Mechanisms for Choosing PV Locations That Allow for the Most Sustainable Usage of Solar Energy

Abstract

The electrical power need in the Kingdom of Saudi Arabia (KSA) has been escalating at a rapid rate of about 7.5% annually. It has the third highest usage rate in the world as stated by World Energy Council statistics. The rising energy demand has a significant impact on the country’s economy since oil is considered to be its mainstay. Additionally, conventional energy production using fossil fuels is a leading contributor to ecological degradation and adversely influences human health. As a result, Saudi Arabia has taken significant steps to shift from its current status of total reliance on oil to new frontiers of exploration of other kinds of renewable energies. Photovoltaic (PV) solar energy is the most preferred renewable energy to be harnessed in Saudi Arabia. In accordance with Vision 2030, the KSA intends to generate at least 9.5 GW of electricity from green sources, a significant portion of which will come from solar PV power. Since the site peculiarities have a huge influence on the project’s technical and economic dimensions, the scaled-up deployment of solar projects calls for a judicious selection of PV sites. Undoubtedly, performing a thorough solar site survey is the foremost step to establishing a financially viable and successful solar project. Multiple criterion decision-making (MCDM) strategies can be very helpful in making judgments, given that a number of criteria might influence PV site selection. The objective of this research is to provide valuable information on various MCDM approaches that can be utilized to select optimal locations for PV solar plants. A number of variables, including topography, air temperature, dust storms, solar radiation, etc., are considered in this analysis. This study has combined various MCDM techniques in order for the strengths of each method to outweigh the weaknesses of the others. It has been deduced from this analysis that the most crucial factors in choosing PV sites are solar radiation and sunshine hours. It has also been concluded that of the surveyed cities, Tabuk is the optimum location for the construction of a solar power plant due to its high GHI value of 5992 W/m²/day and abundant sunshine hours of 12.16 h/day. Additionally, the FAHP-VIKOR method is noted as being the most rigorous, whereas Entropy-GRA is the simplest method.

1. Introduction

The electricity usage in the Kingdom of Saudi Arabia (KSA) is expanding every year, and it has the third-highest utilization rate worldwide as per World Energy Council statistics [1]. In the last ten years or so, KSA’s demand for energy has been growing at a rapid rate of about 7.5% annually [1]. One of the primary issues the power companies in Saudi Arabia are dealing with is the rising energy demand, which leads to the burning of more barrels of carbon-based fuel, which has an impact on the country’s economy since oil is considered to be its mainstay. Furthermore, it is well-known that the use of fossil fuels in traditional energy generation contributes significantly to environmental deterioration and has a detrimental effect on human health. Saudi Arabia has thus made tremendous efforts to move away from its current state of complete reliance on oil and move toward new horizons of exploration of other types of renewable energies.

Solar energy is one of the most prominent sustainable energy sources, which is rising in popularity due to its many benefits, including its ability to reduce reliance on energy sources, e.g., oil and gas [2,3]. In addition, solar energy is a dependable, unending, clean alternative that is safe for the environment. Solar photovoltaic (PV) electricity is the most popular green energy source in Saudi Arabia. Its location in the sun belt and favorable spatial conditions make it one of the world’s top solar energy producers [4]. The KSA is fortunate to have access to a wealth of solar energy resources. There are vast undeveloped land areas that might be used to house solar projects, and Saudi Arabia’s average daily global horizontal irradiance (GHI) receives between 5.7 and 6.7 kWh/m² on average [5]. In accordance with Vision 2030, the KSA intends to generate 9.5 GW of electricity from renewable sources, a significant portion of which will come from solar PV power [6]. Additionally, the cost of PV modules has decreased by an average of 70% over the past ten years, mostly due to improvements in their efficiency [7]. The scaled-up deployment of solar projects necessitates a precise evaluation and analysis of PV site selection. A thorough solar site survey is the first step in ensuring a solar project is both affordable and efficient.

The location of a solar PV power plant is vital since the site’s attributes directly affect the project’s technical and financial elements, and consequently, its viability [8]. The selection of the ideal solar energy site, which is crucial to their installation, is influenced by a variety of variables. These factors should be addressed to obtain more energy while also lowering startup and ongoing operating expenses [9]. These variables must be considered in the first stages of solar energy installation to appropriately locate the plant. Considering that a number of criteria might drive the choice of location, implementing multiple criteria decision-making (MCDM) methodologies can greatly help in making judgments regarding site selection for PV solar energy systems by taking important elements into consideration [10]. Numerous investigations have been conducted in the literature in order to locate solar power plants [11,12,13,14]. MCDM provides a potent decision-making method that may be used in numerous applications, including the site selection for solar power projects [15,16,17]. Seven MCDM techniques were used by Villacreses et al. [18] to identify suitable locations for PV solar farm installation. The analysis used nine variables, four constraints, and the Analytic Hierarchy Process (AHP) approach to weigh the factors. The Pearson correlation coefficient was used to analyze seven MCDM outcomes. Wang et al. [14] consolidated three techniques, such as the fuzzy analytical hierarchy process (FAHP), data envelopment analysis (DEA), and the technique for order of preference by similarity to ideal solution (TOPSIS), to discover the best location for a solar power facility combining both quantitative and qualitative metrics. Similarly, Mirzaei [19] coupled MCDM and fuzzy logic to assess possible Turkish cities for the construction of newer solar power stations. Stepwise Weight Assessment Ratio Analysis (SWARA) was used in the suggested procedure to measure weights, and the Pythagorean Fuzzy Form of TOPSIS was used to grade the locations. A two-step methodology premised on FAHP and DEA was presented by Lee et al. [20] for assessing the viability of potential locations for sustainable energy plant sites. The FAHP was applied as the initial step to establish the assurance region (AR) of the input variables, and the AR was integrated into DEA to evaluate the efficacy of potential plant sites. The best-worst method (BWM) was employed to quantify the criterion and sub-criteria in a hybrid MCDM approach [13]. Grey relational analysis (GRA) and Vlsekriterijumska Optimizacija I Kompromisno Resenje (VIKOR) were used to rank the prospective locations. Additionally, a novel Monte Carlo simulation-based (MCSB) methodology was used to examine the sensitivity of GRA and VIKOR. Similarly, to identify the ideal PV system for Saudi Arabia, Al-Shammari et al. [21] utilized AHP and TOPSIS effectively. An MCDM strategy for the site location of PV charging stations was also used by Dang et al. [22] to choose the optimal alternative in China. The fuzzy VIKOR approach was used to rank the solutions, and the fuzzy measure technique was utilized to calculate the weight of the criteria. AHP, the entropy weight method, the fuzzy measure method, and VIKOR were all successfully used in the hybrid fuzzy approach. Seker and Kahraman [23] established a thorough two-stage MCDM model including the AHP and Multiplicative Multi-Objective Ratio Analysis (MULTIMOORA) approaches to select the best PV panel maker for solar power plants. The robustness and verification of the proposed strategy for the solar power industry were also demonstrated by the sensitivity and comparison analyses. Similarly, Wang et al. [24] established a fuzzy MCDM approach by integrating FAHP with DEA to identify a solar panel supplier for a PV system design. They also used a number of DEA models for rating possible suppliers in the last phases.

The preceding works cited in this research serve as proof of the extensive literature that is available in various domains of solar energy applications employing MCDM. The prior studies provide examples of different MCDM methodology behavior when several conflicting variables are considered. It is true that earlier studies have used MCDM to determine the ideal location for PV installation. None of them have highlighted the basis for choosing an appropriate MCDM approach, rather, they have all emphasized choosing a suitable location utilizing any MCDM based on their convenience. The key challenge for the decision-makers has always been deciding which MCDM method to use. Consequently, this study is an effort in which several MCDM methods have been employed and evaluated. Thus, the objective of this research is to provide valuable information on various MCDM combinations that can be utilized to select optimal locations for PV solar plants. This work explores MCDM methods for evaluating possible solar PV sites by considering a variety of factors, such as solar radiation, air temperature, dust storms, topology, etc. The suggested methodology is based on a combination of different MCDM techniques, such as the group eigenvalue method (GEM), FAHP, entropy method, VIKOR, and GRA. The application of various MCDM techniques, which are coupled and employed as an adaptive approach for the PV site selection problem in Saudi Arabia, is where this research makes a scientific contribution. First, the weights of the criteria are determined using GEM, FAHP, and entropy approaches. Next, the PV sites are ranked and prioritized using VIKOR, and GRA. To demonstrate the reliability of the evaluation methodologies, comparison, and sensitivity analyses are carried out. The investigation of a specific instance in Saudi Arabia serves to support the applicability of the proposed methodology. This study, which is still in its early phases of development in terms of solar energy, is expected to help the authorities gain a better knowledge of the potential investment in solar energy.

2. MCDM Methods

This work intends to put into practice the combination of different MCDM techniques, including GEM, FAHP, Entropy, VIKOR, and GRA approaches. The combinations, including GEM-VIKOR, FAHP-VIKOR, Entropy-VIKOR, GEM–GRA, FAHP–GRA, and Entropy-GRA have been used to obtain the optimal PV site for the generation of solar energy. The following is a succinct summary of various MCDM techniques.

2.1. Determination of Weights

2.1.1. Group Eigen Value Method

GEM [25] is deployed to give attribute weights or assess their significance by developing an expert judgment matrix. Since one expert only has a limited amount of information and experience, a judgment matrix derived from a group of various specialists should be employed. This approach aims to locate the ideal expert, whose evaluation is most reliable, accurate, and repeatable with the opinions of fellow experts in the team. Following are the phases and equations (Equations (1)–(6)) for the GEM technique [25].

