Spatial Decision-Making under Uncertainties for Supporting the Prospection of Sites for Hybrid Renewable Energy Generation Systems

Abstract

This research aims at developing methodological tools for decision makers (DMs) to determine the locations for generating sites for hybrid renewable energy systems, considering complementarity characteristics of the corresponding sources and their seasonal availability. The decision process on new sites starts using Geographic Information Systems for modeling relevant spatial criteria. The problem solution is associated with using the Slide-OWA operator, which permits one to control the inter-criteria compensation levels by adjusting pessimism and optimism parameters. After constructing multicriteria estimates for generating sites, the general scheme of multi-objective decision-making is applied in conditions of uncertainty since these estimates are subject to significant levels of uncertainty. Such a scheme is based on the possibilistic approach, which involves the construction of payoff matrices. At this point, it is possible that some alternatives cannot be distinguished based only on the criteria considered so far, or the DM may want to evaluate the alternatives from the point of view of additional considerations. For this purpose, the framework of multi-attribute decision-making models is applied by employing methods for preference modeling in a fuzzy environment. The present work results are applied to a case study of the Minas Gerais State in Brazil. These results play a strategic role for governments and investors in their decisions while considering the ability to meet a wide range of criteria.

1. Introduction

Several specific peculiarities must be considered in the exploration of renewable energy sources. In particular, their use requires dealing with their intermittent character. One possible way to cope with generation discontinuities is to store energy to ensure energy supply [1]. However, such an alternative turns out to be economically unjustified due to the high levels of production and consumption. An efficient approach to overcome this situation is to form hybrid renewable energy systems [2]. Such systems make use of two or more combined sources. Their operation may be intended to guarantee stipulated demands or, occasionally, to feed the network with any surplus energy [3].

Renewable energy potential has an irregular spatial dispersion determined by the availability of natural resources. Furthermore, in analyzing suitable generating sites, it is necessary to prioritize projects following a large set of considerations, including the strategic goals of governments and investors.

Due to all the preceding considerations, choosing an efficient generation matrix and a location for the power plants demands a meticulous decision-making process. This research is particularly interested in supporting the initial steps of such a decision process; more precisely, decisions on how to carry out the prospection of adequate sites for Hybrid Renewable Energy Generation Systems (HREGSs).

The present work aims at supporting the decision makers (DMs) in the evaluation, comparison, choice, prioritization, and ordering of generation sites in conditions of uncertainty. The main contribution of this work is the adaptation and improvement of models and methods of decision-making in an environment of uncertainty [4,5]. This is an innovative contribution to the definition of implantation sites of renewable energy generation plants, considering characteristics of complementarity of seasonal availability.

The decision process begins with studies on the quantitative spatial criteria relevant to the decision. These criteria are modeled with models of Multicriteria Analysis by using Geographic Information Systems (GISs) to carry out the so-called Spatial Multicriteria Analysis (SMA) [6]. Such SMA allows the identification of exclusion zones based on the corresponding restrictive criteria [7]. The suitability of the remaining area can be assessed according to a second set of criteria. Such SMA enables the construction of maps that highlight the relative suitability of locations for setting up power plants.

Furthermore, SMA allows the application of the fuzzy sets theory to normalize the considered criteria. Thus, it is necessary to construct a fuzzy set-based model defined on georeferenced information domains. This model is to include parameters and options for aggregation operations to be chosen according to the preferences of the DMs [8].

This research defines the employed decision rule applying the Slide-OWA (S-OWA) operator [9]. Using the S-OWA operator is associated with adjusting the optimism and pessimism parameters and allows one to create decision maps. This strategy is complemented by applying the possibilistic approach to considering the uncertainty factor. As proposed and discussed in [10], it is realized within a general scheme of multi-objective decision-making under uncertainties. This scheme relies on constructing payoff matrices and applying the choice criteria of the classic approach to dealing with uncertainty as objective functions to analyze these matrices in order to obtain robust multi-objective solutions [5,10,11].

The analysis performed to date indicates alternative solutions that are indistinguishable from the formal point of view. Moreover, the DM may be interested in assessing the highlighted solution alternatives under additional considerations, which can be reflected using quantitative and qualitative information.

In this sense, multi-attribute decision-making techniques can be employed in evaluating, comparing, choosing, prioritizing, and ordering solution alternatives. These techniques rely on fuzzy preference modeling [5,11,12]. Recognizing that any criterion or any DM may require varying formats for the preference representation [12,13,14], it is necessary to use transformation functions (for instance, [12,13,14]) to allow the reduction of different preference formats to Nonreciprocal Fuzzy Preference Relations (NRFPR) [12]. This provides homogeneous information for decision-making. It can be processed within the framework of the corresponding models [5,11,15].

The employed models and methods deal with the uncertainty factor following the classification given by [16]. They recognize two classes of uncertainties: external uncertainties, which are determined by environmental conditions exceeding the DM’s control, and internal uncertainties, which are related to the DM’s values and judgments.

Considering the uncertainties and the possibility of using the DM’s preferences are crucial in analyzing locational problems, notably those related to renewable generation, since this area is usually seen as quite a risky business, with significant initial investments and extended payback periods [17]. Recognizing this, we propose an approach to support prospective studies which can consider business models and strategic goals, providing comprehensive and robust recommendations on new sites. In addition, the analysis’s versatility makes it suitable for supporting the decisions of different energy sector participants, such as governments, investors, or intensive energy consumers interested in expanding energy systems for their supply. It should also be stressed that the choices and combinations of methodologies used in the present work allow for determining the locational options for HREGSs using a wide range of criteria of spatial, non-spatial, quantitative, and qualitative character; therefore, they are essential originality aspects.

It is standard to use SMA as well as other decision-making models in order to determine satisfactory sites for generation based on renewable sources [18,19,20,21,22,23,24,25]. Nevertheless, the correlated works generally do not systematically handle internal and external uncertainties. Furthermore, these works also do not provide mechanisms for considering spatial qualitative criteria, which can play a vital role in the strategic planning of investors and governments.

Another contribution of this research is associated with the case study of site selection for HREGSs in Minas Gerais, Brazil. Here, renewable energy sources are represented by wind and solar photovoltaic sources. There exist publications related to the survey of the energy generation potential of these sources in Minas Gerais [26,27]. However, these publications do not consider aspects of the complementarity potential between these sources. Currently, Minas Gerais does not have HREGs installed or in operation. The quality of the SMA results was evaluated from the decision maps obtained according to the photovoltaic and wind power plants in Minas Gerais.

It is important to stress that the employed models and methods are flexible and can be extended to deal with other renewable energy sources, different hybridization strategies, and other spatial problems (such as that considered in [28]).

2. Decision Process for Supporting the Prospection of New Sites for HREGS

2.1. Identification of the Criteria Relevant to the Decision

In related works, some common criteria are applied: renewable energy potential, distance from a transmission system, distance from roads, distance and presence of urban centers, relief declivity, the economic value of land, land use, presence of woods and forests, presence of Conservation Units (CUs) and public forests, and presence of water bodies [18,19,20,21,22,23,24,25,26,27].

