Pythagorean Fuzzy SWARA–VIKOR Framework for Performance Evaluation of Solar Panel Selection

Abstract

The age of industrialization and modernization has increased energy demands globally. Solar energy has been recognized as an inexhaustible source of energy and has been applied for desalination and electricity generation. Among different non-conventional energy resources, Solar Energy (SE) is one of the main contributors to the global energy system. A photovoltaic system (PS) is applied to produce SE using photovoltaic cells. The selection of a solar panel includes many intricate factors involving both subjective and quantifiable parameters; therefore, it can be regarded as a complex Multi-Criteria Decision-Making (MCDM) problem. As the uncertainty commonly occurs in the selection of an ideal solar panel, the theory of Pythagorean fuzzy sets has been proven as one of the flexible and superior tools to deal with the uncertainty and ambiguity that arise in real-life applications. The aim of the study is to present an MCDM framework for solving the Solar Panel Selection (SPS) problem within the Pythagorean fuzzy (PF) environment. For this, first, a new integrated method is proposed based on the Stepwise Weight Assessment Ratio Analysis (SWARA) and VlseKriterijumska Optimizcija I Kaompromisno Resenje (VIKOR) approaches in the Pythagorean fuzzy sets (PFSs) context. In the proposed approach, subjective weights of the evaluation criteria are calculated by the SWARA method, and the preference order of alternatives is decided by the VIKOR method in the PF context. The criteria weights evaluated by this approach involve the imprecision of experts’ opinions, which makes them more comprehensible. The computational procedure of the proposed methodology is established through a case study of the SPS problem under PF environment, which proves the applicability and efficiency of the proposed method. Furthermore, this study performs sensitivity analysis to reveal the stability of the developed framework. This analysis signifies that the solar panel option R₄ constantly secures its highest ranking despite how the parameter values vary. In addition, a comparative study is discussed to analyze the validity of the obtained result. The results show that the proposed approach is more efficient and applicable with previously developed methods in the PFS environment.

1. Introduction

Sustainability is defined as an integrated economic, social, environmental, and technological development that meets the needs of the present without compromising the ability of future generations to meet their own needs [1]. The basic principle of sustainable development is that natural resources can be exploited only to the level that provides their reproduction. To ensure a sustainable future, there is an increasing awareness in the world for the development of sustainable energy resources. Sustainable energy is a form of energy that is non-polluted and long-lasting with much less emission of carbon and greenhouse gases. It includes solar, hydroelectricity, geothermal, biomass, wind, wave, and tidal energies. Solar energy is one of the most emerging alternative sources of energy that is widely available, environmentally friendly, and can be used indefinitely without diminishing its future availability [2,3]. Nowadays, solar energy has been recognized as an important sector to support the sustainable development of various countries.

Photovoltaic (PV) systems, which contain photovoltaic cells, are known as a method for generating power in an efficient way. One of the most imperative modules of a solar power plant is the solar panel (SP). SPs have usually been utilized for lower-scale energy production, predominantly for business or residential utilization in multiplexes or individual buildings. They collect clean renewable energy in the form of sunlight and convert that light into electricity, which can be employed to supply power for electrical loads [4]. Solar panels are comprised of several individual solar cells, which are themselves composed of layers of silicon, phosphorous (which provides the negative charge), and boron (which provides the positive charge). The cost of an SP varies with reference to its size, dimension, and strength. As the SP selection (SPS) depends on several tangible and intangible factors/criteria, therefore, it can be scrutinized as a Multi-Criteria Decision-Making (MCDM) problem. Thus, the selection of most suitable solar panel is one of the most significant decisions in the photovoltaic system design. In the recent past, several studies have established diverse decision support systems (DSS) to evaluate solar energy systems. For instance, Ramachandra et al. [5] developed an adaptable DSS to evaluate the solar system potential. Charabi and Gastli [6] proposed an integrated technique to assess the solar PV power plant location selection problem in Oman. Cavallaro [7] established a Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS)-based framework to assess the thermal storage of solar PV plants. Beltran [8] studied an analytic network process for selecting solar projects. Khan and Rathi [9] introduced a method for assessing solar PV plant locations. Kowalski et al. [10] discussed an innovative MCDM procedure for energy problems. Tavana et al. [11] proposed a framework to evaluate the solar farm site selection problem. Ozdemir and Sahin [12] developed an MCDM model to solve the solar PV power plant location assessment problem. Wang et al. [13] studied an MCDM framework to evaluate the SPS problem in Vietnam. The evaluation of the SPS problem involves several uncertain characteristics and a lack of information/data. The conception of fuzzy sets (FSs) originated with Zadeh [14]; they have been widely applied to deal with uncertainties that arise in practical problems. Many scholarly studies have utilized the concept of FSs in order to cope with uncertainties that arise in the ranking of solar panels [15,16,17].

FS theory has successfully been applied in MCDM problems because human judgments are generally imprecise when selecting an alternative concerning multiple criteria with different levels of significance. Later, Atanassov [18] generalized the conception of FSs to intuitionistic fuzzy sets (IFSs), depicted by membership degree (MD) and non-membership degree (ND); this satisfies a constraint that the addition of its MD and ND is less than or equal to 1. After they came into existence, researchers have made their efforts to develop new information measures [19,20,21] and aggregation operators [22,23,24] within the context of IFSs. Due to their potential for solving uncertain real-world problems, IFSs have been broadly utilized in medical diagnosis [25,26], image processing [27,28], and decision-making problems [20,29].

However, there may be a case in the MCDM approach wherein the decision experts (DEs) may give a value, to which an option $R_i$ holds the attribute $T_j,$ of $\frac{\sqrt{3}}{2}$, and the value of an option $R_i$ that nullifies the attribute is $\frac{1}{2}.$ As a result, it can be simply observed that $\frac{\sqrt{3}}{2}+ \frac{1}{2} > 1$, and thus, the IFS failed to deal with these circumstances. Recently, Refs. [4,5] introduced the notion of Pythagorean fuzzy sets (PFSs), depicted by the MD and ND, which satisfies the requirement that the square sum of MD and ND is less than or equal to 1 [30]. Therefore, the above-mentioned situation can be systematically handled by PFSs. Due to the increasing complexity and time limitations, the theory of PFSs has been taken into account by several researchers for handling uncertainty and imprecision in a more adaptable way. Yager [31] and Yager and Abbasov [32] studied the fundamental concepts associated with PFSs and explained a relation between complex numbers and Pythagorean fuzzy numbers (PFNs). Zhang and Xu [33] presented the addition, multiplication, union, and intersection operations for PFNs. Gou et al. [34] defined the subtraction and division operations of PFNs and also studied their properties. Apart from them, several researchers have incorporated the idea of PFS theory into information measures [35,36] and aggregation operators [37,38] and have utilized them to handle real-life MCDM problems. In a similar way, there is no study in the literature regarding the assessment of the solar panel selection problem using PFSs.