Take into account the assessment of n variables N₁, N₂, N₃,…, N_n by a team of m analysts M₁, M₂, M₃,…, M_m. Let x_ij represent the analyst’s opinion of the j-th variable by i-th analyst. Suppose x = (x_ij)_m×n be the m × n order matrix (Equation (1)) assessed by the specialists.

$${x=\left(x_{\mathit{ij}}\right)}_{m\times n}=\left[\begin{array}{cccc}{x}_{11}& x_{12}& \dots & x_{1n}\\ x_{21}& x_{22}& \dots & x_{2n}\\ \dots & \dots & \dots & \dots \\ x_{m1}& x_{m2}& \dots & x_{mn}\end{array}\right]$$

(1)

x_* as shown in Equation (2) can be used to express the evaluation vector of a specialist with the greatest judgment quality and exact assessment.

x_* = (x_*1, x_*2, x_*3,…, x_*n)^T

(2)

The summation of angles at which the preferable expert judgment vector and the assessment vectors of other specialists’ overlap should be as small as possible. It indicates that x_* can be calculated once the function f = ${{\displaystyle \sum }}_{i=1}^m{\left(b^T{x}_i\right)}^2$ reaches its highest value. Consequently, it is possible to determine an idealized expert’s assessment vector, x_* using Equation (3).

$$\underset{{\left|\right|b\left|\right|}_2=1}{\max }{{\displaystyle \sum }}_{i=1}^m{\left(b^T{x}_i\right)}^2={{\displaystyle \sum }}_{i=1}^m{\left(x_{*}^T{x}_i\right)}^2$$

(3)

The expression ∀b = (b₁, b₂,…, b_n)^T is satisfied by the parameter, b^T, which is an eigenvector of F. x_* designates the positive eigenvector equivalent to ρ_max, which is the highest possible positive eigenvalue of the matrix F = x^T.x. After the eigenvector is normalized to match the highest eigenvalue, the standardized weight vector for every specialist can be found. The procedure listed below should be considered to estimate the criterion weights.

• Analysts designate assessment scores to specified criteria.
• Transpose of the assessment matrix and then multiply it by the transposed one as shown in Equation (4).

  F = x^T.x

  (4)
• The power method, as employed by Qiu in 1997 [25], can be utilized to derive the eigenvector x_*.
  • Suppose k = 0, y₀ = (1/n, 1/n,…..,1/n)^T

    $$y_1=F{y}_0;z_1=\frac{y_1}{\left|\right|y_1\left|\right|{}_2}$$

    (5)
  • For k = 1, 2, 3,..., y_k+1 = Fz_k, and z_k+1 = $\frac{y_{k+1}}{\left|\right|y_{k+1}\left|\right|{}_2}$
  • Verify if │z_k→k+1│≤ ɛ, if it does, then, z_k+1 equates to x_*, else return to the prior step (with ɛ representing the precision and │z_k→k+1│is the maximum absolute value of the difference between z_k and z_k+1)
• Normalization of the derived eigenvector using Equation (6).

  $$w_j=x_{*j}/ {{\displaystyle \sum }}_{j=1}^n{x}_{*j}$$

  (6)

  where, j = 1, 2, 3,…,n, so that ${{\displaystyle \sum }}_{j=1}^n{x}_{*j}$ = 1 and x_*j is the weight obtained for criteria utilizing GEM.

2.1.2. Fuzzy Analytic Hierarchy Process

The fuzzy set principle proposed by Zadeh [26] is used to deal with the ambiguity and vagueness of specialist views as well as information gathering. The FAHP can methodically oversee design choices since it is founded on the fundamentals of fuzzy set theory and hierarchical structural analysis. The FAHP, however, is an expansion of a conventional AHP technique using fuzzy numbers in a fuzzy environment [27]. The initial conversion of the typical AHP into FAHP was posited by Van Laarhoven and Pedrycz [28], Buckley [29], and Chang [30]. They used fuzzy numbers with triangular membership functions to indicate the relative importance of the Saaty. In this work, the crisp values are modeled by applying a trapezoidal membership function because of its greater effectiveness over the triangular membership function [31]. The following are the steps used to estimate fuzzy weights.

Step 1: translating crisp values into fuzzy numbers.

The decision matrices generated by many experts are adapted into trapezoidal fuzzy numbers in this stage. The membership function of the F-designated trapezoidal fuzzy number with the characteristics (l, m, n, u) can be depicted using Equation (7) [32].

Membership function,

$${\mu }_F\left(x\right)=\{\begin{array}{c}\begin{array}{cc}0& x\le l\end{array}\\ \begin{array}{cc}\frac{x-l}{m-l}& l\le x\le m\end{array}\\ \begin{array}{cc}1& m\le x\le n\end{array}\\ \begin{array}{cc}\frac{u-x}{u-n}& n\le x\le u\end{array}\\ \begin{array}{cc}0& x\ge u\end{array}\end{array}$$

(7)

The Saaty scale-based crisp values are initially transformed into triangular fuzzy numbers and subsequently into trapezoidal fuzzy numbers to accomplish this conversion.

The triangle fuzzy numbers are converted into trapezoid numbers by consistently retaining the upper and bottom limits of the fuzzy numbers and slightly increasing the center [33]. As an illustration, consider the transformation of the crisp value of x into a triangular fuzzy number (a, b, c) where a = x − 1; b = x and c = x + 1. Following this transformation, the triangular fuzzy numbers have been changed into trapezoidal fuzzy numbers (l, m, n, u), where l = a; u = c; m = l + 0.5 and n = u − 0.5.

Step 2: weights computation.

The Chang extent analysis is adopted in this work to derive the weights of the performance indicators [30]. For computing priori weights, the extent analysis is outlined as follows [34,35].

Suppose X = {x₁, x_2, x₃,…, x_n} symbolize the array of objects and G = {g₁, g_2, g₃,…, g_n} signify the list of objectives. symbolize the collection of objects. Every object must experience extent analysis for each objective of the problem, according to the extent analysis framework. As a result, m extent analysis values for each object can be obtained as M¹_gi, M²_gi, M³_gi, M⁴_gi,…, M^m_gi, i = 1, 2, 3, 4,…, n. Here, M^j_gi (j = 1, 2, 3, 4,…,m) is an illustration of a fuzzy trapezoidal number. The succeeding steps (Equations (8)–(15)) can be undertaken to complete Chang’s extent analysis [30].

The synthetic fuzzy values for i-th object can be derived using Equation (8).

$$S_i={{\displaystyle \sum }}_{j=1}^m\begin{array}{ccc}{M}_{gi}^j& \otimes & {\left[{{\displaystyle \sum }}_{i=1}^n{{\displaystyle \sum }}_{j=1}^m{M}_{gi}^j\right]}^{-1}\end{array}$$

(8)

Because FAHP evaluates the relevant attribute using a trapezoidal fuzzy number with four values. As a result, the fuzzy addition action for a specific matrix can be accomplished by employing the expression in Equation (9) to estimate ${{\displaystyle \sum }}_{j=1}^m{M}_{gi}^j$.

$${{\displaystyle \sum }}_{j=1}^m{M}_{gi}^j=\left({{\displaystyle \sum }}_{j=1}^m{l}_j,{{\displaystyle \sum }}_{j=1}^m{m}_j,{{\displaystyle \sum }}_{j=1}^m{n}_j, {{\displaystyle \sum }}_{j=1}^m{u}_j\right)$$

(9)

The value of ${\left[{{\displaystyle \sum }}_{i=1}^n{{\displaystyle \sum }}_{j=1}^m{M}_{gi}^j\right]}^{-1}$ can be calculated by the fuzzy addition process of M^j_gi (j = 1, 2, 3, 4,…, m) values in the following way by applying Equation (10).

$${{\displaystyle \sum }}_{i=1}^n{{\displaystyle \sum }}_{j=1}^m{M}_{gi}^j=\left({{\displaystyle \sum }}_{i=1}^n{l}_i,{{\displaystyle \sum }}_{i=1}^n{m}_i,{{\displaystyle \sum }}_{i=1}^n{n}_i, {{\displaystyle \sum }}_{i=1}^n{u}_i\right)$$

(10)

Additionally, the vector’s inverse can be defined using Equation (11).

$${\left[{{\displaystyle \sum }}_{i=1}^n{{\displaystyle \sum }}_{j=1}^m{M}_{gi}^j\right]}^{-1}=\left(\begin{array}{cccc}\frac{1}{{{\displaystyle \sum }}_{i=1}^n{u}_i},& \frac{1}{{{\displaystyle \sum }}_{i=1}^n{n}_i},& \frac{1}{{{\displaystyle \sum }}_{i=1}^n{m}_i},& \frac{1}{{{\displaystyle \sum }}_{i=1}^n{l}_i}\end{array}\right)$$

(11)

Suppose M₁ = (l₁, m₁, n₁, u₁) and M₂ = (l₂, m₂, n₂, u₂) denotes two trapezoidal fuzzy numbers. The following condition in Equation (12) affects how likely it is that M₂ = (l₂, m₂, n₂, u₂) ≥ M₁ = (l₁, m₁, n₁, u₁).

$$V(M_2\ge M_1)=\begin{array}{c}sup\\ y\ge x\end{array}\lfloor \min \left({\mu }_{M_1}\left(x\right),{\mu }_{M_2}\left(y\right)\right)\rfloor$$

(12)

where, V (M₂ ≥ M₁) can be expressed using the definition in Equation (13).

$$\begin{gathered} V(M_2\ge M_1)=\mathrm{hgt}(M_1\cap M_2)={\mu }_{M_2}\left(d\right)= \\ \begin{array}{cc}1,& m_1\ge m_2\\ 0,& \left(m_2-n_1\right)>\left(u_1+l_2\right)\\ \frac{\left(\left(n_1-m_2\right)+\left(u_1+l_2\right)\right)}{\left(u_1+l_2\right)},& 0<\left(m_2-n_1\right)<\left(u_1+l_2\right)\\ \frac{\left(\left(m_2-n_1\right)+\left(u_1+l_2\right)\right)}{\left(u_1+l_2\right)},& \left(m_2-n_1\right)<\left(u_1+l_2\right), \mathrm{where} m_2<n_1 \mathrm{and} m_1<m_2 \end{array} \end{gathered}$$

(13)

A fuzzy number with a probability greater than k can be represented as V (M₂ ≥ M₁, M_2, M₃, M₄,…, M_k) = V [(M ≥ M₁) and V [(M ≥ M₂) and V [(M ≥ M₃) and V [(M ≥ M₄) and V [(M ≥ M_k)] = min V (M ≥ M_i), i = 1, 2, 3,…, k.