This work evaluates the spatial distribution of the renewable energy potential considering the time complementarity of the two sources: photovoltaic solar and wind. Time complementarity refers to sources whose seasonal availability fluctuations complement each other over a determined time horizon [29]. Considering that some renewable resources may have a complementary profile over a period, usually one year, [30] we propose a dimensionless index to assess complementarity between two energy sources. This index is based on three components, considered in the same time horizon, and assumes values between 0 and 1, where values closer to 1 indicate greater potential for complementarity. The three components described by [29,30,31,32] are:

2.1.1. Time-Complementarity Partial Index

The following equation

$$I_{\mathrm{time}}=\frac{|d_{1i}-d_{2i}|}{\sqrt{|D_{1i}-d_{1i}||D_{2i}-d_{2i}|}}$$

(1)

permits one to evaluate the time interval between the minimum availability values of two sources. If this interval is half of the considered period, $I_{\mathrm{time}}=1$. If the minimums coincide, $I_{\mathrm{time}}=0$.

In (1), $D_{1i}$ and $D_{2i}$ are the number of days of maximum energy availability of sources 1 and 2, respectively; and $d_{1i}$ and $d_{2i}$ are the number of days of minimum energy availability of sources 1 and 2, respectively, for the $i$th location and in the considered time period. If the differences are equal to a half period, (1) can be rewritten as follows:

$$I_{\mathrm{time}}=\frac{|d_{1i}-d_{2i}|}{\sqrt{\left|180\right|\left|180\right|}}=\frac{|d_{1i}-d_{2i}|}{180}$$

(2)

2.1.2. Energy-Complementarity Partial Index

The following equation

$$I_{\mathrm{energy}}=1-\frac{|E_{1i}-E_{2i}|}{|E_{1i}+E_{2i}|}$$

(3)

evaluates the relation between the average values of energy availability of the sources. If the mean values are equal, $I_{\mathrm{energy}}=1$. If these values are different, the index should be smaller and tend to zero as the differences increase. In (3), $E_{1i}$ and $E_{2i}$ are the total energy available by sources 1 and 2, respectively, in the assessed period in the $i$th location.

2.1.3. Amplitude-Complementarity Partial Index

The following equation

$$I_{\mathrm{amplitude}}=\left\{\begin{array}{c}\left[1-\frac{{({\delta }_{1i}-{\delta }_{2i})}^2}{{(1-{\delta }_{2i})}^2}\right], \mathrm{if} {\delta }_{1i}\le {\delta }_{2i} \\ \left[\frac{{(1-{\delta }_{2i})}^2}{{(1-{\delta }_{2i})}^2+{({\delta }_{1i}-{\delta }_{2i})}^2}\right], \mathrm{if} {\delta }_{1i}\ge {\delta }_{2i} \end{array}\right.$$

(4)

comes from a suitable manipulation between the parameters ${\delta }_{1i}$ and ${\delta }_{2i}$, which are defined as

$${\delta }_i=2-\frac{E_{d \min i}}{E_{d \max i}}$$

(5)

where $E_{d\min i}$ and $E_{d\max i}$ are the maximum and minimum daily energy availability, respectively, of each evaluated source in the considered period for the $i$th location.

This index evaluates the relationship between the minimum and maximum values of the two energy availability functions. If the ratio between the minimum and maximum availability values of source 1 is equal to source 2 (in this case ${\delta }_{1i}={\delta }_{2i}=1$), the index (4) will be equal to 1. If these values are different, the index should be smaller and tend to zero as the differences increase.

Finally, the total complementarity index can be obtained as

$$I_{\mathrm{total}}=I_{\mathrm{time}}{I}_{\mathrm{energy}}{I}_{\mathrm{amplitude}}$$

(6)

2.2. Fuzzy Set-Based Multicriteria Spatial Decision Model

The existing mechanisms for multicriteria decision-making include a complex set of techniques and procedures aimed at evaluating, comparing, choosing, prioritizing, and ordering alternative solutions, considering judgments provided by a single or group of experts, according to multiple criteria, which are usually conflicting [5,6,33]. For example, frequently, despite the occurrence of significant energy potential in certain areas, there may be restrictive factors for the installation of generation plants, such as, for instance, the obligation to safeguard environmental or cultural heritage [34].

There are two main types of criteria:

• Factors: correspond to soft judgmental rules that may not be fully satisfied. This type of criteria is used to identify, within the set of feasible solutions, the ones that better match the DM preferences;
• Constraints: rigorous judgmental rules that can exclude candidate solutions that do not satisfy such conditions from the set of feasible solutions.

Both types of criteria can be represented through fuzzy sets [35]. A fuzzy set is characterized by a Membership Function (MF) $\mu :U\to [0,1]$ that associates each object of interest belonging to the universe of discourse $U$ with a value that reflects the membership level to the fuzzy set. This level ranges from 0, which means lack of membership, to 1, which means full membership [35].

In spatial decision-making, the georeferenced information [36] can be described by geo-objects (vector representation) and geo-fields (matrix representation). In particular, geo-fields represent the geographical space as a continuous surface divided into a regular grid of cells, where the phenomena of interest are described. Following this representation, the spatial variable considered in a model can be defined as $x=(x_1,x_2)$ where $x_1$ and $x_2$ correspond to longitude and latitude dimensions, respectively. The fuzzy sets can be regarded as geo-fields that describe each location as a value from 0 to 1 through the MFs. This value indicates the degree to which the location attends a specific decision criterion. The corresponding mapping depends on the attributes of each locality, which are reflected by the data geo-fields $g_i(x),i=1,\dots ,m$.

Hence, in order to create fuzzy sets $A_i(x),i=1,\dots ,m$ with MFs ${\mu }_{A_i}(g_i(x)),i=1,\dots ,m$ reflecting a degree of achieving the decision criteria, the correlation

$${\mu }_{A_i}\left(g_i\left(x\right)\right)=\frac{\max g_i(x) -g_i(x)}{\max g_i(x) -\min g_i(x)}$$

(7)

can be employed for minimized objective functions; alternatively, the correlation

$${\mu }_{A_i}\left(g_i\left(x\right)\right)=\frac{g_i(x) -\max g_i(x)}{\max g_i(x) -\min g_i(x)}$$

(8)

can be used for maximized ones [5,12]. Other correlations (for example, [37]) can also be employed to construct the fuzzy sets. For instance,

$${\mu }_{A_i}\left(g_i\left(x\right)\right)=\frac{1}{1+{\left(\frac{g_i(x)}{md}\right)}^{-sp}}$$

(9)

can be used for maximized objective functions, while

$${\mu }_{A_i}\left(g_i\left(x\right)\right)=\frac{1}{1+{\left(\frac{g_i(x)}{md}\right)}^{sp}}$$

(10)

can be applied for minimized objective functions. The parameters $sp$ and $md$ are related to the spread and the midpoint, respectively, of the MFs. The choice of the expressions that best describe the analyzed criteria is a prerogative of the DMs.

The fuzzy sets corresponding to constraints can delineate the set of feasible locations. Often, locations with zero membership degrees are considered unfeasible alternatives.

When multiple criteria are to be considered, multiple fuzzy sets $A_i(x),i=1,\dots ,m$ are constructed, and an aggregation function $h:{[0,1]}^m\to [0,1]$ can be applied to generate a fuzzy set named fuzzy decision, which can be presented in the following form [12]:

$$A_D(x)=h(A_1(x),\dots ,A_m(x))$$

(11)

The Ordered Weighted Averaging (OWA) operator [38] can be used to implement (11). For a given set of criteria or geo-fields (fuzzy sets), it associates a vector of order weights ${\omega }_1,\dots ,{\omega }_m$ $({\omega }_j\in [0,1]$ and ${\displaystyle \sum _{j=1}^m{\omega }_j=1})$ to the ith location. This operator permits the realization of a family of multicriteria aggregation procedures [38,39,40]. Therefore, the expression (11) can be rewritten as

$$A_D(x)=\mathrm{OWA}(A_1(x),\dots ,A_m(x))={\displaystyle \sum _{j=1}^m{\omega }_j{c}_j}$$

(12)

where $c_j$ is the jth highest element of the $A_i(x),i=1,\dots ,m$.