The significance degrees/weights of the criteria are one of the important concerns during the process of MCDM. In the literature, two types of criteria weights are discussed, which are objective and subjective weights [39]. The objective weights are obtained from the information of decision matrices, while the subjective weights are assessed through the knowledge presented by the DEs [40]. For evaluating objective criteria weights, various authors have developed different procedures [41,42,43]. For determining subjective criteria weights, Kersuliene et al. [44] proposed a new efficient method, named the Stepwise Weight Assessment Ratio Analysis (SWARA) approach. Karabasevic et al. [45] discussed a structure for the assessment of personnel based on the Additive Ratio Assessment (ARAS) and SWARA approaches within the FS context. Mardani et al. [46] discussed a thorough review of the SWARA and Weighted Aggregated Sum Product Assessment (WASPAS) approaches and their applications in diverse fuzzy environments. Maghsoodi et al. [47] suggested a hybrid approach by combining the SWARA and Multi-Objective Optimization on the basis of Ratio Analysis plus full multiplicative form (MULTIMOORA) methods for the evaluation of the renewable energy technology selection problem. Ghorabaee et al. [48] introduced a fuzzy hybrid method based on the SWARA, Criteria Importance Through Intercriteria Correlation (CRITIC) method, and Evaluation Based on Distance from Average Solution (EDAS) techniques to handle MCDM problems and then them applied to assess construction equipment in view of sustainability dimensions. Rani and Mishra [49] studied a hybrid method by employing the SWARA and VlseKriterijumska Optimizcija I Kaompromisno Resenje (VIKOR) approaches to deal with the eco-industrial thermal power plant selection problem within a single-valued neutrosophic fuzzy environment.

The VIKOR approach, which originated with Opricovic [6], is a useful and flexible compromise programming-based framework to tackle MCDM problems. The main objective of the VIKOR technique is to present compromise solution(s) from the $L_p-$ metric, utilized as an aggregation function [50]. Numerous authors have applied the conventional VIKOR approach in various fields [51,52,53]. Mardani et al. [54] presented a comprehensive review on the VIKOR method and also discussed its applications. To deal with the uncertainty that arises in MCDM problems, Zhang and Xing [55] suggested an innovative probabilistic linguistic VIKOR methodology to evaluate green supply chain initiatives. Suh et al. [56] discussed an innovative fuzzy VIKOR model to solve the mobile service quality assessment problem. Krishankumar et al. [57] studied a transformation-procedure-based VIKOR method and applied it to solve a personnel selection problem in an IFS context. Rani et al. [43] presented an integrated VIKOR framework based on entropy and divergence measures within a PFSs environment and then employed it in renewable energy technology selection in India. Phochanikorn and Tan [58] suggested a hybrid approach based on the Decision-Making Trial and Evaluation Laboratory (DEMATEL), Analytic Hierarchical Process (AHP), and VIKOR methods, and applied it to evaluate a sustainable supplier selection problem within an intuitionistic fuzzy environment. Salimi et al. [59] recommended an integrated AHP- and VIKOR-based approach to explore the role of mass media advertising types in improving the water consumption pattern in Iran. Recently, several existing studies [60,61,62,63] have extended the VIKOR technique in different fuzzy environments.

Nevertheless, the VIKOR technique has not been combined with the SWARA method within the context of PFSs, though PFSs have been proven as one of the valuable tools to handle with the uncertainty and vagueness that occur in real-life concerns. Consequently, the present study focuses on PFSs. Though various authors have concentrated on the selection of renewable energy resources in PFS contexts, none have studied the SPS problem in this environment. Existing literature shows that there is a need to select the appropriate type of solar panel to generate electricity. The solar panel selection process consists of many objective and subjective attributes that have conflicting goals. In addition, the precision of the assessment procedure is dependent on the nature of the solution methodology implemented. Thus, the above-mentioned problem requires a systematic and suitable approach to evaluate the solar panels. To address this concern, an integrated PF–SWARA–VIKOR method is developed that can successfully tackle the inherent uncertainty and the hesitancy in DEs’ opinions in the evaluation of the solar panel selection problem. The contributions of the present study are:

(a)

An integrated Pythagorean fuzzy–SWARA–VIKOR (PF–SWARA–VIKOR) framework is proposed.

(b)

The PFS-based SWARA method is utilized to assess the criteria weights.

(c)

A problem regarding the selection of solar panels is presented and evaluated by utilizing the proposed PF–SWARA–VIKOR method, which reveals the applicability of the introduced approach.

(d)

A comparative study and sensitivity analysis are also discussed to show the usefulness of the introduced approach.

The rest of the work is constructed as follows. Section 2 describes the elementary concepts associated with PFSs. Section 3 proposes the new PF–SWARA–VIKOR framework to tackle the MCDM problems with PFSs. Section 4 implements the developed framework in an empirical study of solar panel selection, which shows the applicability and strength of the developed framework. In addition, sensitivity analysis and a comparative study are presented to validate the stability of the outcomes. Section 5 discusses the concluding remarks of the whole study.

2. Preliminaries

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Here, we mention some essential definitions of PFSs.

Definition 1 [30,31].

A PFS$Y$in a finite universal set$V$is presented as

$$Y = \left\{\langle v_i, Y\left({\mu }_Y(u_i), {\nu }_Y(v_i)\right)\rangle | v_i \in V\right\},$$

(1)

where${\mu }_Y : V \to \left[0, 1\right]$and${\nu }_Y : V\to \left[0, 1\right]$represent the MD and ND of an object$v_i \in V$to$Y,$respectively, which satisfies a condition$0 \le {\left({\mu }_Y\left(v_i\right)\right)}^2 + {\left({\nu }_Y\left(v_i\right)\right)}^2 \le 1.$For each$v_i \in V,$the function${\pi }_Y(v_i) = \sqrt{1 - {\mu }_Y^2(v_i) - {\nu }_Y^2(v_i) }$is called the hesitation degree. The Pythagorean fuzzy number (PFN) [33] is defined by$\eta = Y\left({\mu }_{\eta }, {\nu }_{\eta }\right)$, which holds${\mu }_{\eta }, {\nu }_{\eta } \in \left[0, 1\right]$and$0 \le {\mu }_{\eta }^2 + {\nu }_{\eta }^2 \le 1.$

Definition 2 [33].

Suppose$\eta = Y\left({\mu }_{\eta }, {\nu }_{\eta }\right)$to be a PFN. The score function and the accuracy function of$\eta$is described as

$$\mathbb{S}\left(\eta \right)={\left({\mu }_{\eta }\right)}^2-{\left({\nu }_{\eta }\right)}^2,\hslash \left(\eta \right)={\left({\mu }_{\eta }\right)}^2+{\left({\nu }_{\eta }\right)}^2, \mathit{where} \mathbb{S}\left(\eta \right)\in \left[-1, 1\right] \mathit{and} \hslash \left(\eta \right)\in \left[0,1\right].$$

(2)

Since$\mathbb{S}\left(\eta \right)\in \left[-1,1\right],$therefore, an improved score function of PFN is presented.