Assume that d (A_i) = min V (S_i ≥ S_k) for k = 1, 2, 3, 4,…, n; k ≠ i. The weight vector can be estimated as depicted in Equation (14).

$$w^{\prime }={\left(d^{\prime }\left(A_1\right), d^{\prime }\left(A_2\right),d^{\prime }\left(A_3\right),d^{\prime }\left(A_4\right),\dots ,d^{\prime }\left(A_n\right) \right)}^T$$

(14)

where A_i (i = 1, 2, 3, 4, …, n) indicates n elements. Normalization is needed to obtain the weight vector for the individual elements. The weight vector is normalized to get the vector normalized as in Equation (15).

$$w={\left(d\left(A_1\right), d\left(A_2\right),d\left(A_3\right),d\left(A_4\right),\dots ,d\left(A_n\right) \right)}^T$$

(15)

where w is a vector of non-fuzzy numbers or crisp values.

2.1.3. Entropy Method

Entropy weight calculation is an empirical approach that uses the attribute’s innate knowledge to compute its weights. Accordingly, noise in the resultant weights can be reduced, leading to the production of impartial outcomes [36]. The attribute’s higher relevance is indicated by its greater entropy weight. The entropy weight approach makes use of the attribute’s intrinsic information to make the resulting weights more realistic than subjective [37].

If there are m alternatives and n criteria, x_ij reflects the value for j-th criterion and i-th alternative. The entropy weight can be calculated as seen below using Equations ((16)–(21)) [36,37]:

Standardization of attribute values, r_ij using Equations (16)–(17).

$$\mathrm{Benefit}\mathrm{attribute}=\frac{x}{x_{max}} (i=1,2,3,4\dots ,m;j=1,2,3,\dots ,n)$$

(16)

$$\mathrm{Cos}\mathrm{t}\mathrm{attribute}=\frac{x_{min}}{x}(i=1,2,3,4\dots ,m;j=1,2,3,\dots ,n)$$

(17)

Quantification of entropy using Equation (19).

$$H_j=-\frac{{{\displaystyle \sum }}_{i=1}^m{P}_{ij}\ln P_{ij}}{\ln m}(i=1,2,3,4\dots ,m;j=1,2,3,\dots ,n)$$

(18)

$$P_{ij}=\frac{r_{ij}}{{{\displaystyle \sum }}_{i=1}^m{r}_{ij}}(i=1,2,3,4\dots ,m;j=1,2,3,\dots ,n)$$

(19)

Estimation of Entropy weight using Equation (20).

$$w_j=1-\frac{1-H_j}{n-{{\displaystyle \sum }}_{j=1}^n{H}_j}$$

(20)

$${{\displaystyle \sum }}_{j=1}^n{w}_j=1(j=1,2,3,\dots ,n)$$

(21)

2.2. Ranking Approaches

2.2.1. VIKOR Method

The various procedures required to establish VIKOR-based MCDM [38] tools are covered in this section. Assume there are m possibilities S_i (i = 1, 2,…, m) and n criteria A_j (j = 1, 2,…, n). The VIKOR approach’s major goal is to assess both the positive and negative ideal positions in the candidate solutions. The applied equations (Equations (22)–(25)) to implement the VIKOR approach are discussed in the following steps [38].

Step 1. The first step is to create the decision matrix, which is written as X = (x_ij)_m×n. Here, x_ij stands for real numbers that represent the values of the j-th criterion for the alternative i.

Step 2. Apply Equation (22) to determine the normalized decision matrix, (r_ij)_m×n.

$${\left(r_{\mathit{ij}}\right)}_{m\times n}=\frac{x_{ij}}{\sqrt{{\displaystyle \sum }{\left(x_{ij}\right)}^2}}$$

(22)

Step 3. Specify the upper and lower bounds for the normalized decision matrix. For the benefit criteria, $v_j^{+}=\underset{j}{\max }{r}_{ij}$ and $v_j^{-}=\underset{j}{\min }{r}_{ij}$. For the non-benefit condition, $v_j^{+}=\underset{j}{\min }{r}_{ij}$ and $v_j^{-}=\underset{j}{\max }{r}_{ij}$.

Step 4. Using Equations (23) and (24), determine how far away each alternative is from the best choice, which is represented by the utility (S_i) and regret (R_i) measures.

$$S_i={{\displaystyle \sum }}_{j=1}^n{w}_j\frac{v_j^{+}-v_{ij}}{v_j^{+}-v_j^{-}}$$

(23)

$$R_i=\underset{j}{\max }(w_j\frac{v_j^{+}-v_{ij}}{v_j^{+}-v_j^{-}})$$

(24)

Step 5. Estimate the values for Q_i (rank indexes) by applying Equation (25).

$$Q_i=\left(\phi * \frac{S_i-S^{+}}{S^{-}-S^{+}}+\left(1-\phi \right)* \frac{R_i-R^{+}}{R^{-}-R^{+}}\right)$$

(25)

where, S⁺ = $\underset{i}{\mathit{min}}{S}_i$, S⁻ = $\underset{i}{\mathit{max}}{S}_i$, R⁺ = $\underset{i}{\mathit{min}}{R}_i$, R⁻ = $\underset{i}{\mathit{max}}{S}_i$

The weight for the “majority criteria” (also known as “the maximum group utility”) technique is represented by $\varnothing$ ∈ [0,1]. The lowest scores are based on R_i values, while the rate of quality is dependent on S_i values.

Step 6. In accordance with the Q_i values, arrange the best options in ascending order.

The VIKOR technique suggests a compromise option for the alternative, that is the best rated by Q (minimum) when the underlying two requirements are met [38,39].

C1. Allowable advantage

Q₂ − Q₁ ≥ DQ, where i = 2 is the second-best choice using Q and DQ = 1/(m − 1).

C2. Allowable stability

Additionally, R and/or S must also rank the option with i = 1 as the highest. It gives a number of potential options in the event that one of the requirements is not met.

• Alternatives i = 1 and i = 2 if only condition C₂ is not achieved or
• Alternatives i = 1, 2, …, m, if the condition C₁ is not attained, where m is specified by the Q_M − Q₁ < DQ relationship, for maximum i.

2.2.2. Grey Relation Analysis

Prof. Deng first established the concept of grey theory from the grey set in conjunction with control theory and space theory [40]. Grey theory is being used with the intention of exploiting its capacity to account for the complexity and ambiguity related to user preferences and data collection. The following method and Equations (26)–(32) can be used to implement GRA [41,42].

If x_i^*(k) denotes the sequence following the data processing, x_i^(O)(k) indicates the original sequence of responses, (where i = 1, 2,…, m and k = 1, 2,…, n), max x_i^(O)(k) and min x_i^(O)(k) denote the highest and lowest values of x_i^(O)(k), respectively. The following is how the data is standardized (see Equations (26)–(28)).

$$x_i^{*}\left(k\right)=\frac{x_i^{\left(O\right)}\left(k\right)-min x_i^{\left(O\right)}\left(k\right)}{\max x_i^{\left(O\right)} \left(k\right)-\min x_i^{\left(O\right)}\left(k\right)}\text{Larger-the-better}$$

(26)

$$x_i^{*}\left(k\right)=\frac{max x_i^{\left(O\right)}\left(k\right)-x_i^{\left(O\right)}\left(k\right) }{\max x_i^{\left(O\right)} \left(k\right)-\min x_i^{\left(O\right)}\left(k\right)}\text{Smaller-the-better}$$

(27)

$$x_i^{*}\left(k\right)=\frac{\left|x_i^{\left(O\right)}\left(k\right)-IV\right|}{\max \{max x_i^{\left(O\right)} \left(k\right)-IV, IV-\min x_i^{\left(O\right)}\left(k\right)\}}\text{Nominal-the-better (define the response as intended value (IV))}$$

(28)

A reference sequence is established employing the comparability sequences. Upon data processing, the preprocessed sequences are used to estimate the grey relational coefficient (GRC). Equation (29) is used to calculate the GRC.

$$\mathit{GRC}(x_O^{*}\left(k\right), x_i^{*}\left(k\right))=\frac{{\Delta }_{min}+\epsilon {\Delta }_{max}}{{\Delta }_{oi}\left(k\right)+ϵ {\Delta }_{max}}$$

(29)

where ∆_oi (k) is the deviation sequence of the reference sequence x_O^*(k) and the comparability x_i^*(k), i.e., ∆_oi (k) = |x_O^*(k) − x_i^*(k)| is the absolute magnitude of the difference between x_O^*(k) and x_i^*(k).

$${\Delta }_{min}=\underset{\forall i}{\min }\underset{\forall k}{\min }{\Delta }_{\mathit{oi}}\left(k\right)$$

(30)

$${\Delta }_{max}=\underset{\forall i}{\max }\underset{\forall k}{\max }{\Delta }_{\mathit{oi}}\left(k\right)$$

(31)

where i = 1, 2,…, m and k = 1, 2,…, n ϵ: distinguishing coefficient, ϵ [0,1]. The value of ϵ is set as 0.5. The grey relational grade (GRG), which is calculated using Equations (30) and (31), can be described as a weighting sum of the GRC.

$$\mathit{GRG}(x_o,{x_i}^{*})={{\displaystyle \sum }}_{k=1}^n{\beta }_k\gamma (x_o*(k),x_i*(k\left)\right)$$

(32)

where β_k represented the weighting value of the k^th performance characteristic, and ${{\displaystyle \sum }}_{k=1}^n{\beta }_k$ = 1.

