (as discussed above). But in practice, it is difficult to obtain precise data on these because factors such as airflow speed, radiation, dust haze levels, and labour availability are variable. We note that these criteria involve greater uncertainty that cannot be effectively captured by simpler fuzzy sets. They do not separately account for non-membership, which limits their ability to model incomplete or ambiguous information. Therefore, pentagonal intuitionistic fuzzy numbers are used to represent these criteria more comprehensively, including both membership and non-membership degrees, which allows uncertainty to be handled within a mathematical framework. By combining the pentagonal intuitionistic fuzzy set (PIFS) with MCDM, the study improves decision-making, providing a more accurate and flexible method for solar PV site selection than conventional models.

1.1. Motivation for This Study

India has enormous potential for solar energy due to its diverse geography and high solar radiation. However, selecting a suitable location for a solar PV power plant is very challenging because of variations in climate, solar radiation, infrastructure, dust and fog levels, wind speed, labour availability, and socio-economic factors. Traditional site selection methods often fail to address the inherent uncertainties and complexities of this multi-dimensional decision-making process. Inspired by the goal of selecting an optimal site while considering these challenges, in this project, the decision-making process incorporates pentagonal intuitionistic fuzzy numbers (PIFNs) to evaluate and rank potential sites, accurately measuring the inherent vagueness and errors in various parameters related to the project, thereby simplifying the site selection process.

1.2. Research Outline

Based on the introduction and research motivation, the preliminary outlines of this study are presented below:

• To construct a solar (PV) power plant in India, choose suitable locations, considering various environmental, economic, and infrastructural factors.
• Identify the various challenges for constructing a solar (PV) power plant in India and their comparative studies.
• Create a decision matrix combining criteria and alternatives using pentagonal intuitionistic fuzzy numbers (PIFNs), allowing decision-makers to effectively capture and address all forms of ambiguity, uncertainty, and vagueness.
• Determine the criteria weights using the MCDM method CRITIC. Based on input from decision experts, this step identifies the most important criteria for constructing the solar (PV) power plant.
• Rank the alternatives by the MCDM method VIKOR. This step consists in evaluating and ordering the alternatives based on their performance relative to the criteria, using the VIKOR method.
• We perform sensitivity and comparative analysis to check the ambiguity and neutrality of the results. Through this, we realize how closely aligned and well matched our results are.

1.3. Novelties

The key innovations of the proposed work are outlined below:

• This study presents an approach using pentagonal intuitionistic fuzzy numbers (PIFNs) integrated with the CRITIC and VIKOR techniques for selecting optimal locations for constructing solar (PV) power plants in India.
• A developed de-i-fuzzification method is proposed, based on the concept of the relative difference between two real numbers.
• A sensitivity analysis is performed for the CRITIC–VIKOR method, considering practical issues (such as downtrends in the stock market, climatic change due to pollution, etc.) to evaluate the stability of the MCDM results.
• The validity of the proposed framework is verified by conducting a comparative analysis with other methods to ensure its effectiveness.

1.4. Structure of This Paper

In this section, we present the overall structure of the present study. The introduction and motivation for this study are presented in Section 1. Section 2 contains the literature survey gathering existing research knowledge. All necessary preliminary mathematical tools, including a new de-i-fuzzification method for this research work, are given in Section 3. Section 4 covers both proposed MCDM methodologies. Criteria and alternatives for this study are selected in a logical way, and they are nicely described in Section 5 and Section 6, respectively. Details of the model structure and the data collection process are described in Section 7. Based on the previous section, a numerical approach is conducted and presented in Section 8. Sensitivity analyses and a comparative analysis are described in Section 9 and Section 10, respectively, to check the stability of the result. The research implication of the study is provided in Section 11. Finally, a conclusion is presented in Section 12, along with the study’s future research scope and limitations. Additionally, the preparation of mathematical tools is presented in Appendix A.

2. Literature Survey of This Study

In this section, we present a literature survey relevant to the present study. The literature survey is divided into three parts, namely, background on application, background on mathematical tools, and literature on methodologies, and each part is neatly described below.

2.1. Background on Application

As a renewable energy source, solar (PV) energy has a wide range of practical applications. We have gained substantial insights from prior studies on solar (PV) energy, and this section discusses the various directions of its applications.

Various farm applications of solar PV systems, along with the associated technologies, have been highlighted in [12]. Recent advances in photovoltaics, including thermal and photovoltaic–thermal panels, are reviewed in [13], while developments in robotics and agricultural automation are discussed in [14]. A photovoltaic water pumping system uses solar energy in [15]. In a rail transit power supply system, the implementation of photovoltaic power generation is notable [16]. In agricultural practices, various energy-saving strategies and applications, as well as the potential for using commercial building rooftops for solar photovoltaics, are highlighted in [17] and [18], respectively. There is a research article on the applications of various solar energy technologies in North Africa, with a focus on Algeria, Egypt, Libya, Morocco, and Tunisia [19]. There is extensive use of solar energy for charging various electric vehicles through TISO DC–DC converters [20]. Biomass–solar PV–battery hybrid power plants effectively support local needs in various remote areas [21]. In addition, numerous further applications of solar photovoltaic power, such as in transportation, the domestic sector, street lighting, water management, and district heating and cooling (DHC), are documented in [22,23].

Recently, the use of solar energy has increased rapidly. The following

Table 1. Recent studies on the application of solar photovoltaic energy.

Authors	Year	Application Area
Maka, A. et al. [24]	2024	Cathodic protection system
Ardeh, S. S. et al. [25]	2024	Production of heat, electricity and desalinated water
Xiao, H. et al. [26]	2024	Solar heat technology
Kok, C. L. et al. [27]	2024	Wireless charging device
Asgari, N. et al. [28]	2025	Heat pump systems for greenhouses
Yang, H. et al. [29]	2025	Rooftop PV projects at elevated metro stations.
González, P. et al. [30]	2025	Space applications
Terkes, M. et al. [31]	2025	Battery control strategies for prosumers
Tripathi, S. et al. [32]	2025	Battery charging system for EV
Saha, S. K. [33]	2025	Agriculture

presents some of the latest research on solar PV energy applications.

2.2. Background on Mathematical Tools

The pentagonal intuitionistic fuzzy number (PIFN) is a valuable tool in decision-making theory, combining the features of both the pentagonal fuzzy set and the intuitionistic fuzzy set. Many researchers have employed the pentagonal intuitionistic fuzzy number (PIFN) to make significant contributions to their work.

A production model is presented using pentagonal intuitionistic fuzzy numbers (PIFNs) [34]. A PIFN is used to select a software engineer for a software company in [35]. A new de-i-fuzzification method for non-linear PIFNs is proposed and applied to a minimum spanning tree problem in [36]. The shortest path of a network is found using an intuitionistic pentagonal fuzzy number in [37]. Pentagonal intuitionistic fuzzy numbers are used to study multi-objective linear fractional programming problems in [38]. There is a detailed explanation of how to evaluate the performance of a fuzzy queue model in an intuitionistic pentagonal fuzzy environment in [39]. A procedure for solving an integral equation with the help of PFNs is given in [40]. A method is developed for solving problems of transportation using an intuitionistic fuzzy environment with PIFNs in [41]. In the paper [42], Yuvashri, P. et al. formulated a multi-objective optimisation problem within a pentagonal intuitionistic fuzzy environment. A study on a green inventory EPQ model applying PIFNs is performed by Dey, A. et al. in [43]. In the paper [44], PIFNs are incorporated for a study on disease awareness campaigns using the PROMETHEE method.

2.3. Literature on Methodologies

Multi-criteria decision-making (MCDM) is a complex decision-making technique and is very effective for identifying the best option from a set of available alternatives where multiple criteria or features are taken into consideration. Two popular MCDM methods, namely, CRITIC and VIKOR, were chosen for this study. Till now, there have been various research work related to real-world problems where these two MCDM methods were applied. A brief literature survey on the CRITIC and VIKOR method is discussed below.

In decision-making problems, the CRITIC method is very effective at assigning the objective weights of criteria [45]. This method provides a systematic and structured framework, which also increases the popularity of this method in different fields.

The CRITIC method is applied under a neutrosophic environment in an aircraft selection process in [46]. In a paper on wearable health technology [47], a new picture fuzzy CRITIC method along with the REGIME method were used. In the paper [48], a new score function-based CRITIC–MARCOS method is introduced, where the considered information is given in a spherical fuzzy environment. The assessment of smartphone-related addiction is performed using the Pythagorean fuzzy CRITIC–TOPSIS method in [49]. A framework using a fuzzy CRITIC–TOPSIS approach is introduced for evaluating the risks during the demolition of building in [50]. The CRITIC and MARCOS methods are applied in a model to rationalise the ranking of zero-carbon measures in [51]. In a problem of selecting sites for a nursing home in [52], the CRITIC method is used under an interval type-2 fuzzy environment. A method for determining the location of an IoT-based product warehouse has been described using a fuzzy CRITIC technique [53]. A strategy for selecting the CCN cache placement has been proposed using fuzzy CoCoSo and CRITIC methodologies [54]. Spherical fuzzy D-CRITIC methods are applied to select suitable locations for electric vehicle charging stations in [55]. A method for selecting a suitable agricultural crop has been analysed using the CRITIC–VIKOR framework [56]. A new approach has been introduced that uses the CRITIC method to select a software reliability growth model [57].

In the current era, numerous studies have been developed using the CRITIC approach.

Table 2. Recent studies on CRITIC methodology with related data.

Authors	Year	Optimization Tools	Application Area
Wang, Y. et al. [58]	2024	CRITIC, MARCOS	Sustainable food supplier selection
Alkan, N. [59]	2024	CRITIC, SWARA, CODAS	Renewable energy systems
Zolfani, S.H. [60]	2024	CRITIC, ITARA, MEREC	Urban planning project
Alkan, N. et al. [61]	2024	CRITIC, WASPAS	Evaluation of a smart city
Kara, K. et al. [62]	2024	CIMAS, CRITIC, RBNAR	Financial performance analysis
Mao, L. et al. [63]	2025	CRITIC	Product development
Nguyen, V. et al. [64]	2025	CRITIC, CoCoSo	Selection of solar-wind energy plant location
Akpınar, M. E. [65]	2025	CRITIC, WASPAS	Logistics 4.0
Fan, J. et al. [66]	2025	CRITIC, TOPSIS	Vehicle Selection
Rasool, Z. et al. [67]	2025	CRITIC, EDAS	Green technology for environmental sustainability

below presents some of the latest research and related application areas utilising the CRITIC approach.

The VIKOR technique provides a compromise solution that ranks and selects the best alternative when multiple conflicting criteria are involved [68]. Numerous researchers have applied the VIKOR method to solve a wide range of multi-criteria decision-making problems. Some notable contributions and studies related to the application of the VIKOR method are described below.

A study applied the VIKOR method with complex Fermatean fuzzy sets to find a suitable location for a nuclear power plant [69]. An MCDM method based on TIVF-VIKOR has been proposed to solve a fuzzy MCGDM problem and applied to machine fault detection [70]. The VIKOR method has been extended with spherical fuzzy sets and applied to supplier selection [71]. The VIKOR method has been used effectively for rail transportation systems and genset selection [72,73]. The VIKOR method has been used for third-party logistics selection in a sustainable supply chain [74]. The application of the VIKOR method in an intuitionistic fuzzy environment for renewable energy source selection has been reported in [75]. The VIKOR approach has been applied in hydrogen production technology using an Intuitionistic Hypersoft Sets (IHSSs) framework [76]. The ITARA and VIKOR techniques are improved using two-tuple linguistic q-rung picture fuzzy sets in [77]. Sustainable carbon dioxide storage in geological formations has been evaluated using a modified Pythagorean fuzzy VIKOR and DEMATEL approach [78].

Recently, many researchers have presented various studies using the VIKOR technique, contributing to a wide range of applications across different fields.

Table 3. Recent studies on the VIKOR method with their related data.

Authors	Year	Optimization Tools	Application Area
Büyüközkan, G. et al. [79]	2024	DEMATEL, ANP, VIKOR	Selection of renewable energy
Mahmudah, R. S. et al. [80]	2024	AHP, VIKOR	Identifying a location for a nuclear power plant in Indonesia
Dağıstanlı, H. A. [81]	2024	VIKOR	Selecting R&D projects for defence investments
Shahidpoorfalah, B. et al. [82]	2024	VIKOR	Assessment of risks in digital technologies
Thakur, P. et al. [83]	2024	VIKOR	Vehicle battery selection process
Kumari, S. et al. [84]	2025	FVIKOR	Evaluation of water treatment plant failures
Runtuk, J. K. et al. [85]	2025	VIKOR, MOORA	Supplier selection
Baki, R. et al. [86]	2025	FVIKOR	Supplier selection for e-commerce supply chains
Wang, T. et al. [87]	2025	GRA, FVIKOR	Designing wickerwork cultural and creative products
Ahıskalı, A. et al. [88]	2025	VIKOR	Evaluation of water quality

below lists some of the most recent works related to the VIKOR approach.

3. Preliminaries on Mathematical Tools

In this section, we provide a comprehensive overview of the mathematical concepts and techniques that constitute the pentagonal intuitionistic fuzzy number (PIFN) framework presented in this paper. The preparation of the mathematical tools, such as fuzzy sets and their extensions, is presented in Appendix A.