Several versions of the OWA operator differ in the methods for determining the order weights ([41]). This research applies the S-OWA operator [39] to model the DM’s optimism/pessimism level. The weights of the S-OWA operator are defined as

$${\omega }_j=\left\{\begin{array}{c}\frac{1}{m}(1-(\alpha +\beta ))+\alpha ,\hspace{1em}j=1\\ \frac{1}{m}(1-(\alpha +\beta )),\hspace{1em}j=2,\dots ,m-1\\ \frac{1}{m}(1-(\alpha +\beta ))+\beta ,\hspace{1em}j=m\end{array}\right.$$

(13)

where $\alpha ,\beta \in [0,1]$ and $\alpha +\beta \le 1$. The parameter $\alpha$ is the degree of optimism to be considered and $\beta$ is the degree of pessimism.

2.3. Uncertainty Treatment of the Initial Data Based on the Possibilistic Approach

The classical approach to dealing with data uncertainty [42,43] is generalized in [12,44] to solve multicriteria problems and is further improved in [5,10]. This research applies a possibilistic approach [10,12] to the HREGSs’ localization problems.

The possibilistic approach is suitable for solving this problem due to its universal character, since it permits the combination and aggregation of information from different sources and of different types (such as deterministic, probabilistic, interval, and fuzzy, among others), improving the results of the decision process. This widespread approach to solving practical problems [5,10,11,12,15] allows for considering assumptions designed by DMs through experience, intuition, and prior knowledge in scenario-building.

Using different combinations of solution alternatives $X_k$, $k=1,\dots ,K$ and scenarios $Y_s,s=1,\dots ,S$, the results of [5,10] can be applied to construct the quantifying effects $F(X_k,Y_s)$ of payoff matrices.

The first six columns of

Table 1. Payoff matrix.

	${\mathit{Y}}_{\mathbf{1}}$	$\mathbf{.}\mathbf{.}$	${\mathit{Y}}_{\mathit{s}}$	$\mathbf{.}\mathbf{.}$	${\mathit{Y}}_{\mathit{S}}$	${\mathit{F}}^{\mathbf{min}}({\mathit{X}}_{\mathit{k}})$	${\mathit{F}}^{\mathbf{max}}({\mathit{X}}_{\mathit{k}})$	$\overline{\mathit{F}}({\mathit{X}}_{\mathit{k}})$	${\mathit{R}}^{\mathbf{max}}({\mathit{X}}_{\mathit{k}})$
$X_1$	$F(X_1,Y_1)$	$..$	$F(X_1,Y_s)$	$..$	$F(X_1,Y_S)$	$F^{\min }(X_1)$	$F^{\max }(X_1)$	$\overline{F}(X_1)$	$R^{\max }(X_1)$
$..$	$..$	$..$	$..$	$..$	$..$	$..$	$..$	$..$	$..$
$X_k$	$F(X_k,Y_1)$	$..$	$F(X_k,Y_s)$	$..$	$F(X_k,Y_S)$	$F^{\min }(X_k)$	$F^{\max }(X_k)$	$\overline{F}(X_k)$	$R^{\max }(X_k)$
$..$	$..$	$..$	$..$	$..$	$..$	$..$	$..$	$..$	$..$
$X_K$	$F(X_K,Y_1)$	$..$	$F(X_K,Y_s)$	$..$	$F(X_K,Y_S)$	$F^{\min }(X_K)$	$F^{\max }(X_K)$	$\overline{F}(X_K)$	$R^{\max }(X_K)$

form a payoff matrix [5,10], and the number of considered criteria of the factor type defines the total quantity of payoff matrices.

2.4. Construction of the Scenarios (or States of Nature)

Possible future situations can be reflected using scenarios which are also known as representative combinations of initial data or states of nature. Several approaches have been utilized for scenario construction [45,46]. The technique of so-called LP_τ-sequences has superior uniformity characteristics among other uniformly distributed sequences [5,10,11,12,44,47].

In essence, these sequences provide points $Q_s,s=1,\dots ,S$ (S is the number of the scenarios) with coordinates $q_{st},s=1,\dots ,S,t=1,\mathrm{\dots },T$ in the corresponding unit hypercube $Q^T$, where T is the number of criteria of the factor type under uncertainty [10]. The construction of scenarios is reduced to selecting points of a uniformly distributed sequence in $Q^T$ and their transformation to the hypercube $C^T$ defined by the lower $c_t^{\prime }$ and upper $c_t^{\prime }$ bounds for the corresponding intervals of uncertainty as follows:

$$c_{st}=c_t^{\prime }+(c_t^{"}-c_t^{\prime })q_{st},\hspace{1em}s=1,\dots ,S, t=1,\dots ,T$$

(14)

For more examples concerning LPτ-sequences, refer to [10,44].

2.5. Choice Criteria and the Modification of the Classical Approach

The so-called choice criteria are used to analyze the payoff matrices and to choose the rational solution alternatives. This work applies characteristic estimates in constructing the choice criteria using the classical approach of Wald, Laplace, Savage, and Hurwicz [42,43].

In particular, for each line (alternative solution) of the payoff matrix (Table 1), the choice criteria of Wald, Laplace, Savage, and Hurwicz are based on the following characteristic estimates:

$$\begin{gathered} F^{\min }(X_k)=\underset{1\le s\le S}{\min }F(X_k,Y_s), F^{\max }(X_k)=\underset{1\le s\le S}{\min }F(X_k,Y_s), \\ \overline{F}(X_k)=\frac{1}{S}{\displaystyle \sum _{s=1}^{S}F(X_k,Y_s)}, \mathrm{and} R^{\max }(X_k)=\underset{1\le s\le S}{\max }R(X_k,Y_s). \end{gathered}$$

To evaluate the maximum risk level R^max(X_k), it is necessary to use the minimum value of the criterion $F^{\min }(Y_s)=\underset{1\le k\le K}{\min }F(X_k,Y_s).$ (for a case of minimization) or $F^{\max }(Y_s)=\underset{1\le k\le K}{\max }F(X_k,Y_s)$ (for a case of maximization) for each column (scenario) of the payoff matrix [42,43,44]:

$$R(X_k,Y_s)=F(X_k,Y_s)-F^{\min }(Y_s)$$

(15)

for minimized criteria and

$$R(X_k,Y_s)=F^{\max }(Y_s)-F(X_k,Y_s)$$

(16)

for maximized criteria.

The characteristic estimates indicated above are presented in Table 1 (7th−10th columns).