Definition 3.

Assume that$\eta = Y\left({\mu }_{\eta }, {\nu }_{\eta }\right)$is a PFN. Then,the normalized score and uncertainty functions of$\eta$are described as

$${\mathbb{S}}^{\ast }\left(\eta \right)=\frac{1}{2}\left(\mathbb{S}\left(\eta \right)+1\right),{\hslash }^{°}\left(\eta \right)=1-\hslash \left(\eta \right),\mathrm{such} \mathrm{that} {\mathbb{S}}^{\ast }\left(\eta \right), {\hslash }^{°}\left(\eta \right)\in \left[0,1\right].$$

(3)

For any two PFNs${\eta }_1= Y\left({\mu }_{{\eta }_1}, {\nu }_{{\eta }_1}\right)$and${\eta }_2= Y\left({\mu }_{{\eta }_2}, {\nu }_{{\eta }_2}\right)$

if${\mathbb{S}}^{\ast }\left({\eta }_1\right)>{\mathbb{S}}^{\ast }\left({\eta }_2\right),$then${\eta }_1>{\eta }_2,$

if${\mathbb{S}}^{\ast }\left({\eta }_1\right)={\mathbb{S}}^{\ast }\left({\eta }_2\right),$then

if${\hslash }^{°}\left({\eta }_1\right)>{\hslash }^{°}\left({\eta }_2\right),$then${\eta }_1<{\eta }_2;$

if${\hslash }^{°}\left({\eta }_1\right)={\hslash }^{°}\left({\eta }_2\right),$then${\eta }_1={\eta }_2.$

Definition 4 [30,31].

Let$\eta = Y\left({\mu }_{\eta }, {\nu }_{\eta }\right),$${\eta }_1 = Y\left({\mu }_{{\eta }_1}, {\nu }_{{\eta }_1}\right)$, and${\eta }_2 = Y\left({\mu }_{{\eta }_2}, {\nu }_{{\eta }_2}\right)$be the PFNs. Then, the following expressions are defined as

$$\begin{array}{c}{\eta }^c = Y\left({\nu }_{\eta }, {\mu }_{\eta }\right);\\ {\eta }_1 \oplus {\eta }_2 = Y\left(\sqrt{{\mu }_{{\eta }_1}^2 + {\mu }_{{\eta }_2}^2 - {\mu }_{{\eta }_1}^2 {\mu }_{{\eta }_2}^2}, {\nu }_{{\eta }_1} {\nu }_{{\eta }_2}\right);\\ {\eta }_1 \otimes {\eta }_2 = Y\left({\mu }_{{\eta }_1} {\mu }_{{\eta }_2}, \sqrt{{\nu }_{{\eta }_1}^2 + {\nu }_{{\eta }_2}^2 - {\nu }_{{\eta }_1}^2 {\nu }_{{\eta }_2}^2}\right);\\ \lambda \eta = Y\left(\sqrt{1-{\left(1- {\mu }_{\eta }^2\right)}^{\lambda }} , {\left({\nu }_{\eta }\right)}^{\lambda }\right), \lambda > 0;\\ {\eta }^{\lambda } = Y\left({\left({\mu }_{\eta }\right)}^{\lambda }, \sqrt{1-{\left(1- {\nu }_{\eta }^2\right)}^{\lambda }}\right), \lambda > 0.\end{array}$$

Definition 5 [33].

Let${\eta }_1 = Y\left({\mu }_{{\eta }_1}, {\nu }_{{\eta }_1}\right)$and${\eta }_2 = Y\left({\mu }_{{\eta }_2}, {\nu }_{{\eta }_2}\right)$be the PFNs. Then, the distance between${\eta }_1$and${\eta }_2$is given by

$$D_h\left({\eta }_1, {\eta }_2\right)=\frac{1}{2}\left(\left|{\mu }_{{\eta }_1}^2-{\mu }_{{\eta }_2}^2\right|+\left|{\nu }_{{\eta }_1}^2-{\nu }_{{\eta }_2}^2\right|+\left|{\pi }_{{\eta }_1}^2-{\pi }_{{\eta }_2}^2\right|\right).$$

(4)

3. Proposed Pythagorean Fuzzy–SWARA–VIKOR Method

Decision-making processes comprise a logical and scientific way for choosing a feasible course of action among multiple options. When we consider only one criterion for each alternative, the problem is referred to as single-criterion decision-making (SCDM); SCDM turns out to be less complicated because the decision can be constructed implicitly by choosing the optimal one under the best single criterion. Nevertheless, numerous real-life decision-making problems are evaluated under multiple criteria. Such problems turn into MCDM processes, where various MCDM approaches utilize the importance (i.e., weights) vectors of criteria.

Next, an integrated framework based on SWARA and VIKOR methods is introduced in a PFS environment and is named as the PF–SWARA–VIKOR method. In this framework, the subjective criteria weights are estimated by the SWARA method. The main advantage of the SWARA procedure is its ability to estimate the accuracy of the opinions of decision experts (DEs) regarding the weights assigned by the SWARA procedure. The VIKOR method [50] is a compromise programming-based technique to evaluate the compromise solution. Thus, the proposed study combines these two methods within the concept of PFSs, which determines the subjective criteria weights and then evaluates the preference order of the options, respectively. Brief descriptions of PF–SWARA–VIKOR are presented below:

Step I: Construct a decision matrix.

In the MCDM procedure, assume that $R = \left\{R_1, R_2, \dots , R_m\right\}$ is a set of ‘m’ alternatives and $T= \left\{T_1, T_2, \dots , T_n\right\}$ is a set of ‘n’ criteria. A set of DEs $E=\left\{E_1, E_2, \dots , E_l\right\}$ has been formed to obtain desirable alternative(s). Let $N = \left(g_{ij}^{(k)}\right), i =1(1)m, j =1(1)n$ be a decision matrix expressed by the DEs, wherein $g_{ij}^{(k)}$ presents the evaluation of an alternative R_i concerning the criteria $T_j; j= 1(1)n$ for the k^th DE.

Step II: Evaluate the DEs’ weights.

The computation of the significance degrees of the DEs is an important concern in the process of MCDM. For evaluation of the k^th DE, let $E_k = Y\left({\mu }_k, {\nu }_k\right)$ be the Pythagorean fuzzy number, then the weight computation formula for k^th DE is presented as follows:

$${\omega }_k = \frac{\left({\mu }_k^2 + {\pi }_k^2 \times \left(\frac{{\mu }_k^2}{{\mu }_k^2 + {\nu }_k^2}\right)\right)}{{\displaystyle \sum _{k = 1}^{\ell }\left({\mu }_k^2 + {\pi }_k^2 \times \left(\frac{{\mu }_k^2}{{\mu }_k^2 + {\nu }_k^2}\right)\right)}}, k=1(1)\ell ; {\omega }_k \ge 0, {\displaystyle \sum _{k =1}^{\ell }{\omega }_k = 1.}$$

(5)

Step III: Construct the aggregated Pythagorean fuzzy decision (APF-D) matrix.