3.1. Pentagonal Intuitionistic Fuzzy Number (PIFN)

Pentagonal intuitionistic fuzzy numbers (PIFNs) serve as a powerful tool for managing uncertainty and ambiguity in applications dealing with complex and uncertain information in real-world environments. The concept of PIFNs is given below.

Definition 1.

Pentagonal Intuitionistic Fuzzy Number (PIFN)

A pentagonal intuitionistic fuzzy number (PIFN) [34] is defined as ${\tilde{\mathcal{P}}}^I=\left[({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,{\zeta }_5);\alpha ,\beta ;\gamma ,\lambda \right]$, where each ${\zeta }_k\in \mathbb{R}$ for $k=1,2,\dots ,5$ and ${\zeta }_1\le {\zeta }_2\le {\zeta }_3\le {\zeta }_4\le {\zeta }_5$. The membership function, ${\eta }_{{\tilde{\mathcal{P}}}^I}:\mathcal{U}\to [0,1]$, and the non-membership function, ${\nu }_{{\tilde{\mathcal{P}}}^I}:\mathcal{U}\to [0,1]$ of ${\tilde{\mathcal{P}}}^I$, are defined as follows:

$${\eta }_{{\tilde{\mathcal{P}}}^I}(s)=\left\{\begin{array}{cc}0\hfill & ;s<{\zeta }_1\hfill \\ \alpha {\displaystyle \frac{s-{\zeta }_1}{{\zeta }_2-{\zeta }_1}}\hfill & ;{\zeta }_1\le s<{\zeta }_2\hfill \\ \alpha +(\beta -\alpha ){\displaystyle \frac{s-{\zeta }_2}{{\zeta }_3-{\zeta }_2}}\hfill & ;{\zeta }_2\le s\le {\zeta }_3\hfill \\ \beta +(\alpha -\beta ){\displaystyle \frac{s-{\zeta }_3}{{\zeta }_4-{\zeta }_3}}\hfill & ;{\zeta }_3<s\le {\zeta }_4\hfill \\ \alpha {\displaystyle \frac{{\zeta }_5-s}{{\zeta }_5-{\zeta }_4}}\hfill & ;{\zeta }_4<s\le {\zeta }_5\hfill \\ 0\hfill & ;{\zeta }_5<s\hfill \end{array}\right.$$

(1)

and

$${\nu }_{{\tilde{\mathcal{P}}}^I}(s)=\left\{\begin{array}{cc}1\hfill & ;s<{\zeta }_1\hfill \\ 1+(\gamma -1){\displaystyle \frac{s-{\zeta }_1}{{\zeta }_2-{\zeta }_1}}\hfill & ;{\zeta }_1\le s<{\zeta }_2\hfill \\ \gamma +(\lambda -\gamma ){\displaystyle \frac{s-{\zeta }_2}{{\zeta }_3-{\zeta }_2}}\hfill & ;{\zeta }_2\le s\le {\zeta }_3\hfill \\ \lambda +(\gamma -\lambda ){\displaystyle \frac{s-{\zeta }_3}{{\zeta }_4-{\zeta }_3}}\hfill & ;{\zeta }_3<s\le {\zeta }_4\hfill \\ \gamma +(1-\gamma ){\displaystyle \frac{s-{\zeta }_4}{{\zeta }_5-{\zeta }_4}}\hfill & ;{\zeta }_4<s\le {\zeta }_5\hfill \\ 1\hfill & ;{\zeta }_5<s\hfill \end{array}\right.,$$

(2)

where $0<\alpha <\beta \le 1$, $0\le \gamma <\lambda <1$, $\beta +\lambda \le 1$ and $\left[{\eta }_{{\tilde{\mathcal{P}}}^I}(s)+{\nu }_{{\tilde{\mathcal{P}}}^I}(s)\right]\le 1$ for all s.

Figure 1 beautifully captures the visual representation of the pentagonal intuitionistic fuzzy number (PIFN).

Remark 1.

To maintain the convexity of the membership function, the interior angles created at points $({\zeta }_2,\alpha )$ and $({\zeta }_4,\alpha )$ must be less than ${180}^{\circ }$. A similar concept is applied to the non-membership function to ensure convexity.

Remark 2.

If $\alpha =\beta$ and $\gamma =\lambda$, the above fuzzy number takes the shape of a trapezoidal intuitionistic fuzzy number (TrIFN). To obtain a triangular-shaped intuitionistic fuzzy number, set $\alpha =0$ and $\gamma =0$.

3.2. Justification for Using Pentagonal Intuitionistic Fuzzy Numbers (PIFNs)

Selecting a suitable site for building a solar PV power plant depends on several criteria, such as high solar radiation, available land, wind speed, labour availability, good transport system, and government approval. In a practical situation, it is hard to obtain exact and accurate data because factors like wind speed, solar radiation, dust haze, and labour availability often change, and their correct values are not always known. The uncertainty in these criteria cannot always be handled by simple fuzzy sets, so more complex fuzzy sets are needed. Pentagonal intuitionistic fuzzy numbers (PIFNs) are one such type, which use both membership and non-membership degrees to show the criteria more clearly and provide a mathematical way to handle such uncertainty. This helps decision-makers use PIFNs in different models to evaluate sites under uncertain conditions and select the most suitable site for the project. For this reason, we adopted PIFNs for this study.

3.3. Operations on Pentagonal Intuitionistic Fuzzy Numbers (PIFNs)

Let ${\tilde{\mathcal{H}}}^I=\left[({\psi }_1,{\psi }_2,{\psi }_3,{\psi }_4,{\psi }_5);{\alpha }_1,{\beta }_1;{\gamma }_1,{\lambda }_1\right]$ and ${\tilde{\mathcal{T}}}^I=\left[({\varphi }_1,{\varphi }_2,{\varphi }_3,{\varphi }_4,{\varphi }_5);{\alpha }_2,{\beta }_2;{\gamma }_2,{\lambda }_2\right]$ be two PIFNs. The mathematical operations between ${\tilde{\mathcal{H}}}^I$ and ${\tilde{\mathcal{T}}}^I$ are outlined below with details on the process and outcome of the interaction between them.

(i) 

Addition:

$${\tilde{\mathcal{H}}}^I\oplus {\tilde{\mathcal{T}}}^I=\left[({\psi }_1+{\mathsf{\Phi }}_1,{\psi }_2+{\mathsf{\Phi }}_2,{\psi }_3+{\mathsf{\Phi }}_3,{\psi }_4+{\mathsf{\Phi }}_4,{\psi }_5+{\mathsf{\Phi }}_5);min({\alpha }_1,{\alpha }_2),min({\beta }_1,{\beta }_2);max({\gamma }_1,{\gamma }_2),max({\lambda }_1,{\lambda }_2)\right]$$

(3)

(ii) 

Subtraction:

$${\tilde{\mathcal{H}}}^I\ominus {\tilde{\mathcal{T}}}^I=\left[({\psi }_1-{\mathsf{\Phi }}_5,{\psi }_2-{\mathsf{\Phi }}_4,{\psi }_3-{\mathsf{\Phi }}_3,{\psi }_4-{\mathsf{\Phi }}_2,{\psi }_5-{\mathsf{\Phi }}_1);min({\alpha }_1,{\alpha }_2),min({\beta }_1,{\beta }_2);max({\gamma }_1,{\gamma }_2),max({\lambda }_1,{\lambda }_2)\right]$$

(4)

(iii) 

Multiplication by a scalar:

$$c{\tilde{\mathcal{T}}}^I=\left\{\begin{array}{cc}\left[(c{\varphi }_1,c{\varphi }_2,c{\varphi }_3,c{\varphi }_4,c{\varphi }_5);{\alpha }_2,{\beta }_2;{\gamma }_2,{\lambda }_2\right]\hfill & ;c\ge 0\hfill \\ \left[(c{\varphi }_5,c{\varphi }_4,c{\varphi }_3,c{\varphi }_2,c{\varphi }_1);{\alpha }_2,{\beta }_2;{\gamma }_2,{\lambda }_2\right]\hfill & ;c<0\hfill \end{array}\right.;$$

(5)

where c is a real number.

(iv) 

Multiplication:

$${\tilde{\mathcal{H}}}^I\otimes {\tilde{\mathcal{T}}}^I=\left[({\psi }_1{\mathsf{\Phi }}_1,{\psi }_2{\mathsf{\Phi }}_2,{\psi }_3{\mathsf{\Phi }}_3,{\psi }_4{\mathsf{\Phi }}_4,{\psi }_5{\mathsf{\Phi }}_5);min({\alpha }_1,{\alpha }_2),min({\beta }_1,{\beta }_2);max({\gamma }_1,{\gamma }_2),max({\lambda }_1,{\lambda }_2)\right]$$

(6)

(v) 

Division:

$${\tilde{\mathcal{H}}}^I\oslash {\tilde{\mathcal{T}}}^I=\left[\left({\displaystyle \frac{{\psi }_1}{{\mathsf{\Phi }}_5}},{\displaystyle \frac{{\psi }_2}{{\mathsf{\Phi }}_4}},{\displaystyle \frac{{\psi }_3}{{\mathsf{\Phi }}_3}},{\displaystyle \frac{{\psi }_4}{{\mathsf{\Phi }}_2}},{\displaystyle \frac{{\psi }_5}{{\mathsf{\Phi }}_1}}\right);min({\alpha }_1,{\alpha }_2),min({\beta }_1,{\beta }_2);max({\gamma }_1,{\gamma }_2),max({\lambda }_1,{\lambda }_2)\right],$$

(7)

where ${\varphi }_j>0$ for $j=1,2,\dots ,5$.

(vi) 

Inverse:

$${\left[{\tilde{\mathcal{H}}}^I\right]}^{-1}=\left[\left({\displaystyle \frac{1}{{\psi }_5}},{\displaystyle \frac{1}{{\psi }_4}},{\displaystyle \frac{1}{{\psi }_3}},{\displaystyle \frac{1}{{\psi }_2}},{\displaystyle \frac{1}{{\psi }_1}}\right);{\alpha }_1,{\beta }_1;{\gamma }_1,{\lambda }_1\right]$$

(8)

Remark 3.

The operations performed above are valid for positive PIFNs. Some modifications are needed to work with negative PIFNs, especially for multiplication, division, and inverse operations.

3.4. De-i-Fuzzification Technique of PIFN

We work with fuzzy concepts in fuzzy set theory, where values lie on a spectrum of possibilities rather than just true or false. These fuzzy representations, including membership levels, are defuzzified to produce a crisp result. There are several common methods for defuzzification, such as the Centroid method, the Weighted Average method, the Mean of Maxima, and the Bisector method. In this section, a novel de-i-fuzzification (the term “de-i-fuzzification” refers to defuzzification with the intuitionistic nature of fuzzy numbers) method is proposed using the concept of the relative difference between two real numbers for the de-i-fuzzification of pentagonal intuitionistic fuzzy numbers (PIFNs). The detailed preliminary concept and the de-i-fuzzification procedure are very well described in the following.

Definition 2.

Absolute difference of two real numbers [89]

The absolute difference between two real numbers is defined as the absolute value of the difference between those numbers. In a mathematical way, if ${\mathcal{D}}_{absolute}(\gamma ,\delta )$ represents the absolute difference between two real numbers, γ and δ, then

$${\mathcal{D}}_{absolute}(\gamma ,\delta )=|\gamma -\delta |$$

(9)

Definition 3.

Relative difference of two real numbers [90]

The relative difference between two real numbers, γ and δ, denoted as ${\mathcal{D}}_{relative}(\gamma ,\delta )$, is defined as follows:

$${\mathcal{D}}_{relative}(\gamma ,\delta )={\displaystyle \frac{|\gamma -\delta |}{max(\left|\gamma \right|,\left|\delta \right|)}}$$

(10)

Remark 4.

The above formula expresses how large the difference between two numbers is in proportion to the size of the larger number.

Remark 5.

Based on the context of the application, $max(|\gamma |,|\delta |)$ can be replaced by $|max(\gamma ,\delta )|$, $min(|\gamma |,|\delta |)$, $|min(\gamma ,\delta )|$, ${\displaystyle \frac{\left|\gamma \right|+\left|\delta \right|}{2}}$, $|{\displaystyle \frac{\gamma +\delta }{2}}|$, etc.

Proposed De-i-Fuzzification of PIFNs

Consider the pentagonal intuitionistic fuzzy number (PIFN) ${\tilde{\mathcal{P}}}^I=\left[({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,{\zeta }_5);\alpha ,\beta ;\right.$$\left.\gamma ,\lambda \right]$ outlined in Section 3.1. To convert the fuzzy number into a crisp value, we introduce a new type of de-i-fuzzification technique based on the concept of the relative difference between two real numbers, as defined in Definition 3. The steps associated with the proposed de-i-fuzzification technique are described below:

I. 