The rating is calculated for each user and also for an imaginary user whose needs are the sum of each of the four unique users.

2.3. Sensitivity Analysis

The consistency and dependability of the ranking produced using the suggested approach are assessed using sensitivity analysis. It can be described as a technique for understanding how changes in input values affect a model’s results [43]. Undoubtedly, it is important to consider the changes in the proposed model’s output parameters brought on by changes in the values of the input variable [44,45]. The model is reliable and trustworthy if the result is not highly sensitive to changes in input. The most popular sensitive analysis method is changing the performance criterion weights and examining how the outcomes evolve [46]. Previous studies have also shown that the ranks of the alternatives strongly depend on the weight coefficients of the criterion [47]. Sensitivity analysis based on variations in the weight coefficients is used in this study to confirm the model’s validity and check the accuracy of the findings. The first stage in sensitivity analysis is the random assignment of a criterion. The chosen criterion’s weight is then changed by a certain percentage (increase or decrease). As a result, Equation (33) is used to calculate the weights for the remaining criteria [48].

$$w_n^{*}=\frac{w_n\left(1-w_i^{*}\right)}{\left(1-w_i\right)}$$

(33)

where,

w_i: original weights for attribute i

w_i^*: weight derived after adjusting the original weight by 10% for attribute i

w_n: original weight for attribute n

w_n^*: recalculated weight for attribute n

To assess the robustness of the proposed method and to examine the similarity of ranks obtained by varying the weights, Kendall’s coefficient (Z) of concordance is used [49,50]. Kendall’s coefficient, whose values range from 0 to 1, illustrates analogies in the ranking of sorted quantities. For instance, a value of 1 indicates that all of the different ranking orders perfectly match each other. It implies that the robustness will be better the closer the value of Z is to 1. Equations (34)–(36) can be used to calculate the value of Z [49,50].

$$Z=\frac{12R}{m^2 \left(k^3-k\right)}$$

(34)

$$R_i={{\displaystyle \sum }}_{j=1}^m{r}_{ij}$$

(35)

$$R={{\displaystyle \sum }}_{i=1}^k{\left(R_i-\overline{R}\right)}^2$$

(36)

where m is the number of scenarios, k is the number of options, and r_ij is the score that scenario i gives to alternative j.

As shown in

Table 1. Scenarios explored in the sensitivity analysis.

Scenario	Description
1	20% increase in weight for average temperature
2	40% increase in weight for the population
3	20% decrease in weight for topography
4	30% increase in weight for global horizontal irradiance
5	40% decrease in weight for dust storm
6	30% decrease in weight for wind speed

, six different scenarios based on weight percentage are devised. These scenarios involve a percentage in weight for a chosen criterion and subsequently changing the weights of the remaining criteria using Equation (36).

3. Data Collection

3.1. Study Area

The KSA is located in the farthest region of southwestern Asia, surrounded by the Red Sea to the west and the Arabian Gulf to the east. The KSA, with a total area of around 2,000,000 square kilometers, makes up approximately four-fifths of the Arab Peninsula [51]. Saudi Arabia has a desert climate, with summers that are exceedingly hot and dry, with temperatures ranging from 27 °C to 43 °C inland and 27 °C to 38 °C by the shore [52].

3.2. Criteria for MCDM

Factors from technological, economical, ecological, social, and risk considerations are examined in order to discover the ideal location for a PV solar power plant to produce electricity [21,53,54,55,56]. The following is a brief description of these criteria.

• Solar radiation (C1). The yearly solar radiation is a meteorological factor that is applied to quantify the intensity of sunlight for a prospective site. The uninterrupted functioning of a PV plant relies upon solar irradiance. The opportunity for producing energy in a location increases with the amount of solar radiation available. In this research, the amount of solar radiation received by a surface is measured using the Global Horizontal Irradiance (GHI) [57]. The unit of measurement for GHI is in Watts per square meter per day (W/m²/day).
• Average air temperature (C2). The efficiency of power generation in PV systems is significantly influenced by ambient temperature [58]. The efficiency of power conversion in solar cells decreases with increasing ambient temperature, resulting in a reduction in the amount of power produced. The air temperature is measured in °C.
• Wind speed (C3). The efficacy of a PV system’s energy production is influenced by wind speed. The rate at which solar systems cool down increases with wind speed, which is accompanied by an increase in power production. It is measured in kilometers per hour (km/h).
• Sunshine hours (C4). The territory with more sun hours has the ability to generate more power when taking into account that various regions receive the same quantity of solar radiation. It is measured in hours (h).
• Sand and dust storm (C5). A suitable parameter for solar PV systems can be sand and dust storm [59]. Among the globe’s places where sand and dust storm existence are particularly intense is the Arabian Peninsula. The extent of radiation reaching the surface of PV panels is decreased with a greater incidence of storm phenomena. As a result, the amount of power generated in locations prone to higher sand storms is also decreased. It is calculated using the average yearly number of storms.
• Topography (C6). Minimal elevation fluctuations aid in lowering the high construction costs and flat topography is generally preferred for the placement of PV plants. Due to low economic viability, more change in topography or terrain is not acceptable. A maximum elevation change in a specific location can be approximated in feet.
• Population (C7). A high population in a region causes both a larger energy demand as well as a higher need for employment. Consequently, job generation increases with an increase in population. It implies that having a higher population in a specific area makes it a good location for a PV installation. As a result, the site of a solar system must be preferable to one where there is enough consumption and trained personnel to run and manage the PV system.

Table 2. Criteria and their values.

Cities	Criteria
	C1	C2	C3	C4	C5	C6	C7
Arar	5392	22	14	12.16	0.30	174	148,540
Al-Jouf	5016	21.87	13.5	12.16	0.57	82	102,903
Tabuk	5992	22.98	10	12.16	0.16	125	455,450
Hail	5359	23.3	12	12.15	0.28	387	267,005
Dhahran	5205	26.4	15	12.16	0.15	525	99,540
Al-Ahsa	5278	26.9	12	12.15	0.91	157	293,179
Taif	5671	22.2	14	12.15	0.08	459	530,848
Makkah	5303	28.6	3	12.15	0.02	1106	1,323,624
Jeddah	5525	28.1	13	12.13	0.17	112	2,867,446
Yanbu	5765	27.7	11	12.14	0.48	190	200,161
Medina	5376	27.2	11	12.14	0.09	276	1,300,000
Riyadh	6040	27	11	12.15	0.16	200	4,205,961
Abha	5674	19.2	11	12.15	0.05	1942	210,886
Jizan	4663	28.55	11	12.13	0.60	190	105,198
Najran	6684	23.9	11	12.15	0.26	1486	258,573

displays the values of the chosen criteria that have been acquired from various sources [21,52,54,60,61,62,63,64,65]. Since the information has been gathered from a number of sources, the authors cannot guarantee that it is updated and highly accurate. The main focus of the authors’ research is the implementation and viability of MCDM techniques in the problem of solar site selection, even though they have made every effort to acquire data that is as accurate as possible.

We have solely taken technical factors into account due to our consideration of cities. It is presumed that economic considerations, such as distance to power lines and distance to urban centers, are not very significant and can be assumed to be the same for all cities. They have not been found to be important as past literature has claimed [21,53,56].

4. Implementation

The effectiveness and efficiency of solar power generation are influenced by the site location. A lower benefit-to-cost ratio may result from the site’s random selection. A variety of MCDM techniques are combined to find the optimal site. The decision-making tools used in this research have advantages and limitations of their own. The various strategies used to determine the ideal location for solar power generation are as follows. Additionally, the estimation of the weight or importance of various criteria is the most significant step in the use of any MCDM method. Numerous techniques including GEM, FAHP, and entropy approaches have been used in this work. The development of the decision matrix comes before the estimate of weights using these techniques. These decision tables are created with the assistance of professionals with extensive experience in the solar energy sector. Three specialists (E1, E2, and E3), including academics, researchers, etc., are taken into consideration. These experts are asked to rate the chosen criteria on a scale of 1 to 10, with 1 representing the lowest priority and 10 the most important.

Table 3. Assessment of criteria by experts.

Criteria	E1	E2	E3
C1	9	10	9
C2	8	7	8
C3	7	8	6
C4	9	9	8
C5	7	8	6
C6	6	5	5
C7	4	6	4

displays the ranking outcomes of the experts.