To form the APF-D matrix, each single decision matrix is required to be united in one decision matrix by using the DEs’ opinions. To do this, a Pythagorean fuzzy weighted averaging (PFWA) [31] operator is utilized, and then $\mathbb{Z}={\left(z_{ij}\right)}_{m\times n}$ is the APF-D matrix, where

$$z_{ij}=Y\left({\mu }_{ij}, {\nu }_{ij}\right)=PFW{A}_{\lambda } \left(g_{ij}^{(1)},g_{ij}^{(2)},\dots ,g_{ij}^{(\ell )}\right)=Y \left(\sqrt{1-{\displaystyle \prod _{k= 1}^{\ell }{\left(1- {\mu }_k^2\right)}^{{\omega }_k}}} , {\displaystyle \prod _{k=1}^{\ell }{\left({\nu }_k\right)}^{{\omega }_k}}\right).$$

(6)

Step IV: Evaluate the normalized APF-D matrix.

In the decision-making process, the APF-D matrix $\mathbb{Z}={\left(z_{ij}\right)}_{m \times n}$ is converted into a normalized APF-D matrix $\mathbb{N}={\left({\tilde{z}}_{ij}\right)}_{m\times n},$ where

$${\tilde{z}}_{ij}=Y\left({\tilde{\mu }}_{ij},{\tilde{\nu }}_{ij}\right)=\{\begin{array}{ll}{z}_{ij}=Y\left({\mu }_{ij},{\nu }_{ij}\right),& j\in T_b\\ {\left(z_{ij}\right)}^c=Y\left({\nu }_{ij},{\mu }_{ij}\right),& j\in T_n\end{array};\hspace{1em}i=1\left(1\right)m,$$

(7)

where T_b and T_n denote the beneficial and non-beneficial criterion sets, respectively.

Step V: Calculate the criteria weights.

The SWARA procedure starts to rank the criteria, and directly compares the upper- to lower-ranking criteria pair-wise. Then, a comparative coefficient is evaluated, and the weight is decided and measured for handling decision-making problems. Estimation of criteria weights using SWARA is done using the following steps:

Step V-A: Calculate the crisp values. Score values ${\mathbb{S}}^{\ast }\left({\tilde{z}}_{kj}\right)$ of PFNs are computed by Equation (3) given in Definition 3.

Step V-B: Preference order of the criteria. The criteria are arranged according to the DE’s preferences from the most to the least significant criterion.

Step V-C: Evaluate the comparative significance of score value. The comparative significance is determined from the criteria that are preferred in the second place, and successive comparative significance is evaluated by differencing criterion j and j − 1.

Step V-D: Compute the comparative coefficient. The coefficient $k_j$ is given by

$$k_j=\{\begin{array}{ll}1,& j=1\\ s_j+1,& j>1,\end{array}$$

(8)

where s_j presents the comparative significance of score value [41].

Step V-E: Estimate the weight. The recalculated weight p_j is defined by

$$p_j=\{\begin{array}{ll}1,& j=1,\\ \frac{k_{j-1}}{k_j},& j>1.\end{array}$$

(9)

Step V-F: Evaluate the criteria weights. The criteria weights are defined by

$$w_j=\frac{p_j}{{\displaystyle {\sum }_{j=1}^n{p}_j}}.$$

(10)

Step VI: Find the best and worst values.

In the developed framework, the best and worst values are computed in terms of the PF-ideal solution (PF-IS) and the PF-anti-ideal solution (PF-AIS). Let ${\sigma }_j^{+}$ and ${\sigma }_j^{-}$ denote the PF-IS and PF-AIS, respectively; they are calculated as follows:

$${\sigma }_j^{+} =Y\left({\tilde{\mu }}_j^{+},{\tilde{\nu }}_j^{+}\right) =\{\begin{array}{ll}Y\left(\max {\tilde{\mu }}_{ij},\min {\tilde{\nu }}_{ij}\right),& j\in T_b\\ Y\left(\min {\tilde{\mu }}_{ij},\max {\tilde{\nu }}_{ij}\right),& j\in T_n\end{array},$$

(11)

$${\sigma }_j^{-} = Y\left({\tilde{\mu }}_j^{-},{\tilde{\nu }}_j^{-}\right)=\{\begin{array}{ll}Y\left(\min {\tilde{\mu }}_{ij},\max {\tilde{\nu }}_{ij}\right),& j\in T_b\\ Y\left(\max {\tilde{\mu }}_{ij},\min {\tilde{\nu }}_{ij}\right),& j\in T_n\end{array}.$$

(12)

Step VII: Calculate the group utility, individual regret, and compromise measure.

In the present method, the group utility and individual regret of each alternative R_i are evaluated by employing the Hamming distance measure given in Equation (4). The group utility, individual regret, and compromise degree of the options are computed by using the following procedures:

$$S_i = L_{1, i} = {\displaystyle \sum _{j =1}^n{w}_j \frac{D_h\left({\sigma }_j^{+}, {\tilde{z}}_{ij}\right)}{D_h\left({\sigma }_j^{+}, {\sigma }_j^{-}\right)}},$$

(13)

$$I_i = L_{\infty , i} = \underset{1 \le j \le n}{\max }\left(w_j \frac{D_h\left({\sigma }_j^{+}, {\tilde{z}}_{ij}\right)}{D_h\left({\sigma }_j^{+}, {\sigma }_j^{-}\right)}\right),$$

(14)

$$Q_i = \tau \frac{\left(S_i- S^{+}\right)}{\left(S^{-} - S^{+}\right)}+ \left(1- \tau \right) \frac{\left(I_i- I^{+}\right)}{\left(I^{-} - I^{+}\right)}.$$

(15)

where $S^{+} = \underset{i}{\min } S_i,$ $S^{-} = \underset{i}{\max } S_i,$ $I^{+} = \underset{i}{\min } I_i,$ $I^{-} = \underset{i}{\max } I_i$, and $\tau$ is the coefficient of the strategy of the majority of criteria (or maximum group utility), while $\left(1-\tau \right)$ is the coefficient of the strategy of the individual regret.

Step VIII: Estimate the ranking of the options.

On the basis of decreasing values of $S_i, I_i$ and $Q_i$ determine the preference values of the alternatives. The minimum value of $Q_i$ denotes the most desirable choice.

Step IX: Find the compromise solution.