Calculate the area $({\mathsf{\Delta }}_{{\eta }_{{\tilde{\mathcal{P}}}^I}})$ bounded by the membership function ${\eta }_{{\tilde{\mathcal{P}}}^I}(s)$ and x-axis:

The area $({\mathsf{\Delta }}_{{\eta }_{{\tilde{\mathcal{P}}}^I}})$ enclosed by the membership function ${\eta }_{{\tilde{\mathcal{P}}}^I}(s)$ of the given pentagonal intuitionistic fuzzy number (PIFN) ${\tilde{\mathcal{P}}}^I=\left[({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,{\zeta }_5);\alpha ,\beta ;\gamma ,\lambda \right]$ and the x-axis is given as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathsf{\Delta }}_{{\eta }_{{\tilde{\mathcal{P}}}^I}}=& {\int }_{{\zeta }_1}^{{\zeta }_5}{\eta }_{{\tilde{\mathcal{P}}}^I}(s)ds\hfill \\ \hfill =& {\int }_{{\zeta }_1}^{{\zeta }_2}{\eta }_{{\tilde{\mathcal{P}}}^I}(s)ds+{\int }_{{\zeta }_2}^{{\zeta }_3}{\eta }_{{\tilde{\mathcal{P}}}^I}(s)ds+{\int }_{{\zeta }_3}^{{\zeta }_4}{\eta }_{{\tilde{\mathcal{P}}}^I}(s)ds+{\int }_{{\zeta }_4}^{{\zeta }_5}{\eta }_{{\tilde{\mathcal{P}}}^I}(s)ds\hfill \\ \hfill =& {\int }_{{\zeta }_1}^{{\zeta }_2}\alpha {\displaystyle \frac{s-{\zeta }_1}{{\zeta }_2-{\zeta }_1}}ds+{\int }_{{\zeta }_2}^{{\zeta }_3}\left[\alpha +(\beta -\alpha ){\displaystyle \frac{s-{\zeta }_2}{{\zeta }_3-{\zeta }_2}}\right]ds+{\int }_{{\zeta }_3}^{{\zeta }_4}\left[\beta +(\alpha -\beta ){\displaystyle \frac{s-{\zeta }_3}{{\zeta }_4-{\zeta }_3}}\right]ds+{\int }_{{\zeta }_4}^{{\zeta }_5}\alpha {\displaystyle \frac{{\zeta }_5-s}{{\zeta }_5-{\zeta }_4}}ds\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}[\alpha ({\zeta }_2-{\zeta }_1)+(\alpha +\beta )({\zeta }_3-{\zeta }_2)+(\alpha +\beta )({\zeta }_4-{\zeta }_3)+\alpha ({\zeta }_5-{\zeta }_4)]\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}[\alpha ({\zeta }_5-{\zeta }_1)+\beta ({\zeta }_4-{\zeta }_2)]\hfill \end{array}\end{array}$$

(11)

II. 

Calculate the area $({\mathsf{\Delta }}_{{\nu }_{{\tilde{\mathcal{P}}}^I}})$ bounded by the non-membership function, ${\nu }_{{\tilde{\mathcal{P}}}^I}(s)$ and the straight line $y=1$.

The area $({\mathsf{\Delta }}_{{\nu }_{{\tilde{\mathcal{P}}}^I}})$ enclosed by the non-membership function, ${\nu }_{{\tilde{\mathcal{P}}}^I}(s)$ of the given pentagonal intuitionistic fuzzy number (PIFN) ${\tilde{\mathcal{P}}}^I=\left[({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,{\zeta }_5);\alpha ,\beta ;\gamma ,\lambda \right]$ and the straight line $y=1$ is taken as follows:

$$\begin{array}{cc}\hfill {\mathsf{\Delta }}_{{\nu }_{{\tilde{\mathcal{P}}}^I}}=& {\int }_{{\zeta }_1}^{{\zeta }_5}{\nu }_{{\tilde{\mathcal{P}}}^I}(s)ds\hfill \\ \hfill =& {\int }_{{\zeta }_1}^{{\zeta }_2}{\nu }_{{\tilde{\mathcal{P}}}^I}(s)ds+{\int }_{{\zeta }_2}^{{\zeta }_3}{\nu }_{{\tilde{\mathcal{P}}}^I}(s)ds+{\int }_{{\zeta }_3}^{{\zeta }_4}{\nu }_{{\tilde{\mathcal{P}}}^I}(s)ds+{\int }_{{\zeta }_4}^{{\zeta }_5}{\nu }_{{\tilde{\mathcal{P}}}^I}(s)ds\hfill \\ \hfill =& {\int }_{{\zeta }_1}^{{\zeta }_2}\left[1+(\gamma -1){\displaystyle \frac{s-{\zeta }_1}{{\zeta }_2-{\zeta }_1}}\right]ds+{\int }_{{\zeta }_2}^{{\zeta }_3}\left[\gamma +(\lambda -\gamma ){\displaystyle \frac{s-{\zeta }_2}{{\zeta }_3-{\zeta }_2}}\right]ds+{\int }_{{\zeta }_3}^{{\zeta }_4}\left[\lambda +(\gamma -\lambda ){\displaystyle \frac{s-{\zeta }_3}{{\zeta }_4-{\zeta }_3}}\right]ds+{\int }_{{\zeta }_4}^{{\zeta }_5}\left[\gamma +(1-\gamma ){\displaystyle \frac{s-{\zeta }_4}{{\zeta }_5-{\zeta }_4}}\right]ds\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}[(1+\gamma )({\zeta }_2-{\zeta }_1)+(\gamma +\lambda )({\zeta }_3-{\zeta }_2)+(\gamma +\lambda )({\zeta }_4-{\zeta }_3)+(1+\gamma )({\zeta }_5-{\zeta }_4)]\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}[(1+\gamma )({\zeta }_5-{\zeta }_1)-(1-\lambda )({\zeta }_4-{\zeta }_2)]\hfill \end{array}$$

(12)

III. 

Determine the proposed de-i-fuzzified value $(\mathcal{D}[{\tilde{\mathcal{P}}}^I])$.

The de-i-fuzzified value $(\mathcal{D}[{\tilde{\mathcal{P}}}^I])$ of the pentagonal intuitionistic fuzzy number (pifn) ${\tilde{\mathcal{P}}}^I=\left[({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,{\zeta }_5);\alpha ,\beta ;\gamma ,\lambda \right]$ is taken as follows:

$$\mathcal{D}[{\tilde{\mathcal{P}}}^I]={\displaystyle \frac{{\zeta }_1+{\zeta }_2+{\zeta }_3+{\zeta }_4+{\zeta }_5}{5}}\times {\displaystyle \frac{|{\mathsf{\Delta }}_{{\eta }_{{\tilde{\mathcal{P}}}^I}}-{\mathsf{\Delta }}_{{\nu }_{{\tilde{\mathcal{P}}}^I}}|}{max({\mathsf{\Delta }}_{{\eta }_{{\tilde{\mathcal{P}}}^I}},{\mathsf{\Delta }}_{{\nu }_{{\tilde{\mathcal{P}}}^I}})}},$$

(13)

where ${\mathsf{\Delta }}_{{\eta }_{{\tilde{\mathcal{P}}}^I}}$ and ${\mathsf{\Delta }}_{{\nu }_{{\tilde{\mathcal{P}}}^I}}$ are outlined in Equations (11) and (12), respectively.

Remark 6.

The main advantage of the proposed method is that it directly incorporates the normalised relative difference between the membership and non-membership areas. In this approach, the difference between the membership and non-membership areas is divided by the maximum value of the two areas. As a result, the generated weight is scale-invariant and bounded, ensuring that the crisp value is not distorted even if the parameters associated with the PIFN are increased or decreased uniformly. It is often observed that some PIFNs are able to produce the same or very close centroids, but with this method, those cases can also be clearly distinguished. Therefore, this new de-i-fuzzification method provides reliable crisp values for decision-making and ranking.

Example 1.

Let ${\tilde{\mathcal{E}}}^I=\left[(2,6,9,11,13);0.5,0.8;0.1,0.2\right]$ be a PIFN, defined as in Definition 1. Clearly, the maximum degree of membership of ${\tilde{\mathcal{E}}}^I$ is 0.8, and the maximum degree of non-membership is 0.2. Also, the sum of the maximum membership and non-membership values is less than or equal to 1. We find the corresponding crisp value of ${\tilde{\mathcal{E}}}^I$ using the de-i-fuzzification formula defined in Equation (13). For this, we have to calculate the area bounded by the membership function ${\eta }_{{\tilde{\mathcal{E}}}^I}$ and the x-axis, and the area bounded by the non-membership function ${\nu }_{{\tilde{\mathcal{E}}}^I}$ and the line $y=1$. Now, the area bounded by the membership function ${\eta }_{{\tilde{\mathcal{E}}}^I}(s)$ and x-axis of the pentagonal intuitionistic fuzzy number ${\tilde{\mathcal{E}}}^I=\left[(2,6,9,11,13);0.5,0.8;0.1,0.2\right]$ is ${\mathsf{\Delta }}_{{\eta }_{{\tilde{\mathcal{E}}}^I}}=4.75$, and the area enclosed by the non-membership function ${\nu }_{{\tilde{\mathcal{E}}}^I}(s)$ and the straight line $y=1$ is ${\mathsf{\Delta }}_{{\nu }_{{\tilde{\mathcal{E}}}^I}}=4.05$. Thus, the de-i-fuzzified value of ${\tilde{\mathcal{E}}}^I=\left[(2,6,9,11,13);0.5,0.8;0.1,0.2\right]$ is determined as

$$\begin{array}{cc}\hfill \mathcal{D}[{\tilde{\mathcal{E}}}^I]=& {\displaystyle \frac{2+6+9+11+13}{5}}\times {\displaystyle \frac{4.75-4.05}{max(4.75,4.05)}}\hfill \\ \hfill =& 1.208\hfill \end{array}$$

(14)

Razzaq, O. A. et al. [44] invented the de-i-fuzzification method on PIFNs, and the de-i-fuzzification value of PIFNs ${\tilde{\mathcal{E}}}^I$ was

$$\begin{array}{cc}\hfill \mathcal{D}[{\tilde{\mathcal{E}}}^I]=& {\displaystyle \frac{{\zeta }_1+{\zeta }_2+{\zeta }_3+{\zeta }_4+{\zeta }_5}{10}}\times (1+\beta -\gamma )\hfill \\ \hfill =& {\displaystyle \frac{2+6+9+11+13}{10}}\times (1+0.8-0.2)={\displaystyle \frac{41}{10}}\times 1.6\hfill \\ \hfill =& 6.56\hfill \end{array}$$

(15)

The de-i-fuzzification value of PIFNs ${\tilde{\mathcal{E}}}^I$ is 1.208 using the proposed method (see in Equation (13)) and 6.56 by Equation (15) (see in Razzaq, O. A. et al. [44]), respectively.

4. Proposed MCDM Methodologies Under Pentagonal Intuitionistic Fuzzy Number (PIFN)

This section explains the MCDM methodologies applied to determine the criteria weight and prioritise the alternatives for the current research. To perform the present research work, we selected two MCDM approaches: Criteria through Inter-Criteria Correlation Technique (CRITIC) and Višestruki Kriterijumska Optimizacija Kompromisno Rešenje (VIKOR). CRITIC was employed to calculate the criteria weights, while VIKOR was utilised to rank the chosen alternatives. The detailed mathematical procedures of the two methodologies are described below.

4.1. Weight Calculation Method: CRITIC

In this study, let f represent the number of criteria and p is the number of alternatives. We assume there are q decision-makers (DMs), each providing data in linguistic terms. Consequently, decision matrices are constructed by each qth DM, with an order of $(f\times p)$. The mathematical formulation of the CRITIC methodology is as follows:

A. 

Formulate the decision matrices $({({\tilde{\mathcal{D}}}^I)}_{\xi })$.

The decision matrix is formed by the $\xi$th DM based on their views in terms of linguistic decisions, where $\xi =1,2,3,\dots ,q$. Here, q decision matrices are created, and all the linguistic terms are converted to pentagonal intuitionistic fuzzy numbers (PIFNs). The $\xi$th decision matrix can be expressed as follows:

$${({\tilde{\mathcal{D}}}^I)}_{\xi }={\left[{({{\tilde{\mathcal{P}}}^I}_{lm})}_{\xi }\right]}_{p\times f}={\left[\begin{array}{cccc}{({\tilde{\mathcal{P}}}^I}_{11}{)}_{\xi }& {({\tilde{\mathcal{P}}}^I}_{12}{)}_{\xi }& \dots & {({\tilde{\mathcal{P}}}^I}_{1f}{)}_{\xi }\\ {({\tilde{\mathcal{P}}}^I}_{21}{)}_{\xi }& {({\tilde{\mathcal{P}}}^I}_{22}{)}_{\xi }& \dots & {({\tilde{\mathcal{P}}}^I}_{2f}{)}_{\xi }\\ ⋮& ⋮& \ddots & ⋮\\ {({\tilde{\mathcal{P}}}^I}_{p1}{)}_{\xi }& {({\tilde{\mathcal{P}}}^I}_{p2}{)}_{\xi }& \dots & {({\tilde{\mathcal{P}}}^I}_{pf}{)}_{\xi }\end{array}\right]}_{p\times f},$$

(16)

where $\xi =1,2,\dots ,q$ and ${({\tilde{\mathcal{D}}}^I)}_{\xi }$ is a matrix of order $p\times f$, and each entry of the decision matrices is expressed as a pentagonal intuitionistic fuzzy number (PIFN), i.e.,

$${({\tilde{\mathcal{P}}}^I}_{lm}{)}_{\xi }=\left[\left({({{\zeta }_1}^{{{\tilde{\mathcal{P}}}^I}_{lm}})}_{\xi },{({{\zeta }_2}^{{{\tilde{\mathcal{P}}}^I}_{lm}})}_{\xi },{({{\zeta }_3}^{{{\tilde{\mathcal{P}}}^I}_{lm}})}_{\xi },{({{\zeta }_4}^{{{\tilde{\mathcal{P}}}^I}_{lm}})}_{\xi },{({{\zeta }_5}^{{{\tilde{\mathcal{P}}}^I}_{lm}})}_{\xi }\right);{({\alpha }^{{{\tilde{\mathcal{P}}}^I}_{lm}})}_{\xi },{({\beta }^{{{\tilde{\mathcal{P}}}^I}_{lm}})}_{\xi };{({\gamma }^{{{\tilde{\mathcal{P}}}^I}_{lm}})}_{\xi },{({\lambda }^{{{\tilde{\mathcal{P}}}^I}_{lm}})}_{\xi }\right]$$

(17)

for each criteria m and alternative l and $\xi =1,2,\dots ,q$.