The authors of [5,10] justify the consideration of the characteristic estimates as objective functions for the pth criterion:

$$F_p^W(X_k)=F_p^{\max }(X_k)=\underset{1\le s\le S}{\max }{F}_p(X_k,Y_s)$$

(17)

$$F_p^L(X_k)=\overline{F_p}(X_k)=\frac{1}{S}{\displaystyle \sum _{s=1}^S{F}_p(X_k,Y_s)}$$

(18)

$$F_p^S(X_k)=R_p(X_k)=\underset{1\le s\le S}{\max }{R}_p(X_k,Y_s)$$

(19)

$$F_p^H(X_k)=\alpha F^{\max }(X_k)+(1-\alpha )F_p^{\min }(X_k)=\alpha \underset{1\le s\le S}{\max }{F}_p(X_k,Y_s)+(1-\alpha )\underset{1\le s\le S}{\min }{F}_p(X_k,Y_s)$$

(20)

In (20), $\alpha \in [0,1]$ is the index “pessimism-optimism” defined by the DM. It is recommended to use values of α between 0.5 and 1 [43].

As aforesaid, the existence of m criteria of the factor type requires the construction of m payoff matrices.

The m matrices with the choice criteria estimates can be obtained using [5,10] results, as shown in

Table 2. Matrix of choice criteria estimates for the pth criteria of the factor type.

	${\mathit{F}}_{\mathit{p}}^{\mathit{W}}({\mathit{X}}_{\mathit{k}})$	${\mathit{F}}_{\mathit{p}}^{\mathit{L}}({\mathit{X}}_{\mathit{k}})$	${\mathit{F}}_{\mathit{p}}^{\mathit{S}}({\mathit{X}}_{\mathit{k}})$	${\mathit{F}}_{\mathit{p}}^{\mathit{H}}({\mathit{X}}_{\mathit{k}})$
$X_1$	$F_p^W(X_1)$	$F_p^L(X_1)$	$F_p^S(X_1)$	$F_p^H(X_1)$
$..$	$..$	$..$	$..$	$..$
$X_k$	$F_p^W(X_k)$	$F_p^L(X_k)$	$F_p^S(X_k)$	$F_p^H(X_k)$
$..$	$..$	$..$	$..$	$..$
$X_K$	$F_p^W(X_K)$	$F_p^L(X_K)$	$F_p^S(X_K)$	$F_p^H(X_K)$
	$\underset{1\le k\le K}{\min }{F}_p^W(X_k)$	$\underset{1\le k\le K}{\min }{F}_p^L(X_k)$	$\underset{1\le k\le K}{\min }{F}_p^S(X_k)$	$\underset{1\le k\le K}{\min }{F}_p^H(X_k)$
	$\underset{1\le k\le K}{\max }{F}_p^W(X_k)$	$\underset{1\le k\le K}{\max }{F}_p^L(X_k)$	$\underset{1\le k\le K}{\max }{F}_p^S(X_k)$	$\underset{1\le k\le K}{\max }{F}_p^H(X_k)$

. They indicate the performance of each solution alternative reflected by the choice criteria of Wald, Laplace, Savage, and Hurwicz.

The information given in Table 2 can be used to construct m multicriteria problems, including four or fewer objective functions (depending on the number of the considered choice criteria) as follows:

$$F_{e,p}(X)\to \underset{X\in L}{\mathrm{ext}},\hspace{1em}e=1,\dots ,E\le 4,p=1,\dots ,m$$

(21)

where $F_{1,p}(X)=F_p^W(X_k)$, $F_{2,p}(X)=F_p^L(X_k)$, $F_{3,p}(X)=F_p^S(X_k)$, and $F_{4,p}(X)=F_p^H(X_k)$.

It is possible to apply the approach of [48] to decision-making in a fuzzy environment to solve problem (21). This is an advantageous approach as it reflects the principle of guaranteed results and produces constructive lines for obtaining harmonious solutions [5,10,12,49,50]. The concept of optimality when using this approach is based on the maximum degree of implementing all objectives. In the Bellman–Zadeh approach [48], the objective functions of interest $F_p(X),p=1,\dots ,m$ are modeled by fuzzy sets

$$A_p=\left\{X,{\mu }_{A_p}(X)\right\},X\in L,p=1,\dots ,m$$

where ${\mu }_{A_p}(X)$ is the MF of $A_p$. A fuzzy solution D is defined [49,50] as $D={\displaystyle \underset{p=1}{\overset{q}{\cap }}{A}_p}$ with the MF

$${\mu }_D(X)=\underset{1\le p\le m}{\min }{\mu }_{A_p}(X),\hspace{1em}X \in L$$

(22)

The following problem can be generated:

$$\max {\mu }_D(X)=\underset{X\in L}{\max }\underset{1\le p\le m}{\min }{\mu }_{A_p}(X)$$

(23)

whose solution is

$$X^0=\arg \underset{X\in L}{\max }\underset{1\le p\le m}{\min }{\mu }_{A_p}(X)$$

(24)

To obtain (24), it is necessary to build ${\mu }_{A_p}(X),p=1,\dots ,m$ to reflect the degree of achieving own optima by $F_p(X),X\in L,p=1,\dots ,m$. This can be reached by applying (7) and (8) [49,50].

The m modified matrices of the choice criteria estimates are based on the m matrices for the choice criteria estimates, having a general form similar to Table 2.

In the modified matrices, applying expression (7), the estimates $F_p^W$, $F_p^L$, $F_p^S$, and $F_p^H$ are replaced, respectively, by the membership values ${\mu }_{A_p}^W(X_k)$, ${\mu }_{A_p}^L(X_k)$, ${\mu }_{A_p}^S(X_k)$, and ${\mu }_{A_p}^H(X_k)$. Therefore, the availability of m modified matrices of the choice criteria estimates can be used to construct the aggregated matrix of the choice criteria estimates. The aggregated payoff matrix also has the same general form as Table 2, where the estimates $F_p^W$, $F_p^L$, $F_p^S$, and $F_p^H$ are replaced, respectively, by the aggregated membership values ${\mu }_D^W(X_k)$, ${\mu }_D^L(X_k)$, ${\mu }_D^S(X_k)$, and ${\mu }_D^H(X_k)$, applying expression (22).

It is noteworthy that the application of (22) can be modified to adjust the optimism and pessimism levels in the decision process by using the OWA operator (12) or its modification (13).

2.6. Analysis of the Alternatives in a Fuzzy Environment

Different alternatives, which, from the formal point of view, cannot be distinguished, may be generated in the analysis so far. In addition, although all considered alternatives have been distinguished, a DM may want to re-examine them by involving additional considerations. Therefore, it is possible to apply multi-attribute decision-making techniques for evaluating, comparing, choosing, prioritizing, and ordering solution alternatives, which are locational alternatives for installing HREGSs $(X_k)$.

These techniques provide a satisfactory and effective manner to consider spatial or non-spatial additional quantitative and qualitative character criteria. The additional criteria can be estimated based on the experts’ experience, knowledge, and intuition. The employed model considers processing the preferences, represented by a pair $<X,R>$, where X is a set of alternatives $\left\{X_1,\dots ,X_k\right\}$ and $R=\left\{R_1,\dots ,R_B\right\}$ is a set of Nonstrict Fuzzy Preference Relations (NSFPRs), which can be presented as

$$R=[X\times X,{\mu }_{R_b}(X_k,X_l),X_k,X_l\in X],b=1, \dots , B$$

(25)

where ${\mu }_{R_b}(X_k,X_l)$ is an MF of the bth NSFPR [51] for the bth additional criterion.

Using a square matrix with MF ${\mu }_{R_b}X\times X\to [0,1]$, an NSFPR $R_b$ can be modeled that associates a number within a unit interval $R_b(X_k,X_l)$ for each pair of alternatives $X_k,X_l\in X$. Thus, $R_b(X_k,X_l)$ indicates the degree that the alternative $X_k$ is at least as good as $X_l$ when the criterion $C_b$ is considered [5,12].