For the uniqueness of the desirable solution, we have to check the following conditions:

(C₁): Acceptable advantage:

$$Q\left(R^{(2)}\right)- Q\left(R^{(1)}\right) \ge \frac{1}{\left(m-1\right)},$$

(16)

where $R^{(1)}$ and $R^{(2)}$ are the options with the initial and subsequent positions in the ranking list, respectively, and m is the number of options.

(C₂): Adequate stability: The option $R^{(1)}$ must also be ranked by $S_i$ and $I_i.$ The compromise solution $Q_i$ is stable within an MCDM procedure, which can be selected with “voting by majority rule $(\tau >0.5)$”, “by consensus $(\tau \approx 0.5)$”, or “by veto $(\tau < 0.5)$”.

If the acceptable advantage (C₁) is not fulfilled, then the extreme value should be inspected by the given relation:

$$Q\left(R^{(J)}\right)-Q\left(R^{(1)}\right) < \frac{1}{\left(m-1\right)}.$$

(17)

In this case, all of the options $R^{(i)} \left(i =1(1)m\right)$ are the compromise solutions.

The options $R^{(1)}$ and $R^{(2)}$ are compromise solutions if the adequate stability (C₂) is not fulfilled.

Step X: End.

4. An Empirical Study: Performance Evaluation of Solar Panel Selection

Here, the developed PF–SWARA–VIKOR framework is implemented to select the most suitable solar panel within the PFS context, which demonstrates the usefulness and feasibility of the introduced approach.

For this, we have selected a decision-making evaluation and selection problem of the performance of five typical solar panel alternatives, which are R₁, R₂, R₃, R₄ and R₅. A team of three DEs is selected to process this solar panel selection problem. This problem associated with the performances of solar panels includes eight attributes or criteria. The facts of the criteria are given in

Table 1. Descriptions of considered criteria for solar panel selection.

Criteria	Descriptions	Type
Peak power rating (T1)	Refers to the maximum output (in Watts) under standard test conditions	Benefit
Peak efficiency (T2)	Refers to the high peak efficiency	Benefit
Maximum power current (T3)	Refers to the high value of current	Benefit
Maximum power voltage (T4)	Refers to the high value of power current	Benefit
Weight (T5)	Prefers to the solar panel with less weight	Cost
Price (T6)	Considers the price of solar panels
Reliability (T7)	Measures the reliability of the solar panel	Benefit
Spare parts availability (T8)	The availability of solar panel (SP) spare parts is one of the factors deciding customer fulfillment	Benefit

.

To start the PF–SWARA–VIKOR approach, first, we assume the weights of the DEs are PFNs, which are given as {Y(0.75, 0.35, 0.5612), Y(0.60, 0.50, 0.6245), Y(0.65, 0.45, 0.6124)}. The PFN decision matrices given by DEs $E_k:1,2,3$ can be obtained in

Table 2. Linguistic values with different experts for solar panel selection.

	R1	R2	R3	R4	R5
T1	E1: (0.29, 0.75) E2: (0.40, 0.70) E3: (0.45, 0.65)	E1: (0.70, 0.45) E2: (0.72, 0.50) E3: (0.65, 0.50)	E1: (0.58, 0.55) E2: (0.55, 0.60) E3: (0.60, 0.55)	E1: (0.55, 0.65) E2: (0.52, 0.66) E3: (0.60, 0.55)	E1: (0.60, 0.55) E2: (0.70, 0.45) E3: (0.65, 0.50)
T2	E1: (0.63, 0.40,) E2: (0.55, 0.60) E3: (0.68, 0.35)	E1: (0.63, 0.45) E2: (0.60, 0.50) E3: (0.55, 0.60)	E1: (0.60, 0.45) E2: (0.65, 0.50) E3: (0.58, 0.44)	E1: (0.60, 0.57) E2: (0.55, 0.60) E3: (0.50, 0.60)	E1: (0.60, 0.50) E2: (0.55, 0.60) E3: (0.55, 0.50)
T3	E1: (0.55, 0.65) E2: (0.60, 0.70) E3: (0.50, 0.70)	E1: (0.70, 0.45) E2: (0.70, 0.50) E3: (0.68, 0.45)	E1: (0.64, 0.55) E2: (0.55, 0.57) E3: (0.60, 0.55)	E1: (0.60, 0.55) E2: (0.70, 0.50) E3: (0.65, 0.55)	E1: (0.70, 0.50) E2: (0.65, 0.50) E3: (0.68, 0.50)
T4	E1: (0.55, 0.60) E2: (0.59, 0.45) E3: (0.60, 0.50)	E1: (0.55, 0.65) E2: (0.50, 0.65) E3: (0.55, 0.60)	E1: (0.50, 0.60) E2: (0.55, 0.60) E3: (0.45, 0.65)	E1: (0.55, 0.65) E2: (0.63, 0.42) E3: (0.60, 0.50)	E1: (0.55, 0.50) E2: (0.60, 0.50) E3: (0.45, 0.65)
T5	E1: (0.51, 0.55) E2: (0.60, 0.50) E3: (0.60, 0.55)	E1: (0.65, 0.48) E2: (0.60, 0.55) E3: (0.66, 0.47)	E1: (0.60, 0.45) E2: (0.65, 0.55) E3: (0.60, 0.50)	E1: (0.65, 0.48) E2: (0.65, 0.50) E3: (0.70, 0.45)	E1: (0.65, 0.58) E2: (0.50, 0.65) E3: (0.65, 0.45)
T6	E1: (0.65, 0.45) E2: (0.60, 0.48) E3: (0.55, 0.50)	E1: (0.62, 0.50) E2: (0.60, 0.52) E3: (0.58, 0.65)	E1: (0.58, 0.49) E2: (0.55, 0.50) E3: (0.68, 0.48)	E1: (0.65, 0.45) E2: (0.57, 0.48) E3: (0.60, 0.50)	E1: (0.62, 0.55) E2: (0.60, 0.55) E3: (0.58, 0.55)
T7	E1: (0.50, 0.58) E2: (0.55, 0.50) E3: (0.52, 0.57)	E1: (0.58, 0.60) E2: (0.50, 0.60) E3: (0.45, 0.60)	E1: (0.55, 0.65) E2: (0.53, 0.64) E3: (0.50, 0.60)	E1: (0.45, 0.55) E2: (0.60, 0.50) E3: (0.65, 0.53)	E1: (0.48, 0.70) E2: (0.50, 0.60) E3: (0.55, 0.60)
T8	E1: (0.67, 0.46) E2: (0.65, 0.45) E3: (0.60, 0.50)	E1: (0.57, 0.68) E2: (0.52, 0.57) E3: (0.50, 0.60)	E1: (0.58, 0.65) E2: (0.55, 0.60) E3: (0.50, 0.62)	E1: (0.60, 0.50) E2: (0.65, 0.55) E3: (0.68, 0.53)	E1: (0.57, 0.58) E2: (0.52, 0.60) E3: (0.45, 0.65)

in the form of $N={\left(g_{ij}^{\left(k\right)}\right)}_{m \times n}, k=1,2,3$ as follows.