B. 

Combine the decision matrices $({\tilde{\mathcal{D}}}^I)$.

Merge the q decision matrices provided by the decision makers (DMs) into a single consolidated decision matrix. To create this, the following format is used:

$$\begin{array}{cc}\hfill {{\tilde{\mathcal{P}}}^I}_{lm}=& \left[\left({{\zeta }_1}^{{{\tilde{\mathcal{P}}}^I}_{lm}},{{\zeta }_2}^{{{\tilde{\mathcal{P}}}^I}_{lm}},{{\zeta }_3}^{{{\tilde{\mathcal{P}}}^I}_{lm}},{{\zeta }_4}^{{{\tilde{\mathcal{P}}}^I}_{lm}},{{\zeta }_5}^{{{\tilde{\mathcal{P}}}^I}_{lm}}\right);{\alpha }^{{{\tilde{\mathcal{P}}}^I}_{lm}},{\beta }^{{{\tilde{\mathcal{P}}}^I}_{lm}};{\gamma }^{{{\tilde{\mathcal{P}}}^I}_{lm}},{\lambda }^{{{\tilde{\mathcal{P}}}^I}_{lm}}\right]\hfill \\ \hfill =& \left[\left(\underset{\xi =1,2,\dots ,q}{min}\left\{{({{\zeta }_1}^{{{\tilde{\mathcal{P}}}^I}_{lm}})}_{\xi }\right\},\underset{\xi =1,2,\dots ,q}{min}\left\{{({{\zeta }_2}^{{{\tilde{\mathcal{P}}}^I}_{lm}})}_{\xi }\right\},{\left(\prod _{\xi =1}^q\left\{{({{\zeta }_3}^{{{\tilde{\mathcal{P}}}^I}_{lm}})}_{\xi }\right\}\right)}^{{\displaystyle \frac{1}{q}}},\underset{\xi =1,2,\dots ,q}{max}\left\{{({{\zeta }_4}^{{{\tilde{\mathcal{P}}}^I}_{lm}})}_{\xi }\right\},\underset{\xi =1,2,\dots ,q}{max}\left\{{({{\zeta }_5}^{{{\tilde{\mathcal{P}}}^I}_{lm}})}_{\xi }\right\}\right);\right.\hfill \\ \hfill & \left.\underset{\xi =1,2,\dots ,q}{min}\left\{{({\alpha }^{{{\tilde{\mathcal{P}}}^I}_{lm}})}_{\xi }\right\},\underset{\xi =1,2,\dots ,q}{min}\left\{{({\beta }^{{{\tilde{\mathcal{P}}}^I}_{lm}})}_{\xi }\right\};\underset{\xi =1,2,\dots ,q}{max}\left\{{({\gamma }^{{{\tilde{\mathcal{P}}}^I}_{lm}})}_{\xi }\right\},\underset{\xi =1,2,\dots ,q}{max}\left\{{({\lambda }^{{{\tilde{\mathcal{P}}}^I}_{lm}})}_{\xi }\right\}\right]\hfill \end{array}$$

(18)

C. 

Form the combined matrix with de-i-fuzzified values $({{\mathcal{D}}^I}_{cv})$.

The de-i-fuzzified combined decision matrix is derived from the combined decision matrix using Equation (13). The de-i-fuzzified aggregated decision matrix is represented by

$${{\mathcal{D}}^I}_{cv}={\left[{{\mathcal{P}}^I}_{lm}\right]}_{p\times f},$$

(19)

where the de-i-fuzzified value ${{\mathcal{P}}^I}_{lm}$ of the pentagonal intuitionistic fuzzy number (PIFN) ${{\tilde{\mathcal{P}}}^I}_{lm}$ is calculated for all $l=1,2,3,\dots ,p$ and $m=1,2,3,\dots ,f$.

D. 

Normalize the updated decision matrix $({{\mathcal{D}}^I}_n)$.

To transform the given fuzzy values into a standardized scale, the updated decision matrix is normalised using the following Equation (20):

$${({{\mathcal{P}}^I}_{lm})}^{\prime }={\displaystyle \frac{{{\mathcal{P}}^I}_{lm}-{{{\mathcal{P}}^I}_m}^{worst}}{{{{\mathcal{P}}^I}_m}^{best}-{{{\mathcal{P}}^I}_m}^{worst}}},$$

(20)

where

$$\left\{\begin{array}{c}{{{\mathcal{P}}^I}_m}^{best}=\underset{l=1,2,3,\dots ,p}{max}\left\{{{\mathcal{P}}^I}_{lm}\right\}\hfill \\ {{{\mathcal{P}}^I}_m}^{worst}=\underset{l=1,2,3,\dots ,p}{min}\left\{{{\mathcal{P}}^I}_{lm}\right\}\hfill \end{array}\right.$$

(21)

E. 

Compute the standard deviation $({\sigma }_m)$ for each criterion.

The standard deviation ${\sigma }_m$ for each criterion is derived from Equation (22) as follows:

$${\sigma }_m=\sqrt{{\displaystyle \frac{\sum {({{\mathcal{P}}^I}_m-{\overline{{\mathcal{P}}^I}}_{lm})}^2}{N-1}}},$$

(22)

where ${\overline{{\mathcal{P}}^I}}_{lm}$ indicates the population mean, N is the size of the population, and $m=1,2,3,\dots ,f$.

F. 

Determine the linear correlation coefficient between the criteria m and $m^{\prime }$.

Let the vectors ${{\mathcal{P}}^I}_m$ and ${{\mathcal{P}}^I}_{m^{\prime }}$ be the performance values of p alternatives for criteria m and $m^{\prime }$, respectively. Each vector contains performance values for p alternatives, i.e.,

$$\left\{\begin{array}{c}{{\mathcal{P}}^I}_m=({{\mathcal{P}}^I}_{m1},{{\mathcal{P}}^I}_{m2},\dots ,{{\mathcal{P}}^I}_{mp})\hfill \\ {{\mathcal{P}}^I}_{m^{\prime }}=({{\mathcal{P}}^I}_{m^{\prime }1},{{\mathcal{P}}^I}_{m^{\prime }2},\dots ,{{\mathcal{P}}^I}_{m^{\prime }p})\hfill \end{array}\right.$$

(23)

so the linear correlation coefficient between the criteria m and $m^{\prime }$ can be defined as follows

$${\theta }_{m{m}^{\prime }}={\displaystyle \frac{{\sum }_{l=1}^p({{\mathcal{P}}^I}_{ml}-{\overline{{\mathcal{P}}^I}}_m)({{\mathcal{P}}^I}_{m^{\prime }l}-{\overline{{\mathcal{P}}^I}}_{m^{\prime }})}{\sqrt{{\sum }_{l=1}^p{({{\mathcal{P}}^I}_{ml}-{\overline{{\mathcal{P}}^I}}_m)}^2}\sqrt{{\sum }_{l=1}^p{({{\mathcal{P}}^I}_{m^{\prime }l}-{\overline{{\mathcal{P}}^I}}_{m^{\prime }})}^2}}},$$

(24)

where

$$\left\{\begin{array}{c}{\overline{{\mathcal{P}}^I}}_m={\displaystyle \frac{1}{p}}\times \sum _{l=1}^p{{\mathcal{P}}^I}_{ml}\hfill \\ {\overline{{\mathcal{P}}^I}}_{m^{\prime }}={\displaystyle \frac{1}{p}}\times \sum _{l=1}^p{{\mathcal{P}}^I}_{m^{\prime }l}\hfill \end{array}\right.$$

(25)

G. 

Measure of the conflict generated by the criteria.

Here, we need to calculate a measure of the conflict created by criterion m about the decision situation defined by the remaining criteria, defined as follows:

$${\rho }_m=\sum _{m^{\prime }=1}^f\left(1-{\theta }_{m{m}^{\prime }}\right)$$

(26)

H. 

Determining the amount of information associated with each criterion.

The amount of information associated with each criterion is taken from Equation (27):

$${\mathcal{C}}_m={\sigma }_m\times {\rho }_m,$$

(27)

where $m=1,2,3,\dots ,f$.

I. 

Calculate the objective weights.

Compute the weight of criterion m, denoted as ${{\mathcal{C}}_m}^{\mathcal{W}}$, as follows:

$${{\mathcal{C}}_m}^{\mathcal{W}}={\displaystyle \frac{{\mathcal{C}}_m}{{\sum }_{m=1}^f{\mathcal{C}}_m}}$$

(28)

Equation (28) determines the criterion weight ${{\mathcal{C}}_m}^{\mathcal{W}}$ for each criterion $m=1,2,3,\dots ,f$. In Figure 2 below, a methodical representation of the CRITIC method is shown.

Remark 7.

The CRITIC method determines the correlation between the criteria. If two criteria are highly correlated, the approach will give one less weight because they provide similar information.

4.2. A Ranking-Oriented MCDM Approach: VIKOR

This study considers f criteria and p alternatives. Opinions are provided by q decision-makers (DMs) based on their experience, ensuring a comprehensive evaluation. The steps of the VIKOR method are given below:

I. 

Establishment of decision matrices $({({\tilde{\mathcal{D}}}^I)}_{\xi })$.

We developed a decision matrix using linguistic terms and converted it into pentagonal intuitionistic fuzzy numbers to capture the opinions of decision makers and handle uncertainty in evaluations. In the previous section, Equation (16) represents the $\xi$th decision matrix for $\xi =1,2,3,\dots ,q$.

II. 

Aggregation of decision matrices.

The process of combining the decision matrix into a single aggregated decision matrix is performed using Equation (18), which provides a clear and structured format for subsequent evaluation.

III. 

Determine the de-i-fuzzified valued aggregated matrix $({{\mathcal{D}}^I}_{cv})$.

The de-i-fuzzified aggregated decision matrix $({{\mathcal{D}}^I}_{cv})$ is obtained from the aggregated decision matrix $({\tilde{\mathcal{D}}}^I)$ by using the method described in Equation (13), and the matrix is shown in Equation (19).

IV. 

Calculate the best value $({{\delta }_m}^{+})$ and worst value $({{\delta }_m}^{-})$.

The best and worst values for each criterion are defined by ${{\delta }_m}^{+}$ and ${{\delta }_m}^{-}$, respectively, in Equations (29) and (30):

$${{\delta }_m}^{+}=\left\{\begin{array}{cc}\underset{l=1,2,\dots ,p}{max}({\mathcal{P}}_{lm}^I)\hfill & ;\mathrm{for}\mathrm{beneficial}\mathrm{criteria}\hfill \\ \underset{l=1,2,\dots ,p}{min}({\mathcal{P}}_{lm}^I)\hfill & ;\mathrm{for}\mathrm{non-beneficial}\mathrm{criteria}\hfill \end{array}\right.$$

(29)

and

$${{\delta }_m}^{-}=\left\{\begin{array}{cc}\underset{l=1,2,\dots ,p}{min}({\mathcal{P}}_{lm}^I)\hfill & ;\mathrm{for}\mathrm{beneficial}\mathrm{criteria}\hfill \\ \underset{l=1,2,\dots ,p}{max}({\mathcal{P}}_{lm}^I)\hfill & ;\mathrm{for}\mathrm{non-beneficial}\mathrm{criteria}\hfill \end{array}\right.,$$

(30)

where $m=1,2,\dots ,f$.

V. 

Evaluate the utility value $({\mathcal{S}}_l)$ and regret value $({\mathcal{R}}_l)$.

The concepts of utility and regret values are used to balance the compromise solutions. The values of ${\mathcal{S}}_l$ and ${\mathcal{R}}_l$ are evaluated from Equations (31) and (32):

$${\mathcal{S}}_l=\sum _{m=1}^f\left\{{{\mathcal{C}}_m}^{\mathcal{W}}\times {\displaystyle \frac{{{\delta }_m}^{+}-{\mathcal{P}}_{lm}^I}{{{\delta }_m}^{+}-{{\delta }_m}^{-}}}\right\}$$

(31)

and

$${\mathcal{R}}_l=\underset{m}{max}\left\{{{\mathcal{C}}_m}^{\mathcal{W}}\times {\displaystyle \frac{{{\delta }_m}^{+}-{\mathcal{P}}_{lm}^I}{{{\delta }_m}^{+}-{{\delta }_m}^{-}}}\right\},$$

(32)

where $l=1,2,\dots ,p$.