The decision matrix obtained from the experts must be analyzed to see whether their subjective assessment is valid. Thus, the consistency ratio-based approach established by Prof. Saaty is adopted in the paper [66]. In this method, the ratio between the consistency index (CI) and random consistency index (RI) yields a consistency ratio (CR). According to this method, if the CR is 10% or less, the inconsistency is acceptable; otherwise, the subjective judgment needs to be changed. An earlier study described in [67] provides an illustration of CR estimation. For experts E1, E2, and E3, the calculated CR is, respectively, 3.91%, 3.97%, and 4.02%, which is substantially less than 10%. This suggests that the expert’s subjective judgment is reasonable.

4.1. GEM-VIKOR

The encoding of ratings by the evaluation matrix x, as illustrated below, is the preliminary step in the use of GEM-VIKOR.

$$x=\left(\begin{array}{ccccccc}9& 8& 7& 9& 7& 6& 4\\ 10& 7& 8& 9& 8& 5& 6\\ 9& 8& 6& 8& 6& 5& 4\end{array}\right);x^T=\left(\begin{array}{ccc}9& 10& 9\\ 8& 7& 8\\ 7& 8& 6\\ 9& 9& 8\\ 7& 8& 6\\ 6& 5& 5\\ 4& 6& 4\end{array}\right)$$

$$F=\left(\begin{array}{ccccccc}262& 214& 197& 243& 197& 149& 132\\ 214& 177& 160& 199& 160& 123& 106\\ 197& 160& 149& 183& 149& 112& 100\\ 243& 199& 183& 226& 183& 139& 122\\ 197& 160& 149& 183& 149& 112& 100\\ 149& 123& 112& 139& 112& 86& 74\\ 132& 106& 100& 122& 100& 74& 68\end{array}\right)$$

The ideal evaluation vector x^* has been derived in

Table 4. Outcomes from the power method.

k	0	1	2	3
ykT	(0.14, 0.14, 0.14, 0.14, 0.14, 0.14, 0.14)	(199.14, 162.71, 150, 185, 150, 113.57, 100.29)	(539.20, 440.77, 406.01, 500.91, 406.01, 307.58, 271.36)	(539.20, 440.77, 406.01, 500.91, 406.01, 307.58, 271.36)
$\left|\right|y_k\left|\right|{}_2$	-	410.262	1110.79	1110.79
zkT	-	(0.4854, 0.3966, 0.3656, 0.4509, 0.3656, 0.2768, 0.2444)	(0.4854, 0.3968, 0.3655, 0.4509, 0.3655, 0.2769, 0.2443)	(0.4854, 0.3968, 0.3655, 0.4509, 0.3655, 0.2769, 0.2443)

by applying the power approach and assuming precision, = 0.0005.

Ideal evaluation vector, x_* = (0.4854, 0.3968, 0.3655, 0.4509, 0.3655, 0.2769, 0.2443)^T

The GHI and sunshine hours, according to experts, are the most important criteria to consider when choosing a location for a solar power plant, whereas population and topology are regarded to be less important. The weights acquired for various criteria after normalization can be rewritten as

w = (0.1877, 0.1535, 0.1414, 0.1744, 0.1414, 0.1071, 0.0945)

The establishment of the decision matrix, as demonstrated in Table 2, is the first step in the VIKOR implementation process. The weights generated from the GEM technique are then used to construct the weighted normalized decision matrix (

Table 5. Weighted normalized decision matrix.

Cities	Criteria
	C1	C2	C3	C4	C5	C6	C7
Arar	0.0471	0.0346	0.0433	0.0451	0.0293	0.0065	0.0026
Al-Jouf	0.0438	0.0344	0.0417	0.0451	0.0557	0.0031	0.0018
Tabuk	0.0523	0.0361	0.0309	0.0451	0.0158	0.0047	0.0078
Hail	0.0468	0.0366	0.0371	0.0450	0.0276	0.0145	0.0046
Dhahran	0.0455	0.0415	0.0464	0.0451	0.0146	0.0197	0.0017
Al-Ahsa	0.0461	0.0423	0.0371	0.0450	0.0886	0.0059	0.0050
Taif	0.0495	0.0349	0.0433	0.0450	0.0076	0.0173	0.0091
Makkah	0.0463	0.0449	0.0093	0.0450	0.0020	0.0416	0.0228
Jeddah	0.0483	0.0441	0.0402	0.0450	0.0166	0.0042	0.0493
Yanbu	0.0504	0.0435	0.0340	0.0450	0.0466	0.0071	0.0034
Medina	0.0470	0.0427	0.0340	0.0450	0.0089	0.0104	0.0224
Riyadh	0.0528	0.0424	0.0340	0.0450	0.0159	0.0075	0.0723
Abha	0.0496	0.0302	0.0340	0.0450	0.0046	0.0730	0.0036
Jizan	0.0407	0.0448	0.0340	0.0450	0.0588	0.0071	0.0018
Najran	0.0584	0.0375	0.0340	0.0450	0.0249	0.0559	0.0044

).

To compute v_j⁺ and v_j⁻ as shown in

Table 6. Estimation of v_j⁺ and v_j⁻.

	Criteria
	C1	C2	C3	C4	C5	C6	C7
vj+	0.0584	0.0302	0.0464	0.0451	0.0020	0.0031	0.0723
vj−	0.0407	0.0449	0.0093	0.0450	0.0886	0.0730	0.0017

, the different attributes have been categorized as benefit and cost attributes.

Set of benefit criteria = C1, C3, C4, and C7

Set of cost criteria = C2, C5, and C6

The values of S_i, R_i, and Q_i are computed in the final steps of the VIKOR technique as shown in

Table 7. Computation of S_i, R_i, and Q_i.

Cities	Criteria							Si	Ri	Qi
	C1	C2	C3	C4	C5	C6	C7
Arar	0.0471	0.0346	0.0433	0.0451	0.0293	0.0065	0.0026	0.3206	0.1200	0.1929
Al-Jouf	0.0438	0.0344	0.0417	0.0451	0.0557	0.0031	0.0018	0.3983	0.1550	0.4497
Tabuk	0.0523	0.0361	0.0309	0.0451	0.0158	0.0047	0.0078	0.2961	0.0863	0.0000
Hail	0.0468	0.0366	0.0371	0.0450	0.0276	0.0145	0.0046	0.4335	0.1231	0.3309
Dhahran	0.0455	0.0415	0.0464	0.0451	0.0146	0.0197	0.0017	0.3955	0.1374	0.3601
Al-Ahsa	0.0461	0.0423	0.0371	0.0450	0.0886	0.0059	0.0050	0.5856	0.1414	0.5866
Taif	0.0495	0.0349	0.0433	0.0450	0.0076	0.0173	0.0091	0.3284	0.0941	0.0737
Makkah	0.0463	0.0449	0.0093	0.0450	0.0020	0.0416	0.0228	0.6066	0.1535	0.6692
Jeddah	0.0483	0.0441	0.0402	0.0450	0.0166	0.0042	0.0493	0.5073	0.1744	0.6644
Yanbu	0.0504	0.0435	0.0340	0.0450	0.0466	0.0071	0.0034	0.5586	0.1388	0.5446
Medina	0.0470	0.0427	0.0340	0.0450	0.0089	0.0104	0.0224	0.5049	0.1306	0.4458
Riyadh	0.0528	0.0424	0.0340	0.0450	0.0159	0.0075	0.0723	0.3219	0.1274	0.2305
Abha	0.0496	0.0302	0.0340	0.0450	0.0046	0.0730	0.0036	0.4024	0.1071	0.2182
Jizan	0.0407	0.0448	0.0340	0.0450	0.0588	0.0071	0.0018	0.7553	0.1877	1.0000
Najran	0.0584	0.0375	0.0340	0.0450	0.0249	0.0559	0.0044	0.3910	0.0908	0.1256

. The options are then ranked in order of decreasing Q values, with the best choice having the lowest Q value.

Tabuk is ranked first in the VIKOR ranking scheme, followed by Taif and Najran. In VIKOR, a compromise solution is put into practice in order to verify the two conditions mentioned in Section 2.2.2 above. The condition 1 (acceptable advantage) is satisfied since Q₂–Q₁ (0.0737) ≥ DQ (0.07143). As a result, it is feasible to differentiate the best one between Tabuk and Taif. Additionally, the alternative that received the highest Q value also received the highest R and S rankings. Thus, it meets both criteria for acceptability. It implies that condition 2 is also satisfied. Consequently, based on the GEM-VIKOR approach, Tabuk is suggested as the ideal location for a solar power plant.

4.2. FAHP-GRA

In this method, the alternatives are ranked using the GRA technique, and weights are estimated using FAHP. Below are the steps for putting this strategy into practice. First, trapezoidal fuzzy numbers are generated from the crisp values collected from each expert in the form of a decision matrix.

Table 8. Conversion of crisp values to trapezoidal fuzzy numbers.