Step I: Since the DEs’ weights as given by the experts are expressed in terms of PFNs, the crisp Des’ weights ${\omega }_k, k=1, 2,3$ have been computed using Equation (5) as {${\omega }_1=0.3934,{\omega }_2=0.2827$, ${\omega }_3=0.3239$}.

Steps II–IV: Judgments provided by three DEs have been aggregated into an APF-D matrix $\mathbb{Z}={\left(z_{ij}\right)}_{m \times n}$ by utilizing Equation (6) and are provided in

Table 3. Aggregated Pythagorean fuzzy (PF) decision matrix for solar panel evaluation.

	R1	R2	R3	R4	R5
T1	Y (0.381,0.702, 0.601)	Y (0.691,0.480, 0.541)	Y (0.579,0.564, 0.589)	Y (0.559,0.618, 0.552)	Y (0.648,0.504, 0.572)
T2	Y (0.628,0.430, 0.649)	Y (0.598,0.509, 0.620)	Y (0.609,0.460, 0.646)	Y (0.556,0.588, 0.587)	Y (0.571,0.526, 0.630)
T3	Y (0.551,0.680, 0.484)	Y (0.680,0.483, 0.551)	Y (0.604,0.556, 0.571)	Y (0.648,0.535, 0.542)	Y (0.680,0.500, 0.536)
T4	Y (0.578,0.521, 0.649)	Y (0.537,0.633, 0.620)	Y (0.500,0.616, 0.646)	Y (0.591,0.528, 0.587)	Y (0.538,0.544, 0.630)
T5	Y (0.568,0.535, 0.625)	Y (0.640,0.495, 0.587)	Y (0.615,0.493, 0.615)	Y (0.667,0.476, 0.573)	Y (0.615,0.552, 0.563)
T6	Y (0.607,0.474, 0.638)	Y (0.602,0.550, 0.579)	Y (0.609,0.490, 0.624)	Y (0.613,0.474, 0.632)	Y (0.602,0.550, 0.579)
T7	Y (0.521,0.553, 0.650)	Y (0.520,0.600, 0.608)	Y (0.529,0.631, 0.568)	Y (0.570,0.529, 0.629)	Y (0.510,0.638, 0.578)
T8	Y (0.643,0.470, 0.605)	Y (0.535,0.621, 0.573)	Y (0.548,0.626, 0.556)	Y (0.642,0.523, 0.560)	Y (0.521,0.608, 0.599)

.

As the criteria $T_5$ and $T_6$ are of cost type and the remaining are of benefit type, it is therefore necessary to form a normalized APF-D matrix $\mathbb{N}={\left({\tilde{z}}_{ij}\right)}_{m\times n}$ using Equation (7). The normalized APF-D matrix is given in

Table 4. Normalized aggregated PF decision matrix for solar panel selection.

	R1	R2	R3	R4	R5
T1	Y(0.381, 0.702, 0.601)	Y (0.691, 0.480, 0.541)	Y (0.579, 0.564, 0.589)	Y (0.559, 0.618, 0.552)	Y (0.648, 0.504, 0.572)
T2	Y (0.628, 0.430, 0.649)	Y (0.598, 0.509, 0.620)	Y (0.609, 0.460, 0.646)	Y (0.556, 0.588, 0.587)	Y (0.571, 0.526, 0.630)
T3	Y (0.551, 0.680, 0.484)	Y (0.680, 0.483, 0.551)	Y (0.604, 0.556, 0.571)	Y (0.648, 0.535, 0.542)	Y (0.680, 0.500, 0.536)
T4	Y (0.578, 0.521, 0.649)	Y (0.537, 0.633, 0.620)	Y (0.500, 0.616, 0.646)	Y (0.591, 0.528, 0.587)	Y (0.538, 0.544, 0.630)
T5	Y (0.535, 0.568, 0.625)	Y (0.495, 0.640, 0.587)	Y (0.493, 0.615, 0.615)	Y (0.476, 0.667, 0.573)	Y (0.552, 0.615, 0.563)
T6	Y (0.474, 0.607, 0.638)	Y (0.550, 0.602, 0.579)	Y (0.490, 0.609, 0.624)	Y (0.474, 0.613, 0.632)	Y (0.550, 0.602, 0.579)
T7	Y (0.521, 0.553, 0.650)	Y (0.520, 0.600, 0.608)	Y (0.529, 0.631, 0.568)	Y (0.570, 0.529, 0.629)	Y (0.510, 0.638, 0.578)
T8	Y (0.643, 0.470, 0.605)	Y (0.535, 0.621, 0.573)	Y (0.548, 0.626, 0.556)	Y (0.642, 0.523, 0.560)	Y (0.521, 0.608, 0.599)

. The linguistic ratings of the criteria are given in

Table 5. Linguistic scale for the rating of criteria.

Linguistic Values	PFNs
Extremely Low (EL)	Y(0.1500, 0.9500)
Very Low (VL)	Y(0.2500, 0.9000)
Low (L)	Y(0.3000, 0.8500)
Medium Low (ML)	Y(0.3500, 0.7500)
Medium (M)	Y(0.4500, 0.6500)
Medium High (MH)	Y(0.6000, 0.5000)
High (H)	Y(0.7000, 0.3500)
Very High (VH)	Y(0.8000, 0.3000)

.

Steps V–IX: In the SWARA approach, the role of the DEs is an important part of the process of evaluation and criteria weighting. Each DE decides the significance of each criterion. Then, the DE provides the rankings of all the criteria based on their own implicit understanding, information, and experiences (see

Table 6. Criteria weights given by the decision experts (DEs) in terms of LVs for solar panel evaluation.

Criteria	E1	E2	E3	Aggregated PFNs	Score Values
T1	H	VH	VVH	Y(0.788, 0.300, 0.538)	0.765
T2	MH	ML	H	Y(0.592, 0.500, 0.633)	0.550
T3	M	M	MH	Y(0.507, 0.597, 0.621)	0.450
T4	ML	ML	MH	Y(0.456, 0.658, 0.633)	0.388
T5	MH	H	ML	Y(0.579, 0.515, 0.632)	0.535
T6	H	M	MH	Y(0.614, 0.468, 0.635)	0.579
T7	VH	H	VH	Y(0.776, 0.313, 0.547)	0.752
T8	VH	H	H	Y(0.745, 0.329, 0.580)	0.723

). From

Table 7. Results obtained by the Stepwise Weight Assessment Ratio Analysis (SWARA) method for solar panel selection.