VI. 

Find out the compromise value $({\mathcal{Q}}_l)$.

The compromise value, ${\mathcal{Q}}_l$ is determined using Equation (33) as follows:

$${\mathcal{Q}}_l=\tau \times {\displaystyle \frac{{\mathcal{S}}_l-{\mathcal{S}}^{-}}{{\mathcal{S}}^{+}-{\mathcal{S}}^{-}}}+(1-\tau )\times {\displaystyle \frac{{\mathcal{R}}_l-{\mathcal{R}}^{-}}{{\mathcal{R}}^{+}-{\mathcal{R}}^{-}}},$$

(33)

where

$$\left\{\begin{array}{c}l=1,2,\dots ,p,\hfill \\ {\mathcal{S}}^{+}=\underset{l}{max}{\mathcal{S}}_l,\hfill & {\mathcal{S}}^{-}=\underset{l}{min}{\mathcal{S}}_l,\hfill \\ {\mathcal{R}}^{+}=\underset{l}{max}{\mathcal{R}}_l,\hfill & {\mathcal{R}}^{-}=\underset{l}{min}{\mathcal{R}}_l,\hfill \end{array}\right.$$

(34)

$\tau$ is the coefficient of the decision mechanism. The compromise solution is chosen based on the value of $\tau$: majority $(\tau >0.5)$, consensus $(\tau =0.5)$, or veto $(\tau <0.5)$.

VII. 

Rank the alternatives.

To rank alternatives using the ${\mathcal{S}}_l$, ${\mathcal{R}}_l$, and ${\mathcal{Q}}_l$ values, first classify these values in ascending order. It produces three separate ranking lists: ${\mathcal{S}}_{[\ast ]}$, ${\mathcal{R}}_{[\ast ]}$, and ${\mathcal{Q}}_{[\ast ]}$, where each list shows the alternatives from smallest to largest value. The alternative, say $a^{\left[1\right]}$, with the smallest ${\mathcal{Q}}_{[\ast ]}$ value is considered the compromise solution.

VIII. 

Validation of compromise solution.

To ensure that a compromise solution is appropriate, it must meet the following two conditions:

(i) 

Acceptable Advantage:

$\mathcal{Q}(a^{\left[2\right]})-\mathcal{Q}(a^{\left[1\right]})\ge {\displaystyle \frac{1}{1-p}}$, where $a^{\left[1\right]}$ and $a^{\left[2\right]}$ are the two best alternatives of ${\mathcal{Q}}_{[\ast ]}$.

(ii) 

Acceptable Stability:

Based on the values of ${\mathcal{S}}_l$ and ${\mathcal{R}}_l$, the alternative $a^{\left[1\right]}$ should be ranked as the best choice. If both ${\mathcal{S}}_l$ and ${\mathcal{R}}_l$ indicate that the alternative $a^{\left[1\right]}$ has the highest performance among all alternatives, it can be confidently considered as the best choice.

Multiple compromise solutions can be observed when not all conditions can be met simultaneously, i.e., no single solution fully satisfies everything. If only condition $(ii)$ is not satisfied, we consider only alternatives $a^{\left[1\right]}$ and $a^{\left[2\right]}$. If condition $(i)$ is not satisfied, we take all the alternatives $a^{\left[1\right]},a^{\left[2\right]},\dots ,a^{\left[n\right]}$, where $a^{\left[n\right]}$ is defined as $\mathcal{Q}(a^{\left[n\right]})-\mathcal{Q}(a^{\left[1\right]})<{\displaystyle \frac{1}{n-1}}$ for maximum n.

A systematic framework of the VIKOR approach is presented in Figure 3 below.

5. Selection of Criteria for Construction of Solar Photovoltaic (PV) Power Plant

In this section, we discuss the different factors (or criteria) for choosing the best location to build a solar PV power plant in India. After consulting several experts, we identified the following twelve criteria.

5.1. Land Accessibility and Geographical Features $(F_1)$

Geographical features and availability of land are essential for setting up solar photovoltaic (PV) power plants [91]. Generally, areas with broad, flat, and gently sloping land are identified. Moreover, sites near rivers or in flood-prone areas should be carefully considered, as solar panels and electrical systems can be damaged by flooding.

5.2. High Solar Radiation $(F_2)$

The impact of high solar radiation on solar photovoltaic (PV) power plant construction is unparalleled [10]. Different regions have different geographical features. Valleys and high mountains, for example, can significantly limit the amount of sunlight that certain areas receive at certain times of the day. Thus, the land intended for installing solar PV power plants should have the maximum exposure to sunlight, as it is essential to optimise energy production from solar panels.

5.3. Ecological Effect $(F_3)$

It is essential to assess the environmental impact of solar PV plants for their construction and operation [92]. The potential impacts on wildlife habitat, local ecosystems, soil, air, water, usable agricultural land, etc., should be properly analysed. Large solar installations can fragment natural habitats, hamper wildlife movement, reduce biodiversity, cause temperature changes, and increase pollution [93]. Special plans should be adopted to minimize negative impacts.

5.4. Availability of Labour and Resources $(F_4)$

Building a solar photovoltaic (PV) power plant is impossible without adequate labour and resources [94]. Adequately skilled workers are needed for planning, construction, installation of solar panels, supply of necessary materials, and management. In addition, the availability of raw materials required for the project, such as photovoltaic cells, inverters, cables and wiring, transformers, battery storage, infrastructure materials, electrical components, protective coatings, and sealants, etc., must be ensured to avoid any interruption of equipment resources.

5.5. Airflow Speed $(F_5)$

To construct a solar power plant, it is necessary to have an ideal wind speed [95]. Medium-velocity airflow is suitable for solar photovoltaic (PV) power plant construction projects because it maintains the optimal operating temperature of the solar panels and improves their performance. High-velocity airflow reduces the stability of the mounting structure. If the project is to be built in areas with high wind speeds, strong structures and precise panel adaptations need to be implemented to ensure wind-speed protection and stability.

5.6. Level of Dust Haze $(F_6)$

Dust and fog levels in areas where solar PV power plants are being constructed should be as low as possible. High levels of dust can disrupt the process of installing solar panels. If excessive dust accumulates on the panels, sunlight cannot penetrate them properly, reducing panel efficiency. Excessive dust is also a major challenge in panel maintenance. If dust levels are too high at a selected location, necessary dust control measures must be taken to maintain panel performance.

5.7. Local Community Support and Approval $(F_7)$

Installing solar photovoltaic (PV) power plants in a region is not possible without the support and approval of the local community [96]. It is essential to increase economic benefits such as job creation, reducing energy costs, and developing the local infrastructure. When the community feels included in the importance and real benefits of the project, the likelihood of approval increases. The greater the positive community participation, the more smoothly the project will run without social conflict.

5.8. Proximity of the Transmission Network $(F_8)$

The proximity of the transmission network is an essential factor in planning the construction of solar photovoltaic (PV) power plants [97]. If the transmission network is not located near the designated area, plans need to be adopted to install new transmission lines, substations, or other grid infrastructure, which increase the project cost. An optimally positioned transmission network ensures that the solar power generated at the plant can be delivered to users cost-effectively.

5.9. Humidity Ratio $(F_9)$

When identifying specific areas for installing solar PV power plants, it is necessary to accurately determine the amount of moisture in the air [98]. This is because excess moisture traps dust on the panels and blocks sunlight from reaching them, thereby reducing the effectiveness of the panels. Problems arise due to corrosion of various electrical components and other metal parts of the infrastructure [99]. Generally, those areas should be chosen where the air moisture content is moderate.

5.10. Grid Reliability $(F_{10})$

Reliability of the electrical grid is an essential factor in the design of solar photovoltaic (PV) power plants [100]. They should be highly reliable to ensure a continuous, uninterrupted supply of electricity. Solar power generation can vary due to weather, time of day, and seasonal changes, so it is necessary to use modern inverters that control frequency and voltage to ensure grid stability. In addition, adding energy storage solutions, such as batteries, is vital to capture the excess energy generated during peak sunlight hours and use it during periods of low solar output [101].

5.11. Seamless Transport Network $(F_{11})$

To implement the solar PV power plant construction project, it is necessary to identify areas with improved transportation facilities [102]. Large and heavy components used in construction must be easily transported from the manufacturing facility to the construction site. It is essential to have reliable connectivity between the railway station and the airport, and to the construction site, so that workers and project personnel can travel easily. In addition, an efficient logistics management system is needed [103] to streamline transportation activities, reduce project timelines, and lower costs.

5.12. Government Assistance and Policies $(F_{12})$

Government support and policies play an essential role in the development and construction of solar photovoltaic (PV) power plants [104]. Government promotion of projects, low-interest loans, and convenient financing schemes (such as power purchase agreements), etc., increases incentives for investment in projects [105]. In the early stages, infrastructure requires various external permits, which should be simplified wherever possible. Additionally, policies such as government subsidies, grants, and tax credits reduce the financial constraints of the project.

6. Selection of Alternatives for Construction of a Solar Photovoltaic (PV) Power Plant

In this section, we selected eight states from the Indian region based on the criteria described in the previous section. The selected states are thoroughly described with careful consideration of these criteria. Based on these criteria and alternatives, the next section addresses a practical problem related to the construction of a solar PV power plant in India.

6.1. Rajasthan $(P_1)$

There are various reasons for choosing this state for setting up a solar PV power plant, one of which is the high solar power predominance in areas such as Jodhpur, Jaisalmer, Barmer, Bikaner, Nagaur, Pushkar, etc. Its semi-arid climate, ample sunlight, high temperatures, and clear skies are ideal for solar energy production [106]. Also, it is well connected to various international airports, including Jaipur, Jodhpur, Udaipur, Ajmer, and Bikaner; railway junctions; and 33 National Highways (NHs) [107]. There are several government initiatives to support solar energy [108]. In addition, the environmental benefits of solar energy in Rajasthan include lower carbon emissions.

6.2. Uttar Pradesh $(P_2)$

Uttar Pradesh is ideal for setting up solar PV power plants due to its dry winters, high solar radiation, and humid subtropical climate. Thus, Agra, Moradabad, Varanasi, Ghaziabad, Allahabad, Kanpur, Lucknow, and Ayodhya regions were selected for setting up solar PV power plants. The cities of Uttar Pradesh (especially major cities like Lucknow, Varanasi, Agra, Jhansi, and Kanpur) are well connected by a strong road, airport, and railway network [109]. The current state government has released the Solar Policy 2022 [110]. The governmental efforts to establish solar PV power plants are very encouraging [105].

6.3. Andhra Pradesh $(P_3)$

The areas adjacent to the coastal plains in Andhra Pradesh are ideal for setting up solar PV power plants. These areas include Srikakulam, Vizianagaram, Parvathipuram, Manyam, Alluri Sitarama Raju, Visakhapatnam, Anakapalli, Kakinada, Konaseema, East Godavari, West Godavari, Eluru, Krishna, NTR, Guntur, Palnadu, Bapatla, Prakasam, and Nellore. The coastal regions of Andhra Pradesh have well-developed rail, bus, port, and air connectivity. Various government-funded solar projects, such as street lighting and solar water heating systems, are being implemented [111] and are accompanied by significant local community support and an abundance of workers.

6.4. Tamil Nadu $(P_4)$

Tamil Nadu has a very high level of solar radiation, which is highly favourable for installing solar PV power plants. The coastal areas around Madurai, Thoothukudi, Theni, and Chennai experience high solar radiation, making these areas ideal for setting up solar PV power plants. The region is well connected by an extensive network of roads, highways, railways, airports, and ports [112]. The region has an abundance of skilled labour, and the local community is enthusiastic about the project. There is considerable government support for the expansion of solar energy in the state [113]. Therefore, keeping these factors in mind, Tamil Nadu was selected for this study.

6.5. Karnataka $(P_5)$

Due to the diverse climate of Karnataka, many regions, especially Tumkur, Chitradurga, Kalaburagi, Kolar, Davanagere, Koppal, Ramanagar, Bagalkot, Belagavi, Vijayapura, and Bidar, are ideal for the installation of solar photovoltaic PV power plants. Karnataka’s infrastructure plays a vital role in supporting renewable energy projects. It includes Bengaluru International Airport, the airports at Mangalore, Belgaum, Bellary, Gulbarga, and Mysore, Mangalore Port, and various railway stations. The state government has promoted the expansion of solar energy through various initiatives [114]. Additionally, significant community involvement and a ready supply of skilled labour make it easier to develop and maintain solar energy projects across the state.

6.6. Maharashtra $(P_6)$

Due to its well-developed infrastructure and favourable climate, Maharashtra is an excellent location for establishing solar PV power plants. Regions such as Solapur, Dhule, Satara, Pune, Nashik, Ahmednagar, Aurangabad, and parts of Vidarbha were identified as ideal sites for solar power plants due to their high solar radiation. Maharashtra has an extensive transportation network, with major cities well connected by roads, railways, and air travel [115]. The state government has shown significant commitment to renewable energy, including the Maharashtra Solar Policy 2021 [116]. Active government support and natural advantages make it a top candidate for solar energy development.