Importance	Explanation	Trapezoidal Fuzzy Number	Importance	Trapezoidal Fuzzy Number
1	Equal importance	(1,1,1,1)	1	(1,1,1,1)
3	Moderate importance	(2, 2.5, 3.5, 4)	0.3333	(0.25, 0.286, 0.4, 0.5)
5	Strong importance	(4, 4.5, 5.5, 6)	0.2	(0.167, 0.182, 0.222, 0.25)
7	Very strong importance	(6, 6.5, 7.5, 8)	0.1429	(0.125, 0.133, 0.154, 0.167)
9	Extreme importance	(9, 9, 9, 9)	0.1111	(0.111, 0.111, 0.111, 0.111)
2	Intermediate values	(1, 1.5, 2.5, 3)	0.5	(0.333, 0.4, 0.667, 1)
4		(3, 3.5, 4.5, 5)	0.25	(0.2, 0.222, 0.286, 0.333)
6		(5, 5.5, 6.5, 7)	0.1667	(0.143, 0.154, 0.182, 0.2)
8		(7, 7.5, 8.5, 9)	0.125	(0.111, 0.118, 0.133, 0.143)

shows the trapezoidal fuzzy numbers for different levels of significance.

After transformation into trapezoidal fuzzy numbers, Chang’s extent analysis is applied to estimate the weights for various criteria for a specific expert. In this calculation, expert 1′s judgment matrix for several criteria is used as an illustration. The first step is to create a fuzzy pairwise matrix as shown in

Table 9. Fuzzy pairwise comparison matrix.

Criteria	C1	C2	C3	C4	C5	C6	C7
C1	(1,1,1,1)	(1, 1.5, 2.5, 3)	(2, 2.5, 3.5, 4)	(1,1,1,1)	(2, 2.5, 3.5, 4)	(3, 3.5, 4.5, 5)	(5, 5.5, 6.5, 7)
C2	(0.333, 0.4, 0.667, 1)	(1,1,1,1)	(1, 1.5, 2.5, 3)	(0.333, 0.4, 0.667, 1)	(1, 1.5, 2.5, 3)	(2, 2.5, 3.5, 4)	(4, 4.5, 5.5, 6)
C3	(0.25, 0.286, 0.4, 0.5)	(0.333, 0.4, 0.667, 1)	(1,1,1,1)	(0.25, 0.286, 0.4, 0.5)	(1,1,1,1)	(1, 1.5, 2.5, 3)	(3, 3.5, 4.5, 5)
C4	(1,1,1,1)	(1, 1.5, 2.5, 3)	(2, 2.5, 3.5, 4)	(1,1,1,1)	(2, 2.5, 3.5, 4)	(3, 3.5, 4.5, 5)	(5, 5.5, 6.5, 7)
C5	(0.25, 0.286, 0.4, 0.5)	(0.333, 0.4, 0.667, 1)	(1,1,1,1)	(0.25, 0.286, 0.4, 0.5)	(1,1,1,1)	(1, 1.5, 2.5, 3)	(3, 3.5, 4.5, 5)
C6	(0.2, 0.222, 0.286, 0.333)	(0.25, 0.286, 0.4, 0.5)	(0.333, 0.4, 0.667, 1)	(0.2, 0.222, 0.286, 0.333)	(0.333, 0.4, 0.667, 1)	(1,1,1,1)	(2, 2.5, 3.5, 4)
C7	(0.143, 0.154, 0.182, 0.2)	(0.167, 0.182, 0.222, 0.25)	(0.2, 0.222, 0.286, 0.333)	(0.143, 0.154, 0.182, 0.2)	(0.2, 0.222, 0.286, 0.333)	(0.25, 0.286, 0.4, 0.5)	(1,1,1,1)

.

Subsequently, the synthetic fuzzy values are measured utilizing Equations (7)–(14).

S₁ = (15, 17.5, 22.5, 25) ⊗ (1/103.982, 1/91.632, 1/69.994, 1/59.751).

S₂ = (9.666, 11.8, 16.334, 19) ⊗ (1/103.982, 1/91.632, 1/69.994, 1/59.751).

S₃ = (6.833, 7.972, 10.467, 12) ⊗ (1/103.982, 1/91.632, 1/69.994, 1/59.751).

S₄ = (15, 17.5, 22.5, 25) ⊗ (1/103.982, 1/91.632, 1/69.994, 1/59.751).

S₅ = (6.833, 7.972, 10.467, 12) ⊗ (1/103.982, 1/91.632, 1/69.994, 1/59.751).

S₆ = (4.316, 5.03, 6.806, 8.166) ⊗ (1/103.982, 1/91.632, 1/69.994, 1/59.751).

S₇ = (2.103, 2.22, 2.558, 2.816) ⊗ (1/103.982, 1/91.632, 1/69.994, 1/59.751).

These fuzzy values are then compared using Equation (12).

V (S₁ ≥ S₂) = 1, V (S₁ ≥ S₃) = 1, V (S₁ ≥ S₄) = 1, V (S₁ ≥ S₅) = 1, V (S₁ ≥ S₆) = 1, V (S₁ ≥ Sc₇) = 1.

V (S₂ ≥ S₁) = 1, V (S₂ ≥ S₃) = 1, V (S₂ ≥ S₄) = 1, V (S₂ ≥ S₅) = 1, V (S₂ ≥ S₆) = 1, V (S₁ ≥ S7) = 1.

V (S₃ ≥ S₁) = 0.8799, V (S₃ ≥ S₂) = 1, V (S₃ ≥ S₄) = 0.8799, V (S₃ ≥ S₅) = 1, V (S₃ ≥ S₆) = 1, V (S₃ ≥ S₇) = 1.

V (S₄ ≥ S₁) = 1, V (S₄ ≥ S₂) = 1, V (S₄ ≥ S₃) = 1, V (S₄ ≥ S₅) = 1, V (S₄ ≥ S₆) = 1, V (S₄ ≥ S₇) = 1.

V (S₅ ≥ S₁) = 0.8799, V (S₅ ≥ S₂) = 1, V (S₅ ≥ S₃) = 0.8799, V (S₅ ≥ S₄) = 1, V (S₅ ≥ S₆) = 1, V (S₅ ≥ S₇) = 1.

V (S₆ ≥ S₁) = 0.6663, V (S₆ ≥ S₂) = 0.8627, V (S₆ ≥ S₃) = 1, V (S₆ ≥ S₄) = 0.6663, V (S₆ ≥ S₅) = 1, V (S₆ ≥ S₇) = 1.

V (S₇ ≥ S₁) = 0.1931, V (S₇ ≥ S₂) = 0.3416, V (S₇ ≥ S₃) = 0.5529, V (S₇ ≥ S₄) = 0.1931, V (S₇ ≥ S₅) = 0.5529, V (S₇ ≥ S₆) = 0.7930.

The weights for various criteria can be estimated by using Equation (13).

d′(C1) = min (1, 1, 1, 1, 1, 1) = 1.

d′(C2) = min (1, 1, 1, 1, 1, 1) = 1.

d′(C3) = min (0.8799, 1, 0.8799, 1, 1, 1) = 0.8799.

d′(C4) = min (1, 1, 1, 1, 1, 1) = 1.

d′(C5) = min (0.8799, 1, 0.8799, 1, 1, 1) = 0.8799.

d′(C6) = min (0.6663, 0.8627, 1, 0.6663, 1, 1) = 0.6663.

d′(C7) = min (0.1931, 0.3416, 0.5529, 0.1931, 0.5529, 0.7930) = 0.1931.

The weight vector can now be expressed as below.

w′ = (1, 1, 0.8799, 1, 0.8799, 0.6663, 0.1931).

Upon normalization, the weight vector can be inferred as

w = (0.1780, 0.1780, 0.1566, 0.1780, 0.1566, 0.1186, 0.0344).

The remaining weights for the other decision matrices are calculated in a similar manner. The individual weights are also aggregated by applying the geometric mean (GM) method and normalized as indicated in

Table 10. Weights computed using FAHP.

Criteria	E1	E2	E3	Aggregated	Normalized
C1	0.1780	0.1888	0.1835	0.1834	0.1862
C2	0.1780	0.1384	0.1835	0.1654	0.1679
C3	0.1566	0.1762	0.1476	0.1597	0.1622
C4	0.1780	0.1888	0.1835	0.1834	0.1862
C5	0.1566	0.1762	0.1476	0.1597	0.1622
C6	0.1186	0.0404	0.1042	0.0794	0.0806
C7	0.0344	0.0910	0.0500	0.0539	0.0547

.

After computing the weights, the best location for a solar power plant is determined using the GRA approach. Data normalization is the first step in the GRA implementation process to produce comparability sequences, followed by deviation sequences and GRC. The comparability sequences, deviation sequences, and GRC are shown in

Table 11. Comparability sequences after data normalization.

C1	C2	C3	C4	C5	C6	C7
0.3607	0.7021	0.9167	1.0000	0.6857	0.9505	0.0119
0.1747	0.7160	0.8750	1.0000	0.3799	1.0000	0.0008
0.6576	0.5979	0.5833	1.0000	0.8415	0.9769	0.0867
0.3444	0.5638	0.7500	0.6667	0.7045	0.8360	0.0408
0.2682	0.2340	1.0000	1.0000	0.8546	0.7618	0.0000
0.3043	0.1809	0.7500	0.6667	0.0000	0.9597	0.0472
0.4988	0.6809	0.9167	0.6667	0.9353	0.7973	0.1050
0.3167	0.0000	0.0000	0.6667	1.0000	0.4495	0.2981
0.4265	0.0532	0.8333	0.0000	0.8321	0.9839	0.6740
0.5453	0.0957	0.6667	0.3333	0.4859	0.9419	0.0245
0.3528	0.1489	0.6667	0.3333	0.9203	0.8957	0.2923
0.6813	0.1702	0.6667	0.6667	0.8396	0.9366	1.0000
0.5002	1.0000	0.6667	0.6667	0.9700	0.0000	0.0271
0.0000	0.0053	0.6667	0.0000	0.3443	0.9419	0.0014
1.0000	0.5000	0.6667	0.6667	0.7364	0.2452	0.0387

,

Table 12. Deviation sequences after data pre-processing.