Criteria	Crisp Values	$\mathbf{Comparative} \mathbf{Significance} \mathbf{of} \mathbf{Criteria} \mathbf{Value} \left({\mathit{s}}_{\mathit{j}}\right)$	$\mathbf{Coefficient} \left({\mathit{k}}_{\mathit{j}}\right)$	$\mathbf{Recalculated} \mathbf{Weight} \left({\mathit{p}}_{\mathit{j}}\right)$	$\mathbf{Criteria} \mathbf{Weight} \left({\mathit{w}}_{\mathit{j}}\right)$
T1	0.765	-	1.000	1.000	0.1463
T7	0.752	0.013	1.013	0.987	0.1444
T8	0.723	0.029	1.029	0.959	0.1403
T6	0.579	0.144	1.144	0.838	0.1226
T2	0.550	0.029	1.029	0.814	0.1191
T5	0.535	0.015	1.015	0.802	0.1173
T3	0.450	0.085	1.085	0.739	0.1081
T4	0.388	0.062	1.062	0.696	0.1019

, the most important criterion is presented as rank 1 and the least important criterion is presented as the last one. Then, the final criteria weights are evaluated and given in Table 7 as follows:

$$w_j=(0.1463, 0.1191, 0.1081, 0.1019, 0.1173, 0.1226, 0.1444, 0.1403).$$

By employing Equations (11)–(12), the best and worst values of the solar panel alternatives are estimated as follows:

$$\begin{array}{c}{\sigma }_j^{+}=\left\{Y\right(0.691, 0.480, 0.541), Y(0.628, 0.430, 0.649), Y(0.680, 0.483, 0.551),\\ Y(0.591, 0.528, 0.587), Y(0.476, 0.667, 0.573), Y(0.474, 0.613, 0.632),\\ Y(0.570, 0.529, 0.629), Y(0.643, 0.470, 0.605)\},\end{array}$$

$$\begin{array}{c}{\sigma }_j^{-}=\left\{Y\right(0.381, 0.702, 0.601), Y(0.556, 0.588, 0.587), Y(0.551, 0.680, 0.484),\\ Y(0.500, 0.616, 0.646), Y(0.552, 0.615, 0.563), Y(0.550, 0.602, 0.579),\\ Y(0.510, 0.638, 0.578), Y(0.535, 0.621, 0.573)\}.\end{array}$$

With the use of Equations (13)–(15), the values of S_i, I_i, and Q_i are calculated. The obtained results are given in

Table 8. Group utility, individual regret, and compromise measure of each solar panel selection.

	Si	Ii	Qi
R1	0.547	0.183	0.857
R2	0.545	0.140	0.519
R3	0.634	0.145	0.669
R4	0.262	0.119	0.000
R5	0.661	0.144	0.695
Ranking order	$S_4 \succ S_2 \succ S_1 \succ S_3 \succ S_5$	$I_4 \succ I_2 \succ I_5 \succ I_3\succ I_1$	$Q_4 \succ Q_2 \succ Q_3 \succ Q_5\succ Q_1$

. By employing the decreasing values of S_i, I_i, and Q_i, the preference order of the solar panel alternatives is acquired in Table 8. The lowest value of Q_i denotes the optimal solar panel, i.e., R₄ is the best solar panel alternative.

4.1. Sensitivity Analysis

This section discusses sensitivity analysis over different values of parameter $\tau$. The values of $\tau$ vary from 0.0 to 1.0, but the preference order $R_4 \succ R_2 \succ R_3 \succ R_5\succ R_1$ of the five preferred solar panel alternatives is the same in each case. Consequently, this study proves that the obtained outcome by employing PF–SWARA–VIKOR is more consistent and effective.

It can also be observed from Figure 1 that the compromise solution Q_i of R₁ decreases when the value of $\tau$ increases, while R₂, R₃, and R₅ increases when the value of $\tau$ increases. Meanwhile, the fourth alternative R₄ is stable in each set. Accordingly, despite the change of weights in the criterion set, the preference order of the five solar power alternatives remains the same. The final ranking of solar panel alternatives is presented with respect to following performance scores in

Table 9. Different values of compromise solutions over various values of parameter $\tau$.

$\mathit{\tau }$	R1	R2	R3	R4	R5
0.0	1.000	0.328	0.406	0.000	0.391
0.1	0.971	0.366	0.459	0.000	0.452
0.2	0.943	0.404	0.511	0.000	0.512
0.3	0.914	0.442	0.564	0.000	0.573
0.4	0.886	0.481	0.617	0.000	0.634
0.5	0.857	0.519	0.669	0.000	0.695
0.6	0.829	0.557	0.722	0.000	0.756
0.7	0.800	0.595	0.775	0.000	0.817
0.8	0.771	0.633	0.827	0.000	0.878
0.9	0.743	0.671	0.880	0.000	0.939
1.0	0.714	0.709	0.932	0.000	1.000

, and it is observed that solar panel R₄ is best of all options.

4.2. Comparative Study

Here, a comparison was done between the results attained from the PF–SWARA–VIKOR method and those of another approach. To show the efficiency and display the irreplaceable merits of the PF–SWARA–VIKOR framework, the PF–TOPSIS method [33] is implemented to handle the decision-making problem.

PF-TOPSIS Method

Steps I–VI: Same as the previous method.

Step VII: Compute the degree of distances from PF-PIS and PF-NIS.

With the use of Equation (1), calculate the degree of weighted distance $D_h\left( {\tilde{z}}_{ij}, {\sigma }_j^{+}\right)$ among the alternatives $R_i\left(i=1\left(1\right)m\right)$ and the PF-IS${\sigma }_j^{+}:$

$$D_h\left( {\tilde{z}}_{ij}, {\sigma }_j^{+}\right)=\frac{1}{2}{\displaystyle \sum _{j=1}^n\left[ w_j\left(\left|{\mu }_{{\tilde{z}}_{ij}}^2-{\mu }_{{\sigma }_j^{+}}^2\right|+\left|{\nu }_{{\tilde{z}}_{ij}}^2-{\nu }_{{\sigma }_j^{+}}^2\right|+\left|{\pi }_{{\tilde{z}}_{ij}}^2-{\pi }_{{\sigma }_j^{+}}^2\right|\right)\right]}.$$

(18)

Usually, the smaller $D_h\left( {\tilde{z}}_{ij}, {\sigma }_j^{+}\right)$ is, the better the alternative $R_i,$ and let

$$D_{\min }\left( {\tilde{z}}_{ij}, {\sigma }_j^{+}\right)=\underset{1\le i\le m}{\min }{D}_h\left( {\tilde{z}}_{ij}, {\sigma }_j^{+}\right),$$

(19)

and the degree of distance $D_h\left( {\tilde{z}}_{ij}, {\sigma }_j^{-}\right)$ among the alternatives $R_i\left(i=1\left(1\right)m\right)$ and the PF-AIS ${\sigma }_j^{-}$ is given as follows:

$$D_h\left( {\tilde{z}}_{ij}, {\sigma }_j^{-}\right)=\frac{1}{2}{\displaystyle \sum _{j=1}^n\left[ w_j\left(\left|{\mu }_{{\tilde{z}}_{ij}}^2-{\mu }_{{\sigma }_j^{-}}^2\right|+\left|{\nu }_{{\tilde{z}}_{ij}}^2-{\nu }_{{\sigma }_j^{-}}^2\right|+\left|{\pi }_{{\tilde{z}}_{ij}}^2-{\pi }_{{\sigma }_j^{-}}^2\right|\right)\right]}.$$

(20)

The bigger the $D_h\left( {\tilde{z}}_{ij}, {\sigma }_j^{-}\right),$ the better the alternative $R_i,$ and let

$$D_{\max }\left( {\tilde{z}}_{ij}, {\sigma }_j^{-}\right)=\underset{1\le i\le m}{\max }{D}_h\left( {\tilde{z}}_{ij}, {\sigma }_j^{-}\right).$$

(21)

Step VIII: Evaluate the relative closeness coefficient (CC).