While dealing with $<X,R>$ models, it is important to address how fuzzy preference relations should be constructed to reflect the corresponding DM’s preferences. The majority of real decision situations can be covered by five preference formats: utility values, multiplicative preference relations, ordering of the alternatives, fuzzy estimates, and fuzzy preference relations (FPRs) [12]. Following this classification, it is worth noting that fuzzy estimates and the FPR, particularly NRFPRs, are equivalent.

In fact, if two alternatives $X_k,X_l\in X$ have a fuzzy estimate $\mu [F_b(X_k)] \mathrm{and} \mu [F_b(X_l)]$ regarding criterion $C_b$, then the quantity ${\mu }_{R_b}(X_k,X_l)$ is the degree of preference $\mu [F_b(X_k)]$$\succcurlyeq$$\mu [F_b(X_l)]$, while the quantity ${\mu }_{R_b}(X_l,X_k)$ is the degree of preference $\mu [F_b(X_l)]$$\succcurlyeq$$\mu [F_b(X_k)]$.

Based on the conception of an MF of a generalized preference relation [52] and considering the minimization of $C_b$, the quantities ${\mu }_{R_b}(X_k,X_l)$ and ${\mu }_{R_b}(X_l,X_k)$, respectively, can be written as [12,52]

$${\mu }_{R_b}(X_k,X_l)=\underset{\begin{array}{l}{X}_k,X_l\in X\\ F_b(X_k)\le F_b(X_l)\end{array}}{\sup }\min \left\{\mu [F_b(X_k)],\mu [F_b(X_l)]\right\}$$

(26)

and

$${\mu }_{R_b}(X_l,X_k)=\underset{\begin{array}{l}{X}_k,X_l\in X\\ F_b(X_l)\le F_b(X_k)\end{array}}{\sup }\min \left\{\mu [F_b(X_k)],\mu [F_b(X_l)]\right\}$$

(27)

For the maximization of $C_b$, (26) and (27) should be written, respectively, as $F_b(X_k)\ge F_b(X_l)$ and $F_b(X_l)\ge F_b(X_k)$.

The transformation functions can be applied to reduce all used formats into a unique one [5,11,12,13,14]. Thus, the preference information presented within other formats can be employed to construct FPRs, providing homogeneous preference information which can be analyzed and processed. Moreover, the existence of criteria of quantitative character in dealing with $<X,R>$ models requires the reduction of the corresponding quantitative estimates to FPRs.

In particular, since for Additive Reciprocal Fuzzy Preference Relations (ARFPRs), the conditions ${\overline{\mu }}_{R_b}(X_k,X_k)=0.5$ and ${\overline{\mu }}_{R_b}(X_k,X_l)+{\overline{\mu }}_{R_b}(X_l,X_k)=1$ are met, it is possible to use the [5,11,12] results:

$${\overline{\mu }}_{R_b}(X_k,X_l)=\frac{F_b(X_k)-F_b(X_l)}{2[\max F_b(X)-\min F_b(X)]}+0.5$$

(28)

for minimizing $F_b(X)$ and

$${\overline{\mu }}_{R_b}(X_k,X_l)=\frac{F_b(X_l)-F_b(X_k)}{2[\max F_b(X)-\min F_b(X)]}+0.5$$

(29)

for maximizing $F_b(X)$.

The ARFPR can be converted to NRFPR [5,12] as follows:

$${\mu }_{R_b}(X_k,X_l)=\left\{\begin{array}{ll}1+{\overline{\mu }}_{R_b}(X_k,X_l)-{\overline{\mu }}_{R_b}(X_l,X_k), & \mathrm{if} {\overline{\mu }}_{R_b}(X_k,X_l)<0.5\\ 1,& \mathrm{if} {\overline{\mu }}_{R_b}(X_k,X_l)\ge 0.5\end{array}\right.$$

(30)

The information given in the form of $R=\left\{R_1,\dots ,R_B\right\}$ can be processed and applied to narrow the set $X$, leaving as alternatives only the other ones do not dominate from $X$.

Six techniques of analyzing $<X,R>$ models are described in [12] based on the Orlovski choice function [53]. The results of [12] can be used to transform NSFPRs into Strict Fuzzy Preference Relations (SFPRs) as

$${\mu }_R^S(X_k,X_l)=\max ({\mu }_R(X_k,X_l)-{\mu }_R(X_l,X_k),0)$$

(31)

Since ${\mu }_R^S(X_l,X_k),\forall X_k\in X$ is the MF of the fuzzy set of all alternatives $X_k$ that are strictly dominated by $X_l$, its complement $1-{\mu }_R^S(X_l,X_k),\forall X_k\in X$ provides the fuzzy set of the alternatives that are not dominated by $X_l$. The MF of this set can be stated as

$${\mu }_R^{ND}(X_k)=\underset{X_l\in X}{\min }[1-{\mu }_R^S(X_l,X_k)]=1-\underset{X_l\in X}{\max } {\mu }_R^S(X_l,X_k),\forall X_l\in X$$

(32)

The alternatives which can be chosen as the problem solution, according to the highest non-dominance level, are the following:

$$X^{ND}=\left\{X_k^{ND}|X_k^{ND}\in X,{\mu }_R^{ND}(X_k^{ND})=\underset{X_k\in X}{\max } {\mu }_R^{ND}(X_k)\right\}$$

(33)

The expressions (31)–(33) are also applicable when $R$ is a vector of FPRs. For example, they serve for the first method of [12], considering $R={\displaystyle \underset{b=1}{\overset{B}{\cap }}{R}_b}$ with the MF [52,53]:

$${\mu }_R(X_k,X_l)=\underset{1\le b\le B}{\min }{\mu }_{R_B}(X_k,X_l),X_k,X_l\in X$$

(34)

When applying (34), the set $X^{ND}$ [53] performs the role of a Pareto set [54].

A flexible approach for adjusting the degree of optimism using the OWA operator [38,39,40,41] is presented in [12]. Following this approach, (12) can be rewritten as

$${\mu }_R(X_k,X_l)=\mathrm{OWA}({\mu }_{R_1}(X_k,X_l),{\mu }_{R_2}(X_k,X_l),\dots ,{\mu }_{R_B}(X_k,X_l))={\displaystyle \sum _{b=1}^B{\omega }_b{c}_b}$$

(35)

where $c_b$ is the $i$th largest value among ${\mu }_{R_1}(X_k,X_l),\dots ,{\mu }_{R_B}(X_k,X_l)$. Again, the set of weights ${\omega }_i,i=1,\dots ,b$ in (35) is to satisfy the conditions ${\omega }_i\in [0,1]$ and ${\displaystyle \sum _{i=1}^B{\omega }_i=1}$.