C1	C2	C3	C4	C5	C6	C7
0.6393	0.2979	0.0833	0.0000	0.3143	0.0495	0.9881
0.8253	0.2840	0.1250	0.0000	0.6201	0.0000	0.9992
0.3424	0.4021	0.4167	0.0000	0.1585	0.0231	0.9133
0.6556	0.4362	0.2500	0.3333	0.2955	0.1640	0.9592
0.7318	0.7660	0.0000	0.0000	0.1454	0.2382	1.0000
0.6957	0.8191	0.2500	0.3333	1.0000	0.0403	0.9528
0.5012	0.3191	0.0833	0.3333	0.0647	0.2027	0.8950
0.6833	1.0000	1.0000	0.3333	0.0000	0.5505	0.7019
0.5735	0.9468	0.1667	1.0000	0.1679	0.0161	0.3260
0.4547	0.9043	0.3333	0.6667	0.5141	0.0581	0.9755
0.6472	0.8511	0.3333	0.6667	0.0797	0.1043	0.7077
0.3187	0.8298	0.3333	0.3333	0.1604	0.0634	0.0000
0.4998	0.0000	0.3333	0.3333	0.0300	1.0000	0.9729
1.0000	0.9947	0.3333	1.0000	0.6557	0.0581	0.9986
0.0000	0.5000	0.3333	0.3333	0.2636	0.7548	0.9613

, and

Table 13. GRC computed depending on the deviation sequences.

C1	C2	C3	C4	C5	C6	C7
0.4389	0.6267	0.8571	1.0000	0.6141	0.9100	0.3360
0.3773	0.6377	0.8000	1.0000	0.4464	1.0000	0.3335
0.5935	0.5542	0.5455	1.0000	0.7593	0.9558	0.3538
0.4327	0.5341	0.6667	0.6000	0.6285	0.7530	0.3426
0.4059	0.3950	1.0000	1.0000	0.7747	0.6773	0.3333
0.4182	0.3790	0.6667	0.6000	0.3333	0.9254	0.3442
0.4994	0.6104	0.8571	0.6000	0.8854	0.7116	0.3584
0.4225	0.3333	0.3333	0.6000	1.0000	0.4759	0.4160
0.4658	0.3456	0.7500	0.3333	0.7486	0.9688	0.6054
0.5237	0.3561	0.6000	0.4286	0.4931	0.8960	0.3389
0.4358	0.3701	0.6000	0.4286	0.8625	0.8274	0.4140
0.6108	0.3760	0.6000	0.6000	0.7571	0.8874	1.0000
0.5001	1.0000	0.6000	0.6000	0.9434	0.3333	0.3395
0.3333	0.3345	0.6000	0.3333	0.4326	0.8960	0.3336
1.0000	0.5000	0.6000	0.6000	0.6548	0.3985	0.3422

, respectively.

The weights acquired through FAHP are then used to obtain the GRG and the corresponding ranking after the GRC has been estimated.

Table 14. GRG and ranking of the cities for installation of the solar power plant.

Cities	C1	C2	C3	C4	C5	C6	C7	Sum	Rank
Arar	0.0817	0.1052	0.1390	0.1862	0.0996	0.0733	0.0184	0.7035	1
Al-Jouf	0.0703	0.1071	0.1297	0.1862	0.0724	0.0806	0.0183	0.6645	6
Tabuk	0.1105	0.0931	0.0884	0.1862	0.1231	0.0770	0.0194	0.6978	2
Hail	0.0806	0.0897	0.1081	0.1117	0.1019	0.0607	0.0188	0.5715	9
Dhahran	0.0756	0.0663	0.1622	0.1862	0.1256	0.0546	0.0182	0.6887	3
Al-Ahsa	0.0779	0.0636	0.1081	0.1117	0.0541	0.0746	0.0188	0.5088	13
Taif	0.0930	0.1025	0.1390	0.1117	0.1436	0.0573	0.0196	0.6668	5
Makkah	0.0787	0.0560	0.0541	0.1117	0.1622	0.0384	0.0228	0.5237	12
Jeddah	0.0867	0.0580	0.1216	0.0621	0.1214	0.0781	0.0331	0.5611	10
Yanbu	0.0975	0.0598	0.0973	0.0798	0.0800	0.0722	0.0185	0.5051	14
Medina	0.0812	0.0621	0.0973	0.0798	0.1399	0.0667	0.0227	0.5496	11
Riyadh	0.1137	0.0631	0.0973	0.1117	0.1228	0.0715	0.0547	0.6349	8
Abha	0.0931	0.1679	0.0973	0.1117	0.1530	0.0269	0.0186	0.6685	4
Jizan	0.0621	0.0562	0.0973	0.0621	0.0702	0.0722	0.0183	0.4382	15
Najran	0.1862	0.0840	0.0973	0.1117	0.1062	0.0321	0.0187	0.6362	7

shows the GRG and the city’s respective rankings.

The Arar and Tabuk as stated in Table 14 can be recommended for the construction of a solar power plant based on the study utilizing FAHP and GRA.

4.3. Entropy-VIKOR

The criteria weights are computed in the consolidated Entropy-VIKOR scheme and VIKOR is used to rank the alternatives. The weighted normalized decision matrix is computed first, as depicted in

Table 15. Normalized decision matrix.

Cities	Criteria
	C1	C2	C3	C4	C5	C6	C7
Arar	0.8067	1.1458	0.9333	1.0000	14.4000	2.1220	0.0353
Al-Jouf	0.7504	1.1391	0.9000	1.0000	27.4400	1.0000	0.0245
Tabuk	0.8965	1.1969	0.6667	1.0000	7.7600	1.5244	0.1083
Hail	0.8018	1.2135	0.8000	0.9992	13.6000	4.7195	0.0635
Dhahran	0.7787	1.3750	1.0000	1.0000	7.2000	6.4024	0.0237
Al-Ahsa	0.7896	1.4010	0.8000	0.9992	43.6400	1.9146	0.0697
Taif	0.8484	1.1563	0.9333	0.9992	3.7600	5.5976	0.1262
Makkah	0.7934	1.4896	0.2000	0.9992	1.0000	13.4878	0.3147
Jeddah	0.8266	1.4635	0.8667	0.9975	8.1600	1.3659	0.6818
Yanbu	0.8625	1.4427	0.7333	0.9984	22.9200	2.3171	0.0476
Medina	0.8043	1.4167	0.7333	0.9984	4.4000	3.3659	0.3091
Riyadh	0.9037	1.4063	0.7333	0.9992	7.8400	2.4390	1.0000
Abha	0.8489	1.0000	0.7333	0.9992	2.2800	23.6829	0.0501
Jizan	0.6976	1.4870	0.7333	0.9975	28.9600	2.3171	0.0250
Najran	1.0000	1.2448	0.7333	0.9992	12.2400	18.1220	0.0615

, before implementing the entropy technique.

After standardization, the entropy H_j and weights are determined for different criteria as presented in

Table 16. Estimation of entropy and weight.

Criteria	C1	C2	C3	C4	C5	C6	C7
Hj	0.9987	0.9975	0.9873	1.0000	0.8778	0.8237	0.7382
m	15
ln m	2.7081
1 − Hj	0.0013	0.0025	0.0127	0.0000	0.1222	0.1763	0.2618
n − ∑ Hj	0.5767
wj	0.0022	0.0043	0.0219	0.0000	0.2118	0.3057	0.4540
1 − wj	0.9978	0.9957	0.9781	1.0000	0.7882	0.6943	0.5460
Normalized wj	0.1663	0.1659	0.1630	0.1667	0.1314	0.1157	0.0910

.

After the entropy weights are determined, the choices are ranked using the VIKOR based on various criteria. The VIKOR approach indicates that Tabuk or Taif are both viable options because only condition 2 (Section 2.2.1) is met. Tabuk and Taif are therefore the best candidates for the location of a solar power plant according to the results of the Entropy-VIKOR technique.

Similarly, the remaining approaches are applied and the findings are listed in

Table 17. Findings from several MCDM techniques.

MCDM Approach	Ideal Location
GEM-VIKOR	Tabuk
FAHP-VIKOR	Tabuk
Entropy-VIKOR	Tabuk or Taif
GEM-GRA	Tabuk
FAHP-GRA	Arar (Tabuk is second-ranked)
Entropy-GRA	Arar (Tabuk is second-ranked)

.

Kendall’s coefficient of concordance, W (described in Section 2.3) is calculated to assess the degree of ranking agreement between various MCDM approaches. The W is calculated to be 0.8741 for this site selection problem for solar power plants. It shows a high degree of agreement amongst the various methodologies.

The sensitivity analysis is undertaken by varying the weights in accordance with the scenarios in Table 1 and calculating the rank using each approach. The ranking order for the six scenarios generated using each approach is examined for similarity in order to obtain the values of W. The greater values of W imply that the established ranking order is dependable and not overly sensitive to changes in weight. The outputs of the sensitivity study, which are presented in

Table 18. Outcomes obtained in the sensitivity analysis.

MCDM Technique	GEM-VIKOR	FAHP-VIKOR	Entropy-VIKOR	GEM-GRA	FAHP-GRA	Entropy-GRA
W	0.9598	0.9759	0.9694	0.9752	0.9821	0.9841

show that the ranking order ascertained using various methodologies is reliable and stable. It makes sense given that each method’s W value is more closely related to 1.