The formula for the computation of the relative CC of each solar panel alternative is given as

$$ℂ\left(R_i\right)=\frac{D_h\left( {\tilde{z}}_{ij}, {\sigma }_j^{-}\right)}{D_h\left( {\tilde{z}}_{ij}, {\sigma }_j^{-}\right)+ D_h\left( {\tilde{z}}_{ij}, {\sigma }_j^{+}\right)},\hspace{1em}i=1\left(1\right)m.$$

(22)

In accordance with the closeness index $ℂ\left(R_i\right),$ the suitable solar panel alternative and the rankings of all options are decided. However, Hadi-Vencheh and Mirjaberi [64] explained that, in many circumstances, the relative CC cannot attain the goal that the most suitable solution should have the minimum distance from the PF-IS and the maximum distance from the PF-AIS, concurrently. Therefore, the revised CC of each alternative is defined by

$$ℝ\left(R_i\right)=\frac{D_h\left({\tilde{z}}_{ij},{\sigma }_j^{-}\right)}{\max D\left({\tilde{z}}_{ij},{\sigma }_j^{-}\right)}-\frac{D_h\left({\tilde{z}}_{ij},{\sigma }_j^{+}\right)}{\min D_h\left({\tilde{z}}_{ij},{\sigma }_j^{+}\right)}$$

(23)

Step IX: Choose the highest value, $ℝ\left(R_k\right),$ among the values $ℝ\left(R_i\right),i=1\left(1\right)m.$ Hence, $R_k$ is the optimal choice.

From Table 3 and Equations (11) and (12), PF-IS and PF-AIS are evaluated. Now, all computational results of the PF–TOPSIS [33] method are depicted in

Table 10. Computational results of the PF–Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method for solar panel selection.

Alternative	${\mathit{D}}_{\mathit{h}}\left({\tilde{\mathit{z}}}_{\mathit{i}\mathit{j}}, {\mathit{\sigma }}_{\mathit{j}}^{+}\right)$	${\mathit{D}}_{\mathit{h}}\left({\tilde{\mathit{z}}}_{\mathit{i}\mathit{j}}, {\mathit{\sigma }}_{\mathit{j}}^{-}\right)$	$\mathit{ℂ}\left({\mathit{R}}_{\mathit{i}}\right)$	Ranking	$\mathit{ℝ}\left({\mathit{R}}_{\mathit{i}}\right)$	Ranking
R1	0.101	0.085	0.457	4	−1.0446	4
R2	0.068	0.102	0.598	2	−0.3036	2
R3	0.099	0.081	0.449	5	−1.0446	4
R4	0.056	0.112	0.665	1	0.0000	1
R5	0.086	0.083	0.491	3	−0.7946	3

.

Finally, the final ranking of the solar panel alternative is obtained as $R_4 \succ R_2 \succ R_5 \succ R_1 \succ R_3$ and $R_4 \succ R_2 \succ R_5 \succ R_1 \approx R_3 .$ Therefore, the most suitable solar panel alternative is $R_4.$ Obviously, the results slightly vary with different types of methods. To this point, the PF–SWARA–VIKOR method is more robust than the PF–TOPSIS method [33] and thus has wider applicability.

From Figure 2, it is determined that the developed framework is highly consistent with the existing method with PFSs. To retain homogeneity in the method-related comparison, we consider the methods of Zhang and Xu [33] and Hadi-Vencheh and Mirjaberi [64]. The Spearman correlation values with the compromise solution are given by (0.70, 0.90, 1.00, 0.70, 0.825). Spearman correlation is utilized for these rank values to determine the consistency of the developed framework. In addition, Figure 3 depicts the prioritization orders of different methods with PFSs and makes a discussion of different factors to understand the strengths of the proposed framework.

Furthermore, compared with the PF–TOPSIS method, the PF–SWARA–VIKOR approach has the following advantages:

(a)

The PF–SWARA–VIKOR method represents the Pythagorean fuzzy information, which can depict the MD, ND, and hesitation degree with an effortless mathematical description. Based on it, we can determine the significance degree of the DEs without any modification and, therefore, the developed method can successfully avoid the loss of information.

(b)

As some of the previous measures under the PFSs [33] have been incapable of providing the preference order of the alternatives accurately, thus, their consequent methods may not present relevant outcomes. Alternatively, the proposed approach has the capability to prevail over their limitations and is therefore able to order the alternatives appropriately, which makes it a more desirable approach to solving MCDM problems.

(c)

The SWARA approach is utilized to compute the subjective weights of criteria in the process of performance evaluation of solar panels, which makes the developed PF–SWARA–VIKOR approach more sensible, flexible, and efficient.

(d)

The developed framework has the following benefits when choosing solar panels:

• An innovative procedure is utilized to enumerate tangible sub-criteria successfully.
• The integrated approach eradicates the subjective estimation of indistinct sub-criteria.
• Pythagorean fuzzy SWARA is used to achieve appropriate harmonizing of criteria.

5. Conclusions

Recently, the selection of most appropriate solar panel has been a significant concern in the development of the sustainable era [65,66,67,68]. Owing to the occurrence of multiple conflicting criteria, the SPS problem can be considered as a complex MCDM problem. To handle this problem, an integrated decision-making framework has been introduced based on the SWARA and VIKOR approaches within a PFS context. In the developed framework, the criteria weights are computed by aggregating the subjective weights calculated by the SWARA method. Next, the VIKOR approach is used to evaluate the preference order of the alternatives. To exemplify the applicability and feasibility of the developed framework, a case study of solar panel selection has been presented, which confirmed its effectiveness and usefulness. Sensitivity analysis has also been discussed to show the stability of the introduced approach with respect to different sets of criteria weights. A comparative analysis has been presented to prove the strength of the outcomes obtained by the developed approach. In the future, we will expand our research by integrating objective and subjective criteria weight information within PFSs and q-rung orthopair fuzzy environments. Apart from the criteria used in this approach, the proposed model will be implemented in various selection scenarios, such as suitable locations of plants, sustainable suppliers, green suppliers, healthcare management, and others.