Different adjustments of the vector of ordered weights reflect different decision attitudes, characterized by different levels of mutual compensation between criteria. These weights can be naturally determined by using Fuzzy Linguistic Quantifiers (FLQs) [40]. These quantifiers are fuzzy sets $Q(x)$ that reflect the level $x\in [0,1]$ of the criteria portion that satisfies a concept represented by Q. The weights can be obtained, after choosing the appropriate FLQ, by

$${\omega }_i=Q\left(\frac{i}{q}\right)-Q\left(\frac{i-1}{q}\right),i=1, \dots , q.$$

(36)

With the global nonstrict preference matrix, a global ranking of the alternatives can be obtained by using the results of [12,55] related to the Quantifier Guided Dominance Degree (QGDD) as follows:

$$\mathrm{QGDD}(X_k)=\mathrm{OWA}({\mu }_R(X_k,X_l)),l=1, \dots , b,l\ne k$$

(37)

$\mathrm{QGDD}(X_k)$ reflects the dominance level of $X_k$ over other alternatives in the sense implemented by the chosen quantifier. It is also possible to evaluate the Quantifier Guided Nondominance Degree (QGNDD) [12] as

$$\mathrm{QGNDD}(X_k)=\mathrm{OWA}(1-{\mu }_R^S(X_k,X_l)),l=1, \dots , b,l\ne k$$

(38)

To achieve this result, one has to apply the global nonstrict preference matrix to compute the global strict preference relation ${\mu }_R^S(X_k,X_l)$ [55].

Considering that choosing alternatives according to the highest level of non-dominance is reasonable, the alternatives $X^{ND}$ can be found following an expression similar to (33).

Lastly, it should be pointed out that QGNDD can be employed for choosing the best alternative, and, in the case of indistinguishability of decisions between two or more alternatives, QGDD can be used to try to distinguish them.

The main steps in using the models and methods that compose the decision process described in this paper are summarized by the flowchart of Figure 1. The highlighted block in the flowchart refers to the SMA’s realization, which is detailed in Figure 2.

3. Case Study

The proposed analysis has been applied to the prospection of renewable energy sources in Minas Gerais State, Brazil, aimed at determining locational alternatives for generation plants of solar and wind sources.

The criteria, which have been defined in Section 2.1, are summarized in Table 3. They are characterized by the labels, types, and geo-fields utilized in the normalization process, models applied in the normalization process, and general descriptions of the original data.

The georeferenced data for constructing fuzzy sets were converted to a Lambert Conformal Conic Projection [56], specifically for mapping South America. The resolution was set at 500 × 500, in meters, for all data geo-fields (according to [57,58]).

Table 3. Decision criteria.

Criterion Label	Criterion Type	Data Geo-Field Utilized to Define the Fuzzy Set Domain	Normalization Model	General Description of Data Geo-Field
$f_1$—Potential energy of a hybrid power generation system.	Factor	$g_1$—Estimate of potential energy of hybrid power system.	Fuzzy set expressed by (8).	The available hybrid potential in Minas Gerais was estimated through the model described in Section 2.1.1. The wind speed data at 50 meters of height and solar radiation data are provided by [59,60]. These data were generated with a unit degree of spatial resolution.
$f_2$—Economic value of bare land.	Factor	$g_2$—Estimate of land cost per hectare by land use class and municipality.	Fuzzy set expressed by (7).	The economic value of the land was estimated through the cost data and provided by [61]. Land use and occupation data were provided by [62].
$f_3$—Low-slope areas.	Factor	$g_3$—Terrain slope.	Fuzzy set expressed by (10) with $sp=10$ and $md=8$.	The gradient map (expressed as percentages) of Minas Gerais was provided by [63].
$f_4$—Proximity to water bodies.	Factor	$g_4$—Distance from water bodies.	Fuzzy set expressed by (7).	The geodesic distance field to the nearest water bodies, calculated based on a hydrography map, was provided by [64].
$f_5$—Water bodies and swamps.	Constraint	$g_5$—Presence of water bodies.	Boolean.	Map of water bodies and swamps; provided by [64].
$f_6$—Connection to the transmission system.	Factor	$g_6$—Distance from existing transmission lines (TLs).	Fuzzy set expressed by (7).	The geodesic distance field to existing TLs, calculated based on a map, was provided by [64].
$f_7$—Site accessibility.	Factor	$g_7$—Distance from access roads.	Fuzzy set expressed by (7).	The geodesic distance field to the nearest access roads, calculated based on a map, was provided by [64].
$f_8$—Quilombo lands.	Constraint	$g_8$—Location of quilombo lands.	Boolean.	Map of quilombola units; provided by [64].
$f_9$—Indigenous lands.	Constraint	$g_9$—Location of indigenous lands.	Boolean.	Map of indigenous lands; provided by [64].
$f_{10}$—Conservation Units (CUs).	Constraint	$g_{10}$—Location of CUs.	Boolean.	Map of CU; provided by [64].
$f_{11}$—Non-urbanized areas.	Constraint	$g_{11}$—Location of urbanized centers.	Boolean.	Map of cities of Minas Gerais; provided by [62].

Table 3. Decision criteria.

Criterion Label	Criterion Type	Data Geo-Field Utilized to Define the Fuzzy Set Domain	Normalization Model	General Description of Data Geo-Field
$f_1$—Potential energy of a hybrid power generation system.	Factor	$g_1$—Estimate of potential energy of hybrid power system.	Fuzzy set expressed by (8).	The available hybrid potential in Minas Gerais was estimated through the model described in Section 2.1.1. The wind speed data at 50 meters of height and solar radiation data are provided by [59,60]. These data were generated with a unit degree of spatial resolution.
$f_2$—Economic value of bare land.	Factor	$g_2$—Estimate of land cost per hectare by land use class and municipality.	Fuzzy set expressed by (7).	The economic value of the land was estimated through the cost data and provided by [61]. Land use and occupation data were provided by [62].
$f_3$—Low-slope areas.	Factor	$g_3$—Terrain slope.	Fuzzy set expressed by (10) with $sp=10$ and $md=8$.	The gradient map (expressed as percentages) of Minas Gerais was provided by [63].
$f_4$—Proximity to water bodies.	Factor	$g_4$—Distance from water bodies.	Fuzzy set expressed by (7).	The geodesic distance field to the nearest water bodies, calculated based on a hydrography map, was provided by [64].
$f_5$—Water bodies and swamps.	Constraint	$g_5$—Presence of water bodies.	Boolean.	Map of water bodies and swamps; provided by [64].
$f_6$—Connection to the transmission system.	Factor	$g_6$—Distance from existing transmission lines (TLs).	Fuzzy set expressed by (7).	The geodesic distance field to existing TLs, calculated based on a map, was provided by [64].
$f_7$—Site accessibility.	Factor	$g_7$—Distance from access roads.	Fuzzy set expressed by (7).	The geodesic distance field to the nearest access roads, calculated based on a map, was provided by [64].
$f_8$—Quilombo lands.	Constraint	$g_8$—Location of quilombo lands.	Boolean.	Map of quilombola units; provided by [64].
$f_9$—Indigenous lands.	Constraint	$g_9$—Location of indigenous lands.	Boolean.	Map of indigenous lands; provided by [64].
$f_{10}$—Conservation Units (CUs).	Constraint	$g_{10}$—Location of CUs.	Boolean.	Map of CU; provided by [64].
$f_{11}$—Non-urbanized areas.	Constraint	$g_{11}$—Location of urbanized centers.	Boolean.	Map of cities of Minas Gerais; provided by [62].

3.1. Scenario Definitions

Two spatial criteria described in the previous section have deeply associated uncertainties: the potential energy of a hybrid power system and the economic value of bare land. Other criteria do not have significant uncertainties since they are fundamentally related to the existence or absence of well-defined physical entities, such as roads and cities. Thus, the $f_3$–$f_{11}$ criteria can be considered to be well known and determined.