6.7. Gujarat $(P_7)$

Gujarat is an ideal location for setting up solar PV power plants due to its rapidly developing infrastructure and favourable climate. The western part of the state, especially Kutch and the Saurashtra region (Jamnagar, Porbandar, Rajkot, and Junagadh), and nearby areas receive high solar radiation, making them suitable for solar energy production. Cities like Ahmedabad, Surat, Rajkot, Bharuch, Ankleshwar, Valsad, and Bhavnagar were identified as key sites because they receive sunlight throughout the year. Gujarat has a well-developed network of roads, railways, and airports, providing excellent connectivity for transporting solar equipment. The state government is highly committed to renewable energy [117].

6.8. West Bengal $(P_8)$

The state has a moderate climate and receives enough sunlight throughout the year, especially in Purulia, Bankura, Bardhaman (East and West), and Birbhum, making these areas suitable for solar energy production. The state is well connected with major transport facilities, making it easier to transport solar equipment and technology [118]. The West Bengal government is especially focused on developing renewable energy, particularly solar energy [119]. With government support, natural resources, and strategic efforts to create a solar-friendly environment, West Bengal is on track to play an important role in India’s renewable energy sector.

In Figure 4 below, we have marked the aforementioned alternatives on the map of India. Each black logo with the symbol of a solar panel represents a different state in India.

The study has been carried out in Indian regions, but the proposed method and analytical framework are not limited to any specific geographical area. With suitable adaptation based on relevant data and context, it can be applied in any country or region of the world and used effectively for any type of industrial location decision problem.

7. Model Structure and Data Collection

Here, we consider the problem of selecting a suitable site for constructing a solar (PV) power plant in India, taking into account economic, environmental, technical, socio-economic, and political factors. This section describes the model structure and the data collection process used in the study.

For setting up the model, we considered the twelve criteria for the present study, which were described elaborately in Section 5. Also, eight states located in India were selected as alternatives. The states were Rajasthan $(P_1)$, Uttar Pradesh $(P_2)$, Andhra Pradesh $(P_3)$, Tamil Nadu $(P_4)$, Karnataka $(P_5)$, Maharashtra $(P_6)$, Gujarat $(P_7)$, and West Bengal $(P_8)$. The purposes for selecting these states for our study were nicely described in Section 6. The proposed problem was structured in an uncertain environment. Pentagonal intuitionistic fuzzy numbers (PIFNs) was employed here to manage uncertainty-related issues. For selecting the optimal location (or alternative), two multicriteria decision-making methods were adopted: the Criteria through Inter-Criteria Correlation Technique (CRITIC) and VIsekriterijumsko Kompromisno Rangiranje (VIKOR). The CRITIC method was used to calculate the weight of the criteria, while the VIKOR method was used to rank the eight alternatives. Based on the data provided by decision makers related to this research study, the decision matrices were constructed by combining the twelve criteria and the eight alternatives. Each of the decision matrices was of order $8\times 12$ and was defined according to Equation (16).

The valuable data related to this problem in our proposed study were collected by considering several steps. Four decision-makers were employed for this study. Each decision-maker was highly experienced and well qualified in their field. For collecting data, we informed them about the main perspective, our proposed planning, and a detailed description of selected criteria and alternatives connected to the study. We arranged an interview session in either physical or online mode (if the candidate was unavailable in person), through questionnaires or verbal conversations. They provided their valuable information about our decision-making problem in linguistic terms or descriptive languages (such as highly important, very important, minor important, etc.). The decision-makers ensured that they maintained impartiality in the information they provided. Since our problem was structured in an uncertain environment, for evaluation and analysis purposes, we transformed the linguistic information provided by the decision-makers into pentagonal intuitionistic fuzzy numbers (PIFNs).

In the following, we list the designations and related fields of decision-makers (DMs) involved in our study:

(a) 

A project manager for solar power plant development;

(b) 

A senior environmental consultant with 15 years of experience;

(c) 

A Chief Executive Officer (CEO) of a renewable energy company;

(d) 

A senior energy analyst.

In

Table 4. Translation table between linguistic terms and PIFNs.

Linguistic Terms	Pentagonal Intuitionistic Fuzzy Number (PIFN)	De-i-Fuzzified Value of PIFN
Exceptionally Important (EI)	$\left[\left(7,8,9,10,11\right);0.80,0.90;0.05,0.10\right]$	$4.68$
Highly Important (HI)	$\left[\left(6,7,8,9,10\right);0.75,0.90;0.05,0.10\right]$	$4.00$
Very Important (VI)	$\left[\left(5,6,7,8,9\right);0.82,0.88;001,0.12\right]$	$3.33$
Important (I)	$\left[\left(4,5,6,7,8\right);0.70,0.90;0.05,0.10\right]$	$2.87$
Minor Important (MI)	$\left[\left(3,4,5,6,7\right);0.82,0.86;0.10,0.14\right]$	$2.32$
Weakly Important (WI)	$\left[\left(2,3,4,5,6\right);0.60,0.90;0.05,0.10\right]$	$1.71$
Negligibly Important (NI)	$\left[\left(1,2,3,4,5\right);0.69,0.85;0.10,0.15\right]$	$1.20$

, we present the converted form of the linguistic terms (or descriptive language) provided by the decision-makers mentioned above. In addition, the de-i-fuzzified values of the PIFNs obtained from the linguistic terminology provided by the decision-makers are listed in the table. These crisp values were later used for evaluation purposes.

Table 5. Decision matrices between criteria and alternatives in linguistic terms provided by four decision-makers (DMs).

Criteria vs. Alternative

$F_1$

$F_2$

$F_3$

$F_4$

$F_5$

$F_6$

$F_7$

$F_8$

$F_9$

$F_{10}$

$F_{11}$

$F_{12}$

DM 1

Rajasthan $(P_1)$

HI

EI

I

VI

MI

EI

MI

HI

MI

HI

I

HI

Uttar Pradesh $(P_2)$

I

VI

VI

HI

WI

I

I

HI

HI

I

HI

VI

Andhra Pradesh $(P_3)$

I

HI

VI

I

HI

MI

VI

VI

HI

HI

VI

VI

Tamil Nadu $(P_4)$

I

HI

VI

VI

I

I

HI

HI

I

VI

HI

HI

Karnataka $(P_5)$

VI

HI

I

HI

HI

I

HI

HI

VI

HI

VI

HI

Maharashtra $(P_6)$

HI

I

I

VI

I

I

VI

HI

VI

HI

HI

VI

Gujarat $(P_7)$

HI

EI

MI

I

I

EI

I

HI

I

HI

I

HI

West Bengal $(P_8)$

EI

HI

HI

I

MI

HI

I

VI

HI

I

HI

HI

Criteria vs. Alternative

$F_1$

$F_2$

$F_3$

$F_4$

$F_5$

$F_6$

$F_7$

$F_8$

$F_9$

$F_{10}$

$F_{11}$

$F_{12}$

DM 2

Rajasthan $(P_1)$

HI

HI

VI

I

I

HI

HI

HI

VI

HI

VI

HI

Uttar Pradesh $(P_2)$

VI

VI

I

HI

I

I

VI

HI

HI

HI

HI

HI

Andhra Pradesh $(P_3)$

HI

VI

I

I

HI

MI

VI

HI

HI

I

VI

I

Tamil Nadu $(P_4)$

I

EI

MI

HI

HI

MI

VI

VI

VI

HI

I

VI

Karnataka $(P_5)$

HI

HI

VI

VI

I

I

HI

HI

I

HI

HI

HI

Maharashtra $(P_6)$

HI

I

VI

HI

VI

VI

I

VI

MI

VI

HI

HI

Gujarat $(P_7)$

EI

HI

I

HI

HI

I

HI

HI

I

HI

HI

HI

West Bengal $(P_8)$

HI

I

VI

VI

VI

I

HI

VI

HI

I

VI

VI

Criteria vs. Alternative

$F_1$

$F_2$

$F_3$

$F_4$

$F_5$

$F_6$

$F_7$

$F_8$

$F_9$

$F_{10}$

$F_{11}$

$F_{12}$

DM 3

Rajasthan $(P_1)$

EI

HI

MI

I

VI

VI

HI

I

VI

I

MI

HI

Uttar Pradesh $(P_2)$

VI

I

HI

I

VI

I

MI

HI

VI

VI

I

HI

Andhra Pradesh $(P_3)$

HI

EI

VI

MI

VI

I

VI

I

I

I

HI

I

Tamil Nadu $(P_4)$

HI

VI

I

I

I

HI

HI

VI

VI

I

I

HI

Karnataka $(P_5)$

VI

VI

HI

HI

I

I

I

I

VI

HI

I

I

Maharashtra $(P_6)$

VI

I

I

I

HI

VI

MI

HI

I

I

VI

VI

Gujarat $(P_7)$

EI

VI

I

HI

VI

I

VI

EI

VI

VI

I

I

West Bengal $(P_8)$

VI

HI

VI

VI

I

HI

I

MI

EI

I

MI

EI

Criteria vs. Alternative

$F_1$

$F_2$

$F_3$

$F_4$

$F_5$

$F_6$

$F_7$

$F_8$

$F_9$

$F_{10}$

$F_{11}$

$F_{12}$

DM 4

Rajasthan $(P_1)$

HI

HI

EI

I

MI

I

MI

I

HI

I

I

HI

Uttar Pradesh $(P_2)$

VI

I

VI

HI

I

HI

HI

HI

I

VI

HI

I

Andhra Pradesh $(P_3)$

HI

EI

I

HI

VI

I

VI

I

VI

HI

VI

HI

Tamil Nadu $(P_4)$

I

EI

VI

I

HI

VI

I

HI

VI

HI

HI

VI

Karnataka $(P_5)$

VI

HI

I

VI

I

I

HI

VI

I

VI

I

VI

Maharashtra $(P_6)$

HI

I

I

HI

VI

HI

HI

HI

HI

I

HI

HI

Gujarat $(P_7)$

VI

EI

HI

VI

HI

VI

VI

I

VI

HI

HI

I

West Bengal $(P_8)$

EI

VI

VI

I

I

HI

VI

I

HI

I

I

EI

presents the ratings given by the decision-makers (DMs) using linguistic terms, which form the decision matrix that represents the relationship between criteria and alternatives. All the data were collected in the period from July to August 2025. These ratings reflect subjective assessments based on the professional skills and experience of DMs. Linguistic terms were used to represent qualitative assessment, which helped to better understand the decision-making process. The use of linguistic terms increased the flexibility of the model, capturing the uncertainties and nuances in human decision-making.

Remark 8.

All the information provided by the DM presented in Table 5 was unbiased and accurate and is later used in the numerical analysis section. The following steps in the decision-making process were conducted based on this information.

8. Numerical Illustration and Discussion

In this section, a detailed explanation of numerical computation and discussion is given. This section outlines the two MCDM methods CRITIC and VIKOR presented in Section 4, which combine the essential mathematical tools described in Section 3. In addition, to support and perform various analyses and calculations, we used the datasets mentioned in Section 7.

The CRITIC method was used to evaluate the weights of various criteria (described in Section 4.1) related to the current work. According to the personal opinions of the decision-makers (DMs), a set of four decision matrices was constructed using PIFNs (pentagonal intuitionistic fuzzy numbers), as outlined in Equation (16). We then combined the decision matrices into a single decision matrix using Equation (18). Then, the proposed de-i-fuzzification process was applied as described in Equation (13) to convert the PIFNs into crisp values and construct the de-i-fuzzified combined matrix as follows:

$${\mathcal{D}}_{cv}^I=\left[\begin{array}{cccccccccccc}\hfill 5.10798& 5.10798& 4.57817\hfill & 3.44407& 3.48225& 4.82781& 4.16326& 4.56753& 4.20566& 4.56753& 3.51339& 4\\ \hfill 3.49746& 3.47026& 4.20292\hfill & 4.63509& 3.40613& 4.50466& 4.08549& 4& 4.23134& 4.20293& 4.63509& 4.23134\\ \hfill 4.63509& 4.95314& 3.470256\hfill & 4.05497& 4.14135& 2.85619& 3.33333& 4.17129& 4.23134& 4.56753& 4.11420& 4.17129\\ \hfill 4.50467& 4.95314& 3.54084\hfill & 4.17129& 4.56753& 4.08549& 4.23134& 4.14135& 3.49746& 4.23134& 4.56753& 4.14135\\ \hfill 4.11420& 4.16942& 4.17128\hfill & 4.14135& 4.50467& 2.86957& 4.63509& 4.23134& 3.47026& 4.16942& 4.17129& 4.23134\\ \hfill 4.16942& 2.86957& 3.44408\hfill & 4.23134& 4.20293& 4.20293& 4.08549& 4.16942& 4.08549& 4.17129& 4.16942& 4.14135\\ \hfill 4.95314& 4.95314& 4.05497\hfill & 4.23134& 4.23134& 4.76142& 4.20293& 5.24939& 3.47026& 4.16942& 4.56753& 4.56753\\ \hfill 4.95314& 4.23134& 4.11420\hfill & 3.47026& 3.51339& 4.63509& 4.17129& 3.54084& 5.10799& 2.86957& 4.08549& 4.95314\end{array}\right]$$

(35)

In the next step, we normalised the updated decision matrix using Equation (20) and then calculated the standard deviation $({\sigma }_m)$ for each criterion. The standard deviations for each criterion are provided in

Table 6. Standard deviation $({\sigma }_m)$ for each criterion.