5. Results and Discussion

The combinations of MCDM techniques used in this study qualify as logical, practical, and systematic approaches. They are effective in assisting to choose the best location for the generation of solar energy. Because there are so many MCDM strategies already in use, each with advantages and disadvantages, it is necessary to choose the right approach. The effort required and the outcomes produced by various MCDM strategies vary. As a result, the selection of a suitable MCDM approach has a big impact on the decision’s quality as well as the work that must be conducted. As illustrated in

Table 19. Comparative analysis of MCDM methods.

Technique	Combination		Ideal Alternative	Benefits	Limitations	Difficulty	Problem Size
	Weights	Ranking
GEM-VIKOR	GEM	VIKOR	Tabuk	Higher decision reliability. Precise and consistent evaluation. Suitable for medium-scale problems.	Generation of the judgment matrix is relatively difficult, i.e., shortlisting experts and collecting precise information from them is a hectic task.	6	Medium-sized
FAHP-VIKOR	FAHP	VIKOR	Tabuk	Higher precision in small data problems. Checks inconsistency through the consistency index. Utilizes inherent information of criteria.	Large-sized problems can be demanding. Rank reversal problem—the final ranking can be reversed with the addition or elimination of an alternative. Geometric aggregation approach is used, so there is a possibility that some information may be lost.	8	Small-sized
Entropy-VIKOR	Entropy	VIKOR	Tabuk or Taif	Unbiased results. Capable of handling multiple inputs and outputs. Flexible to fit small and medium-sized problems.	Sensitive to inconsistent data.	5	Large -sized
GEM-GRA	GEM	GRA	Tabuk	Straightforward and uncomplicated method. Require precise information. Suitable for medium and small-sized problems.	Generation of judgement matrix is relatively difficult. Can be difficult for large-sized problems.	5	Medium-sized
FAHP-GRA	FAHP	GRA	Arar (Tabuk is second-ranked)	Takes into account uncertainty and vagueness. Checks inconsistency through consistency index. Full use of inherent information of criteria. Suitable for small-sized problems.	Complicated and computationally exhaustive even for medium-sized problems. Possibility of rank reversal.	6	Small-sized
Entropy-GRA	Entropy	GRA	Arar (Tabuk is second-ranked)	Suitable for large-sized problems. Simple and easy to use. Easy to understand. Number of steps remains the same regardless of the number of criteria.	Consistency is not controlled. Results may differ from other methods.	4	Large-sized

Large-sized > 15 criteria. Medium-sized = 10–15 criteria. Small-sized < 10 criteria.

, different MCDM techniques can rank identical alternatives in distinct sequences. The various approaches have varied computational needs and varying degrees of difficulty. Some strategies work well for large-sized problems, while others are better suited to small-sized problems.

The degree of difficulty of the various approaches has been graded on a scale from 1 to 10, with 10 denoting the greatest degree of difficulty. These scores may change depending on the user’s comprehension and the execution method. The Entropy-GRA method is the easiest of all the techniques used, whilst the FAHP-VIKOR method is the most difficult. Keep in mind that some strategies work effectively with medium-sized problems while others perform better with larger-sized problems. Large-sized problems are those that involve more than 15 criteria, whereas small-sized problems only require less than ten criteria. Table 19 presents a detailed summary of the advantages and drawbacks of several MCDM techniques. The FAHP-VIKOR method provides a number of advantages, including improved precision, the use of inherent information in the criteria, and checking for inconsistency using the consistency index, despite the fact that it can be challenging, especially for large-scale situations. Entropy-GRA is different in that it is the simplest and best suited for large-scale problems, but consistency cannot be maintained. This shows that each MCDM strategy has its own unique advantages and constraints. Therefore, it is crucial to choose the right strategy based on the problem, the information that is available, the simplicity of use, consistency, etc.

It is evident that choosing the option that can be replicated by a number of MCDM procedures is the best option out of a range of alternatives. Tabuk can therefore be chosen as the best location for the production of solar energy. This outcome is comparable to that of past research. For instance, Al-Shammari et al. [21] ranked Tabuk city best for PV installation after analyzing 17 Saudi Arabian cities using an AHP-TOPSIS-based approach. Similarly, Tabuk was proven to be the optimal site for both PV and concentrated solar power in a study by Awan et al. [68] that compared three cities, namely Majmaah, Najran, and Tabuk. Tabuk station was recognized to be the preferred venue for a PV power plant with an energy power of 110,250 kWh in other investigations by Awan et al. [1]. Moreover, Tabuk was classified as one of the cities with the best prospects for renewable energy plants by Brumana et al. [69] due to the city’s year-round adequate solar radiation and wind velocity. Mohammed et al. [70] also came to the conclusion that Tabuk city is the ideal place to deploy a PV system for residential structures.

The factors responsible for Tabuk’s selection can be traced to its high GHI. However, it can be demonstrated that Riyadh likewise has a high GHI value but also has a greater ambient temperature, making it less suitable than Tabuk for the generation of solar energy. Similarly, Najran has a very high GHI value but is not chosen because it also has a higher ambient temperature, more dust storms, and more topographic variation. The Tabuk region typically features clear days with minimal clouds and low dust, which is another factor contributing to the higher average GHI value [5]. Furthermore, if a simple and a sophisticated method yield similar rankings, the simplest approach should be used because it can reduce computing time and effort. The strategy that has the fewest steps to estimate weight and rank is the simplest. Moreover, the stakeholders or users can easily comprehend the results using the simple and direct approaches, which are only marginally impacted by the problem’s size. From an ecological standpoint, it has been discovered, according to Rehman et al. [71], that on average 8182 tons of greenhouse emissions can be inhibited from invading the local atmosphere every year. This highlights how important it is to use renewable energy sources and to increase their effectiveness and efficiency, especially by choosing an ideal location for the installation of their generation plants and so by employing proper MCDM methodologies.

The established ranking order is dependent and not unduly susceptible to variations in weight, as evidenced by the higher values of W for the various scenarios developed utilizing each approach. Additionally, the results are adequate and steady as indicated by the ranking order from several MCDM techniques, which is implied by a considerably larger value of W that is nearer to 1.

The work is distinctive since it takes into account a number of factors connected to the choice of location for a solar power facility. The selection procedure in this study is made more flexible, accurate, and useful by the employment of numerous MCDM approaches. When choosing locations for solar power plants or other similar problems, where the level of uncertainty is predicted to vary according to environmental factors, the utilized MCDM methodologies are very useful. However, as the number of criteria and alternative locations increases, some of these selection processes, may become repetitive, computationally difficult, and tiresome. Additionally, because the effectiveness of most of these decision-making methods depends on expert assessments, they frequently tend to exaggerate the ranking process. Because the basic problem of solar power plant site selection utilizing consolidated MCDM techniques has not been treated with proactive diligence due to its dynamic nature, this research intends to assist stakeholders in selecting the most appropriate location. The repository needs to be updated frequently with new criteria and reliable data due to the constantly changing solar energy industry and shifting environmental concerns. For this reason, new efforts should investigate more precise information and different sites in order to make a more trustworthy conclusion regarding the location of solar power plants.

6. Conclusions

Many MCDM approaches have been proposed to address the issue of selecting the appropriate site for solar power generation. A concise guideline for choosing an MCDM approach for solar power plant site selection is provided by the examination of several methods. Consequently, the following conclusions about the outcomes of this study can be drawn.

• The fact that different strategies might produce different outcomes when used to solve the same problem is a key critique of MCDM. A decision-maker need to select a course of action that comes the closest to the ideal. As a result, the best solution can be one that is repeated by numerous MCDM procedures.
• This paper emphasizes the significance of evaluating several decision-making strategies and choosing the most suitable methodology for the specific application, without implying that any one MCDM method is superior to other methods.
• The outcomes of the different techniques might not be the same. This is explained by various weights and their distributions, as well as various solution algorithms.
• All three weight calculation methods have established solar radiation and sunshine hours as the most important criteria. For instance, using GEM, solar radiation and sunlight hours are given weights of 0.1877 and 0.1744, respectively, thereby contributing 18.77% and 17.44% to the choice of the PV site. Similarly, solar radiation and sunshine hour receive the same weight value of 0.1862 from the FAHP. Solar radiation and sunlight hours have entropy weights of 0.1663 and 0.1667, respectively.
• The type of MCDM approach selected affects the decision’s quality and the amount of work necessary. The various approaches display differing degrees of difficulty and demand varying degrees of computation.
• It makes sense to employ one of the simplest techniques. However, the use of multiple techniques is suggested in order to verify consistency and improve the trustworthiness of the results.
• It should be noted that some strategies work well for large-sized problems while others are better suited for small-scale problems.
• Of all the methods utilized in this study, the FAHP-VIKOR technique is the most exhaustive with a difficulty level of 8, while Entropy-GRA is the easiest with a complexity degree of 4.
• A decision among numerous options that is replicated by several MCDM approaches can be regarded as the best option. As a result, Tabuk is the ideal location for the construction of a solar power plant. High solar radiation (GHI value of 5992 W/m²/day) and more sunshine hours (12.16 h/day) are the main factors that contribute to its selection.
• The ranking consistency among the various MCDM techniques employed in the study is reasonable, as indicated by Kendall’s coefficient of concordance value of 0.8741, which is very close to 1.
• The likelihood of uncertainty in expert decision-making as well as the lack of precise data collection is the work’s limitations. Future studies will find ways to circumvent these restrictions. Future versions of the work will also be expanded by integrating additional accurate data, expert opinions, different cities or locations, and improvised MCDM procedures.