The criterion “potential energy of a hybrid power system” has been determined following the model described in Section 2.1, using monthly mean values of wind speed and solar radiation [59,60]. These data were collected over ten years of measurements for cells of one degree of resolution, i.e., they have low precision. Thus, to adequately consider the associated uncertainties, we utilize the LPτ-sequences to generate scenarios, per Section 2.4. The minimum and maximum monthly values of the wind speed and solar radiation registered during the 10 years of measurements have been used to construct the uncertainty intervals utilized in (14). We have built five scenarios, and the potential energy of the hybrid system has been determined for each of them.

Another criterion that has related uncertainties is the “Economic value of bare land.” To consider this criterion, we used the data available in [61,62], which provides the bare land costs (R$/hectare) for each municipality of Minas Gerais for four main classes of land: lands with an aptitude for agriculture, planted pasture, silviculture or natural pasture, and fauna and flora preservation areas. Each one of these classes has a minimum and maximum cost value, which has been used in (14) for constructing the scenarios.

3.2. Spatial Multicriteria Analysis under Uncertainty Conditions

The decision problem for each scenario has been resolved by applying the S-OWA operator (12) and the optimism and pessimism parameters $\alpha =0.4$ and $\beta =0.6$, respectively, in (13). Applying these parameters corresponds to the conservative attitude toward risk in decision-making. In this analysis, the six criteria of the factor type presented in Table 3 were aggregated in order to generate a global map (decision map) where the five constraints (Table 3) were applied (through Boolean intersection operations) to exclude unfeasible solutions.

Figure 3 illustrates the results obtained for each scenario, i.e., the decision maps. The color chart expresses values within the unit interval, which represents the aptitude for HREG installation associated with each map region. Regions in orange/yellow shadows have higher attractiveness than regions in blue shadows. The map indicates that Minas Gerais’s northwestern, northern, and northeastern regions are more attractive for installing HREGs. These areas have the highest hybrid potential and lowest slope areas. However, these areas also contain many conservation units, which are restrictive to the installation of HREGS. This fact reveals the advantages of the proposed methodology, which permits one to consider a large set of conflicting criteria. A similar analysis can be performed in the eastern areas (Triângulo Mineiro). These areas have low slope values, good values of hydrography and accessibility, and higher values of solar potential, but their performance was low with respect to the hybrid potential criterion. Thus, considering that the obtained maps present a conservative decision attitude, these areas received low aptitude values.

Currently, Minas Gerais does not have HREGs in operation or construction, making it challenging to use models to validate the AMC results. Despite this, the solar photovoltaic and wind power plants in operation (Figure 4) were used to analyze the results’ quality (Figure 3). The procedures presented by [65] allow the analysis of the percentage of existing plants in areas identified as suitable by the generated decision maps. Figure 5 shows the percentages of solar photovoltaic and wind power plants existing in the different ranges of aptitude of the decision maps (0–0.2; 0.2–0.4; 0.4–0.6; 0.6–0.8; 0.8–1) in each of the five created scenarios. The “optimal areas” for installing power plants are those whose aptitude values are between 0.6–0.8 and 0.8–1 [65]. Since common plants were considered to evaluate the results obtained for the HREGs, it is assumed that the range of 0.4–0.6 can also be classified as an “optimal area.” Thus, it is possible to observe that most of the existing power plants are in regions with aptitude values of 0.4–0.6 for all analyzed scenarios (Figure 5).

The obtained decision maps serve as a basis for appointing ten solution alternatives, i.e., ten possible locations for installing HREGSs $(X_k)$. All these alternatives are located in regions with membership values ≥ 0.4 and are shown in Figure 3.

The payoff matrices corresponding to the criteria are given in

Table A1. Payoff matrix for the criterion “Potential energy of a hybrid power system”.

	${\mathit{Y}}_1$	${\mathit{Y}}_2$	${\mathit{Y}}_3$	${\mathit{Y}}_4$	${\mathit{Y}}_5$
$X_1$	0.7102	0.8233	0.1180	0.8019	0.1178
$X_2$	0.5018	0.0869	0.2914	0.0985	0.2756
$X_3$	0.1302	0.8529	0.1024	0.1142	0.1031
$X_4$	0.5966	0.7787	0.4535	0.6616	0.4641
$X_5$	0.1241	0.0850	0.6834	0.1003	0.6541
$X_6$	0.5323	0.1026	0.5031	0.0303	0.4912
$X_7$	0.6755	0.7527	0.4504	0.6869	0.4682
$X_8$	0.1144	0.1164	0.4646	0.1230	0.4429
$X_9$	0.7457	0.1180	0.4913	0.1247	0.4613
$X_{10}$	0.4738	0.1227	0.3921	0.0568	0.3877

,

Table A2. Payoff matrix for the criterion “Economic value of bare land”.

	${\mathit{Y}}_1$	${\mathit{Y}}_2$	${\mathit{Y}}_3$	${\mathit{Y}}_4$	${\mathit{Y}}_5$
$X_1$	1485	1485	1485	1485	1485
$X_2$	2000	2500	1500	1750	2750
$X_3$	5107	5107	5107	5107	5107
$X_4$	3250	3625	2875	3062	3812
$X_5$	4433	4433	4433	4433	4433
$X_6$	1100	1150	1050	1075	1175
$X_7$	1586	1586	1586	1586	1586
$X_8$	1750	2125	1375	1562	2312
$X_9$	2500	2750	2250	2375	2875
$X_{10}$	3800	4700	2900	3350	5150

,

Table A3. Payoff matrix for the criterion “Low-slope areas”.

	${\mathit{Y}}_1,{\mathit{Y}}_2,{\mathit{Y}}_3,{\mathit{Y}}_4,{\mathit{Y}}_5$
$X_1$	2.81
$X_2$	0.62
$X_3$	1.39
$X_4$	5.12
$X_5$	3.95
$X_6$	1.39
$X_7$	9.54
$X_8$	1.06
$X_9$	4.39
$X_{10}$	1.00

,

Table A4. Payoff matrix for the criterion “Distance from water bodies”.

	${\mathit{Y}}_1,{\mathit{Y}}_2,{\mathit{Y}}_3,{\mathit{Y}}_4,{\mathit{Y}}_5$
$X_1$	30,103.99
$X_2$	17,528.55
$X_3$	2549.51
$X_4$	20,554.80
$X_5$	9013.88
$X_6$	20,886.60
$X_7$	34,354.77
$X_8$	20,155.64
$X_9$	8845.90
$X_{10}$	33,120.99

,

Table A5. Payoff matrix for the criterion “Connection to transmission system”.

	${\mathit{Y}}_1,{\mathit{Y}}_2,{\mathit{Y}}_3,{\mathit{Y}}_4,{\mathit{Y}}_5$
$X_1$	23,537.21
$X_2$	62,699.68
$X_3$	14,008.93
$X_4$	10,062.31
$X_5$	43,133.51
$X_6$	71,562.91
$X_7$	62,980.16
$X_8$	97,464.09
$X_9$	107,872.60
$X_{10}$	79,210.16

and

Table A6. Payoff matrix for the criterion “Site accessibility”.

	${\mathit{Y}}_1,{\mathit{Y}}_2,{\mathit{Y}}_3,{\mathit{Y}}_4,{\mathit{Y}}_5$
$X_1$	2000.0
$X_2$	0
$X_3$	500.0
$X_4$	3000.0
$X_5$	500.0
$X_6$	707.1
$X_7$	500.0
$X_8$	2500.0
$X_9$	1000.0
$X_{10}$	2061.6

(all tables of the Case Study are presented in Appendix A).