${\mathit{\sigma }}_1$	${\mathit{\sigma }}_2$	${\mathit{\sigma }}_3$	${\mathit{\sigma }}_4$	${\mathit{\sigma }}_5$	${\mathit{\sigma }}_6$	${\mathit{\sigma }}_7$	${\mathit{\sigma }}_8$	${\mathit{\sigma }}_9$	${\mathit{\sigma }}_{10}$	${\mathit{\sigma }}_{11}$	${\mathit{\sigma }}_{12}$
0.33694	0.36518	0.36493	0.338	0.40499	0.40615	0.27697	0.28752	0.34129	0.31409	0.32788	0.32321

below.

We then calculated the linear correlation coefficients between each pair of criteria and assessed the measure of conflict $({\rho }_m)$ generated by the criteria. The conflict measure for each criterion is presented in the following

Table 7. Measure of the conflict $({\rho }_m)$ generated by the criteria.

${\mathit{\rho }}_1$	${\mathit{\rho }}_2$	${\mathit{\rho }}_3$	${\mathit{\rho }}_4$	${\mathit{\rho }}_5$	${\mathit{\rho }}_6$	${\mathit{\rho }}_7$	${\mathit{\rho }}_8$	${\mathit{\rho }}_9$	${\mathit{\rho }}_{10}$	${\mathit{\rho }}_{11}$	${\mathit{\rho }}_{12}$
10.6921	10.385	10.7277	11.976	11.6684	10.4379	10.7615	9.6294	13.4387	12.0056	11.0783	11.0846

.

Then, the amount of information associated with each criterion $({\mathcal{C}}_m)$ was calculated. The amount of information for each criterion is listed in

Table 8. The amount of information associated with each criterion $({\mathcal{C}}_m)$.

${\mathcal{C}}_1$	${\mathcal{C}}_2$	${\mathcal{C}}_3$	${\mathcal{C}}_4$	${\mathcal{C}}_5$	${\mathcal{C}}_6$	${\mathcal{C}}_7$	${\mathcal{C}}_8$	${\mathcal{C}}_9$	${\mathcal{C}}_{10}$	${\mathcal{C}}_{11}$	${\mathcal{C}}_{12}$
3.6026	3.79234	3.91486	4.04786	4.72564	4.23935	2.98065	2.76863	4.58644	3.7708	3.63238	3.58266

below.

Lastly, the factor weight $({{\mathcal{C}}_m}^{\mathcal{W}})$ for each factor was calculated using Equation (28), and the resulting factor weights are shown in

Table 9. List of criteria weights calculated using the CRITIC method.

Criteria	Weight
Land accessibility and geographical features $(F_1)$	$0.07893$
High solar radiation $(F_2)$	$0.08308$
Ecological effect $(F_3)$	$0.08577$
Availability of labour and resources $(F_4)$	$0.08868$
Airflow speed $(F_5)$	$0.10353$
Level of dust haze $(F_6)$	$0.09288$
Local community support and approval $(F_7)$	$0.0653$
Proximity of the transmission network $(F_8)$	$0.06066$
Humidity ratio $(F_9)$	$0.10048$
Grid reliability $(F_{10})$	$0.08261$
Seamless transport network $(F_{11})$	$0.07958$
Government assistance and policies $(F_{12})$	$0.07849$

.

In Figure 5, we show the pie diagram of the criteria weights. Based on the above discussion, and as can be seen from Table 9 and Figure 5, it is clear that the most important criterion for building a solar PV power plant in India was the airflow speed $(F_5)$, with a weight of $0.10353$. This indicates that the speed of the airflow plays an important role in the efficiency and performance of the power plant.

The second most important criterion was the humidity ratio $(F_9)$ with a weight of $0.10048$, followed by the level of dust haze $(F_6)$, availability of labour and resources $(F_4)$, ecological effect $(F_3)$, high solar radiation $(F_2)$, grid reliability $(F_{10})$, seamless transport network $(F_{11})$, land accessibility and geographical features $(F_1)$, government assistance and policies $(F_{12})$, local community support and approval $(F_7)$, and proximity of the transmission network $(F_8)$.

Remark 9.

Figure 5 presents a visual representation of the criteria weights for the current study in the form of a pie chart. This diagram highlights that airflow speed $(F_5)$ is the most significant criterion for constructing a solar PV power plant in India.

The VIKOR method was applied to determine the rank of the selected alternatives or states located in the Indian territory, which was neatly described in Section 4.2. The decision matrices were formulated based on the information provided by four decision-makers. These matrices were then aggregated using Equation (18), and a de-i-fuzzified-valued aggregated matrix was constructed by de-i-fuzzifying the PIFN through the proposed de-i-fuzzification method. The de-i-fuzzified aggregated matrix is presented in Equation (19). The criteria weights are listed in Table 9.

In this study, there were four non-beneficial criteria, ecological effect $(F_3)$, airflow speed $(F_5)$, level of dust haze $(F_6)$, and humidity ratio $(F_9)$, while the remaining eight were beneficial criteria. We calculated the best and worst values by taking these beneficial and non-beneficial criteria into account. The best and worst values for each criterion are provided in

Table 10. The best values $({\delta }^{+})$ and worst values $({\delta }^{-})$ for each criterion.

	${\mathit{F}}_1$	${\mathit{F}}_2$	${\mathit{F}}_3$	${\mathit{F}}_4$	${\mathit{F}}_5$	${\mathit{F}}_6$	${\mathit{F}}_7$	${\mathit{F}}_8$	${\mathit{F}}_9$	${\mathit{F}}_{10}$	${\mathit{F}}_{11}$	${\mathit{F}}_{12}$
${\delta }^{+}$	5.10798	5.10798	3.44408	4.63509	3.40613	2.85619	4.63509	5.24938	3.47026	4.56753	4.63509	4.95314
${\delta }^{-}$	3.49746	2.86957	4.57817	3.44408	4.56753	4.82782	3.33333	3.54084	5.10798	2.86957	3.51339	4

below.

Next, we evaluated the utility value $({\mathcal{S}}_l)$, regret value $({\mathcal{R}}_l)$, and compromise value $({\mathcal{Q}}_l)$. After calculating these values, we generated the ranking of the alternatives, determining which state was most suitable for constructing the solar PV power plant. The associated data for ranking the alternatives are provided in

Table 11. Ranking of alternatives using the VIKOR technique and their corresponding data.

Alternative	${\mathcal{S}}_{\mathit{l}}$	${\mathcal{R}}_{\mathit{l}}$	${\mathcal{Q}}_{\mathit{l}}$	Ranking
Rajasthan $(P_1)$	$0.52518$	$0.09288$	$0.75482$	7
Uttar Pradesh $(P_2)$	$0.47056$	$0.07893$	$0.46039$	3
Andhra Pradesh $(P_3)$	$0.39125$	$0.06554$	$0.12326$	1
Tamil Nadu $(P_4)$	$0.38787$	$0.10353$	$0.61640$	6
Karnataka $(P_5)$	$0.42172$	$0.09793$	$0.61132$	5
Maharashtra $(P_6)$	$0.51644$	$0.08308$	$0.60819$	4
Gujarat $(P_7)$	$0.33051$	$0.08975$	$0.31862$	2
West Bengal $(P_8)$	$0.57691$	$0.10048$	$0.95986$	8
	${\mathcal{S}}^{+}=0.57961$		${\mathcal{S}}^{-}=0.33051$
	${\mathcal{R}}^{+}=0.10353$		${\mathcal{R}}^{-}=0.06554$

below.

In Figure 6, we present a graphical representation of ranking the alternatives in a column chart format. Based on the previous discussion, Table 11 and Figure 6 indicate that the state of Andhra Pradesh $(P_3)$ was the most favourable option with rank one for constructing a solar power plant in India.

Gujarat $(P_7)$, with its strategic geographical location and selected criteria, emerged as the second state for setting up power plants and obtained the second rank. It was followed by Uttar Pradesh $(P_2)$, Maharashtra $(P_6)$, Karnataka $(P_5)$, Tamil Nadu $(P_4)$, Rajasthan $(P_1)$, and West Bengal $(P_8)$. Despite being ranked lower, West Bengal $(P_8)$ retained its position as a valuable participant in the selection process. Together, these states represent a spectrum of opportunities, each contributing strengths to a broader development vision.

Remark 10.

Figure 6 provides a visual depiction of the alternative rankings for the current study, presented as a column chart. This chart clearly illustrates that Andhra Pradesh $(P_3)$ emerged as the most suitable state for establishing a solar PV power plant in India, followed by the other states in the ranking.

9. Sensitivity Analysis

In this section, we conducted a sensitivity analysis for the present work considering the following cases, which are discussed in detail.

9.1. Case 1: Removing the Criterion Local Community Support and Approval $(F_7)$

Community involvement and approval are sometimes not required due to existing regulatory frameworks or legal provisions. Thus, in this section, the criterion “Local community support and approval” was removed, and we proceeded to rank the alternatives based on the remaining criteria. The updated rankings of the alternatives (or selected states) are presented in

Table 12. Updated ranking list considering cases 1, 2, and 3 along with their initial rankings.

Alternatives	Case 1	Case 2	Case 3	Initial
Rajasthan $(P_1)$	7	7	7	7
Uttar Pradesh $(P_2)$	3	5	3	3
Andhra Pradesh $(P_3)$	1	4	1	1
Tamil Nadu $(P_4)$	5	3	5	6
Karnataka $(P_5)$	6	2	4	5
Maharashtra $(P_6)$	4	6	6	4
Gujarat $(P_7)$	2	1	2	2
West Bengal $(P_8)$	8	8	8	8

and Figure 7. As a result of the removal of the criterion “Local community support and approval $(F_7)$”, Figure 7 shows that Tamil Nadu’s rank went up by one position to fifth, while Karnataka’s rank dropped by one position to sixth.

9.2. Case 2: Impact Due to a Stock Market Crash

At the initial stage of site selection, scenarios such as a stock market crash can directly affect the financial viability of a solar PV project. Since a solar PV project requires substantial upfront investment, the impact of a stock market crash generally changes investor sentiment associated with the project and can significantly alter the financial cost structure. As a result, the attractiveness of the potential sites may also be affected. Therefore, this section describes how the impact of a stock market crash could influence the proposed project.

A stock market crash has a direct impact on a variety of criteria selected for the current study, including government assistance and policies $(F_{12})$, local community support and approval $(F_7)$, and grid reliability $(F_{10})$. During stock market downturns, economic recessions, budget cuts, subsidy cuts, and changes in government spending priorities often occur. This financial instability could result in reduced investment in community engagement, slower progress in improving grid infrastructure, and delays in integrating renewable energy into the grid.

Considering the above, in this section, the weightage of criteria for government assistance and policies $(F_{12})$, local community support and approval $(F_7)$, and grid reliability $(F_{10})$ was increased, and a modified weightage list was prepared (from Table 9; the weights of the above-mentioned criteria were doubled, while the remaining weights were kept the same, and then normalised, i.e., divided by their sum). The alternatives were then ranked based on the modified weight list. The modified weight list is given in

Table 13. Modified criteria weights based on case 2 and case 3.

Criteria	Case 2	Case 3
Land accessibility and geographical features $(F_1)$	$0.06436$	$0.05111$
High solar radiation $(F_2)$	$0.06775$	$0.10761$
Ecological effect $(F_3)$	$0.06994$	$0.11108$
Availability of labour and resources $(F_4)$	$0.07231$	$0.05743$
Airflow speed $(F_5)$	$0.08442$	$0.13409$
Level of dust haze $(F_6)$	$0.07573$	$0.12029$
Local community support and approval $(F_7)$	$0.10649$	$0.04229$
Proximity of the transmission network $(F_8)$	$0.04946$	$0.03928$
Humidity ratio $(F_9)$	$0.08193$	$0.13014$
Grid reliability $(F_{10})$	$0.13472$	$0.0535$
Seamless transport network $(F_{11})$	$0.06489$	$0.05153$
Government assistance and policies $(F_{12})$	$0.128$	$0.10166$

, and the updated rank list is presented in Table 12 and Figure 7. There were several changes in the alternative ranking due to the stock market crash. Rajasthan and West Bengal retained the same rank, while the ranks of the remaining selected states changed, either increasing or decreasing.

9.3. Case 3: Impact of Climate Change Caused by Pollution

The impact of climate change caused by pollution directly affects the environmental and technical feasibility of solar PV installations. Since site selection for a solar PV project depends on long-term, stable solar resource availability, this section discusses how pollution-driven climate change may influence PV installations to assess how sustainable and effective a site will remain under such conditions.

Due to climate change, several criteria (selected for the study) were significantly affected during the construction of solar power plants. In the following, we list which criteria strongly influenced the current project and how they affected the current project in terms of climate change due to pollution.

1. 

High solar radiation $(F_2)$: The type of solar radiation can change, which affects the efficiency of solar power generation.

2. 

Ecological effect $(F_3)$: The environmental impact of pollution affects ecosystems and biodiversity, which can affect the development of solar power plants.

3. 

Airflow speed $(F_5)$: Climate change affects the flow of air, which can increase or hinder the efficiency of solar panel cooling and energy production.

4. 

Level of dust haze $(F_6)$: An excessive amount of pollution causes high levels of dust haze in the atmosphere, which reduces the amount of sunlight reaching the solar panels.

5. 

Humidity ratio $(F_9)$: Climate change affects humidity levels, thus disrupting solar panel performance and overall energy production.

6. 

Government assistance and policies $(F_{12})$: The overall cost increases due to the impact of climate change on different aspects, resulting in delays in obtaining various government permits and benefits.

Considering the above situation, in this section, the importance of the mentioned criteria was increased, and a modified weight list was created (from Table 9, the weights of the above-mentioned criteria were doubled, while the remaining weights were kept the same, and then normalised, i.e., divided by their sum). The alternatives were ranked according to the adjusted weight list. The updated weight list is shown in Table 13, while the revised ranking is presented in Table 12 and Figure 7. The impact of climate change caused by pollution yielded significant changes in alternative rankings. Rajasthan $(P_1)$, Uttar Pradesh $(P_2)$, Andhra Pradesh $(P_3)$, Gujarat $(P_7)$, and West Bengal $(P_8)$ retained their original rank, while the rank of other states changed, either rising or falling.

Figure 7 illustrates the graphical representation of the above three cases, displaying their original rankings alongside comparisons with themselves.

10. Comparative Analysis

For the comparative analysis, we introduced two techniques. One could change the fuzzy number without altering the MCDM methodologies, i.e., we simply replaced pentagonal intuitionistic fuzzy numbers (PIFNs) by adopting conventional pentagonal fuzzy numbers (PFNs). On the other hand, we employed an MCDM technique, namely COmplex PRoportional ASsessment (COPRAS), in place of the VIKOR method, without altering the initial fuzzy numbers (PIFNs) and weight calculation method (CRITIC method). For both scenarios, we listed the new rankings of the alternatives and compared them with our initial results. The detailed description of the scenarios mentioned above is provided below.

10.1. Scenario I: Comparative Analysis When Changing the Fuzzy Numbers While Keeping the Method Unchanged

Here, we employed pentagonal fuzzy numbers (PFNs) in place of pentagonal intuitionistic fuzzy numbers (PIFNs). The MCDM methods used in the initial stage of the current study were kept as the CRITIC method and the VIKOR technique. In this scenario, we kept the methods unchanged and collected a new ranking list of the alternatives.

To convert pentagonal fuzzy numbers (PFNs) into a crisp quantity, we employed the following de-fuzzification technique for PFNs, as described in the research article by Onyenike, K. and Ojarikre, H. I. [121].

Definition 4.

De-fuzzification of PFNs:

Consider a pentagonal fuzzy number (PFN) $\tilde{\mathcal{F}}=\left[{\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,{\zeta }_5\right]$. Then, the de-fuzzified value [121], $D[\tilde{\mathcal{F}}]$ of $\tilde{\mathcal{F}}$ is given as follows:

$$\begin{array}{c}\hfill D[\tilde{\mathcal{F}}]={\displaystyle \frac{{\zeta }_1+2{\zeta }_2+2{\zeta }_3+2{\zeta }_4+{\zeta }_5}{8}}\end{array}$$

(36)

A detailed description of the data collection process for the current study is provided in Section 7. In

Table 14. Conversion table from linguistic information to pentagonal fuzzy numbers along with their de-fuzzified values.

Linguistic Information	Pentagonal Fuzzy Number (PFN)	De-Fuzzified Value
Exceptionally Important (EI)	$\left[12,13,14,15,16\right]$	14
Highly Important (HI)	$\left[11,12,13,14,15\right]$	13
Very Important (VI)	$\left[10,11,12,13,14\right]$	12
Important (I)	$\left[9,10,11,12,13\right]$	11
Minor Important (MI)	$\left[8,9,10,11,12\right]$	10
Weakly Important (WI)	$\left[7,8,9,10,11\right]$	9
Negligibly Important (NI)	$\left[6,7,8,9,10\right]$	8

below, we present the data that have been transformed into PFNs from the linguistic information provided by the selected decision-makers.

Using the CRITIC method (described in Section 4.1), the weights of the criteria were calculated. This method objectively assigns weights utilising the intensity of the contrast and conflict between criteria. The resulting criteria weights are presented in

Table 15. Comparison of initial and current criteria weights calculated using the CRITIC method.

Criteria	Weight
	Current	Initial
Land accessibility and geographical features $(F_1)$	0.078	0.079
High solar radiation $(F_2)$	0.082	0.083
Ecological effect $(F_3)$	0.080	0.086
Availability of labour and resources $(F_4)$	0.083	0.089
Airflow speed $(F_5)$	0.081	0.104
Level of dust haze $(F_6)$	0.082	0.093
Local community support and approval $(F_7)$	0.098	0.065
Proximity of the transmission network $(F_8)$	0.076	0.061
Humidity ratio $(F_9)$	0.088	0.100
Grid reliability $(F_{10})$	0.070	0.083
Seamless transport network $(F_{11})$	0.081	0.080
Government assistance and policies $(F_{12})$	0.102	0.078

below, along with their initial weights. These weights were used to determine the ranking of the alternatives in the following step.

For the decision analysis, the VIKOR method was applied to rank the available alternatives. A detailed explanation of the methodological framework of the VIKOR strategy is given in Section 4.2. Based on the VIKOR method, the alternatives were ranked, and the updated rankings are listed in

Table 16. Comparison between initial and updated rankings of alternatives in the PIFN-based VIKOR framework.

Alternatives	Rank
	Updated	Initial
Rajasthan $(P_1)$	5	7
Uttar Pradesh $(P_2)$	7	3
Andhra Pradesh $(P_3)$	6	1
Tamil Nadu $(P_4)$	1	6
Karnataka $(P_5)$	2	5
Maharashtra $(P_6)$	4	4
Gujarat $(P_7)$	3	2
West Bengal $(P_8)$	8	8

, alongside the initial rankings obtained using PIFNs within the VIKOR framework.

By referencing Table 16, we created a chart for comparing the initial and updated rankings of the alternatives, as shown in Figure 8. The updated rankings are displayed in columns, whereas the initial rankings are arranged in a single line.

Remark 11.

Table 16 and Figure 8 in the above discussion represent a comparison between the initial and updated rankings of eight alternatives within the PIFN-based VIKOR framework. Some changes can be noticed in the ranking positions. Tamil Nadu $(P_4)$ displayed a significant improvement, rising from an initial rank of six to the top position. Karnataka $(P_5)$ also improved significantly, moving from the fifth to second place. In contrast, Andhra Pradesh $(P_3)$, which was initially in first place, dropped to sixth place. Similarly, Uttar Pradesh $(P_2)$ fell from third to seventh. Maharashtra $(P_6)$ retained its fourth position, showing consistency in its assessment. Gujarat $(P_7)$ slipped slightly from second to third place, and Rajasthan $(P_1)$ improved slightly from seventh to fifth place. West Bengal $(P_8)$ remained at the eighth position, showing no change.

10.2. Scenario II: Comparative Analysis Using the COPRAS Method Instead of the VIKOR Technique

Here, we introduced the multi-criteria decision-making COPRAS methodology, instead of the VIKOR technique. The adopted fuzzy numbers (PIFNs) and weight calculation method (CRITIC) were maintained as in the original setup.

To determine the rank of the alternatives, we utilised the initial weights of the criteria, as shown in Table 9. For the COPRAS approach, we followed the steps described in several established research works [122,123]. Applying the COPRAS method, we obtained a new ranked list of alternatives, as presented in

Table 17. Comparison of current and initial ranks of alternatives performed by the COPRAS method.

Alternatives	Rank
	Current	Initial
Rajasthan $(P_1)$	4	7
Uttar Pradesh $(P_2)$	2	3
Andhra Pradesh $(P_3)$	7	1
Tamil Nadu $(P_4)$	6	6
Karnataka $(P_5)$	5	5
Maharashtra $(P_6)$	1	4
Gujarat $(P_7)$	8	2
West Bengal $(P_8)$	3	8

below.

Using the data in Table 17, we created a graphical representation comparing the ranks of the alternatives obtained by the COPRAS method with the initial ranks determined by the VIKOR technique, as shown in Figure 9. In the graph, the current ranks are displayed in columns, and a single line indicates the initial ranks.

Remark 12.

From the above discussion, Table 17 and Figure 9 illustrate the differences between the current and initial rankings of various alternatives using the COPRAS method. Maharashtra $(P_6)$ improved the most, moving from fourth place to first. West Bengal $(P_8)$ also showed significant improvement, moving from last place (eighth) to third place. Rajasthan $(P_1)$ improved slightly from seventh to fourth place, and Uttar Pradesh $(P_2)$ improved slightly from third to second place. However, some changes were also negative. Andhra Pradesh $(P_3)$ dropped sharply from first place to seventh. Based on the new assessment, Gujarat $(P_7)$ dropped from second to last place (eighth), showing a decline in performance. Tamil Nadu $(P_4)$ and Karnataka $(P_5)$ retained their previous positions, remaining in sixth and fifth place, indicating that their performance was stable.

11. Research Implication

There are different angles from which the implications of this work can be analysed. This research work gives us a clear view of how effectively the pentagonal intuitionistic fuzzy numbers (PIFNs) can be incorporated in a CRITIC–VIKOR based approach. This model offers an approach to site selection problems in which uncertainty can be incorporated into decision-making. From this study, a scientific and realistic method was developed for identifying the most suitable site for establishing a new solar PV power plant in India. This work also promotes the sustainable expansion of solar energy through the installation of new solar power plants, which enhances energy security. It also helps increase employability and overall economic growth. This study helps develop renewable energy policies in India and mitigate environmental impacts.

Our proposed method includes an MCDM-based model that incorporates pentagonal intuitionistic fuzzy numbers (PIFNs) and a new de-i-fuzzification technique. It helps convert the judgments of decision experts into actionable site scores to establish a new solar photovoltaic (PV) power plant in India. This method improves the ranking process, which becomes more flexible and reliable. It also makes the result capable of integrating with GIS layers and being used by policymakers.

12. Conclusions and Future Research Scope

This research developed a framework for selecting and ranking locations for constructing solar photovoltaic (PV) power plants in various states in India, each with distinct regional characteristics and solar energy potential. The weights of the criteria were determined using the CRITIC method, and the optimisation of the alternatives was performed using the VIKOR-based MCDM methodology. To address internal uncertainties, pentagonal intuitionistic fuzzy numbers (PIFNs) were employed, along with a novel de-i-fuzzification approach based on the relative difference between two real numbers. All necessary data were collected from four decision-makers (DMs) or experts, in linguistic form, to ensure impartial assessments. Thus, this study is not only relevant to India but also applicable to industrial location planning worldwide, offering an effective and practical approach to selecting optimal locations for solar photovoltaic (PV) power plants.

We observe that airflow speed $(F_5)$ had the highest weight among the criteria, while the proximity of the transmission network $(F_8)$ was the least weighted criterion in this study. After ranking the alternatives according to the aforementioned criteria, it was observed that Andhra Pradesh $(P_3)$ received the highest rank, whereas West Bengal $(P_8)$ retained the lowest rank. To assess the stability of the results, a sensitivity analysis was performed across three practical cases. A comparative analysis was also conducted. Lastly, we can talk about the fact that according to the DM’s information, Andhra Pradesh $(P_3)$ was an ideal state for constructing a solar PV power plant. Other states were also suitable for the construction of solar PV power plants. Since West Bengal $(P_8)$ retained the lowest rank, the state has better opportunities in the future.

12.1. Limitation of the Study

The present study identified a suitable location for constructing a solar PV power plant in India but did not consider countries with different geographical environments. The proposed method and analytical framework can be appropriately adapted to relevant data and context and applied in any country or region of the world, and they can be effectively used for any type of industrial location decision problem. The study was conducted using a few decision-makers and a small dataset. Only 12 criteria and 8 alternatives were considered in the survey, and some less essential criteria were omitted. Integrating PIFN causes some complexity, but it helps manage uncertainty. The study could not discuss future changes such as climate change or technological improvements and focused on a few stakeholders.

12.2. Future Research Scope

In the future, various avenues exist to further develop and improve this research. The present study employed two MCDM methods, namely, CRITIC and VIKOR. In addition to these two approaches, researchers can apply other MCDM methodologies, such as AHP, entropy, TOPSIS, MARCOS, MABAC, MAIRCA, etc. Additionally, a combination of genetic algorithms and fuzzy logic can yield improved results. Researchers can use geospatial information systems (GISs) and remote sensing data to select the correct location. The study can be extended by adding supplementary criteria and sub-criteria related to the model, as well as different types of fuzzy numbers, such as spherical fuzzy numbers, hexagonal fuzzy numbers, and heptagonal fuzzy numbers.