Table A7. Matrix of choice criteria estimates for “Potential energy of a hybrid power system”.

	${\mathit{F}}_1^{\mathit{W}}({\mathit{X}}_{\mathit{k}})$	${\mathit{F}}_1^{\mathit{L}}({\mathit{X}}_{\mathit{k}})$	${\mathit{F}}_1^{\mathit{S}}({\mathit{X}}_{\mathit{k}})$	${\mathit{F}}_1^{\mathit{H}}({\mathit{X}}_{\mathit{k}})$
$X_1$	0.8233	0.5142	0.5654	0.6469
$X_2$	0.5018	0.2509	0.7660	0.3981
$X_3$	0.8529	0.2606	0.6877	0.6652
$X_4$	0.7787	0.5909	0.2300	0.6974
$X_5$	0.6834	0.3294	0.7679	0.5338
$X_6$	0.5323	0.3319	0.7716	0.4068
$X_7$	0.7527	0.6067	0.2330	0.6771
$X_8$	0.4646	0.2522	0.7365	0.3770
$X_9$	0.7457	0.3882	0.7348	0.5888
$X_{10}$	0.4738	0.2866	0.7450	0.3695
$\min$	0.4646	0.2509	0.2300	0.3695
$\max$	0.8529	0.6067	0.7716	0.6974

,

Table A8. Matrix of choice criteria estimates for “Economic value of bare land”.

	${\mathit{F}}_2^{\mathit{W}}({\mathit{X}}_{\mathit{k}})$	${\mathit{F}}_2^{\mathit{L}}({\mathit{X}}_{\mathit{k}})$	${\mathit{F}}_2^{\mathit{S}}({\mathit{X}}_{\mathit{k}})$	${\mathit{F}}_2^{\mathit{H}}({\mathit{X}}_{\mathit{k}})$
$X_1$	1485	1485	435	1485
$X_2$	2750	2100	1575	2437
$X_3$	5107	5107	4057	5107
$X_4$	3812	3325	2637	3578
$X_5$	4433	4433	3383	4433
$X_6$	1175	1110	0	1144
$X_7$	1586	1586	536	1586
$X_8$	2312	1825	1137	2078
$X_9$	2875	2550	1700	2719
$X_{10}$	5150	3980	3975	4587
$\min$	1175	1110	0	1144
$\max$	5150	5107	4057	5107

,

Table A9. Matrix of choice criteria estimates for “Low-slope areas”.

	$\begin{array}{l}{\mathit{F}}_3^{\mathit{W}}({\mathit{X}}_{\mathit{k}}),{\mathit{F}}_3^{\mathit{L}}({\mathit{X}}_{\mathit{k}}),\\ {\mathit{F}}_3^{\mathit{H}}({\mathit{X}}_{\mathit{k}})\end{array}$	${\mathit{F}}_3^{\mathit{S}}({\mathit{X}}_{\mathit{k}})$
$X_1$	2.81	2.19
$X_2$	0.62	0.00
$X_3$	1.39	0.77
$X_4$	5.12	4.50
$X_5$	3.95	3.33
$X_6$	1.39	0.77
$X_7$	9.54	8.91
$X_8$	1.06	0.44
$X_9$	4.39	3.77
$X_{10}$	1.00	0.38
$\min$	0.62	0.00
$\max$	9.54	8.91

,

Table A10. Matrix of choice criteria estimates for “Distance from water bodies”.

	$\begin{array}{l}{\mathit{F}}_3^{\mathit{W}}({\mathit{X}}_{\mathit{k}}),{\mathit{F}}_3^{\mathit{L}}({\mathit{X}}_{\mathit{k}}),\\ {\mathit{F}}_3^{\mathit{H}}({\mathit{X}}_{\mathit{k}})\end{array}$	${\mathit{F}}_4^{\mathit{S}}({\mathit{X}}_{\mathit{k}})$
$X_1$	30,103.99	27,554.48
$X_2$	17,528.55	14,979.04
$X_3$	2549.51	0
$X_4$	20,554.80	18,005.29
$X_5$	9013.88	6464.37
$X_6$	20,886.60	18,337.09
$X_7$	34,354.77	31,805.26
$X_8$	20,155.64	17,606.13
$X_9$	8845.90	6296.39
$X_{10}$	33,120.99	30,571.48
$\min$	2549.51	0
$\max$	34,354.77	31,805.26

,

Table A11. Matrix of choice criteria estimates for “Connection to transmission system”.

	$\begin{array}{l}{\mathit{F}}_6^{\mathit{W}}({\mathit{X}}_{\mathit{k}}),{\mathit{F}}_6^{\mathit{L}}({\mathit{X}}_{\mathit{k}}),\\ {\mathit{F}}_6^{\mathit{H}}({\mathit{X}}_{\mathit{k}})\end{array}$	${\mathit{F}}_6^{\mathit{S}}({\mathit{X}}_{\mathit{k}})$
$X_1$	23,537.21	13,474.90
$X_2$	62,699.68	52,637.37
$X_3$	14,008.93	3946.62
$X_4$	10,062.31	0
$X_5$	43,133.51	33,071.20
$X_6$	71,562.91	61,500.60
$X_7$	62,980.16	52,917.85
$X_8$	97,464.09	87,401.78
$X_9$	107,872.60	97,810.29
$X_{10}$	79,210.16	69,147.85
$\min$	10,062.31	0
$\max$	107,872.60	97,810.29

and

Table A12. Matrix of choice criteria estimates for “Site accessibility”.

	$\begin{array}{l}{\mathit{F}}_7^{\mathit{W}}({\mathit{X}}_{\mathit{k}}),{\mathit{F}}_7^{\mathit{L}}({\mathit{X}}_{\mathit{k}}),\\ {\mathit{F}}_7^{\mathit{S}}({\mathit{X}}_{\mathit{k}}),{\mathit{F}}_7^{\mathit{H}}({\mathit{X}}_{\mathit{k}})\end{array}$
$X_1$	2000.00
$X_2$	0
$X_3$	500.00
$X_4$	3000.00
$X_5$	500.00
$X_6$	707.11
$X_7$	500.00
$X_8$	2500.00
$X_9$	1000.00
$X_{10}$	2061.55
$\min$	0
$\max$	3000.00

provide the matrices of the choice criteria estimates. The modified matrices of choice criteria estimates are presented in

Table A13. Modified matrix of choice criteria estimates for “Potential energy of a hybrid power system”.

	${\mathit{\mu }}_{{\mathit{A}}_1}^{\mathit{W}}({\mathit{X}}_{\mathit{k}})$	${\mathit{\mu }}_{{\mathit{A}}_1}^{\mathit{L}}({\mathit{X}}_{\mathit{k}})$	${\mathit{\mu }}_{{\mathit{A}}_1}^{\mathit{S}}({\mathit{X}}_{\mathit{k}})$	${\mathit{\mu }}_{{\mathit{A}}_1}^{\mathit{H}}({\mathit{X}}_{\mathit{k}})$
$X_1$	0.9238	0.7400	0.3807	0.8460
$X_2$	0.0959	0	0.0104	0.0870
$X_3$	1	0.0273	0.1549	0.9019
$X_4$	0.8090	0.9555	1	1
$X_5$	0.5637	0.2207	0.0069	0.5011
$X_6$	0.1745	0.2278	0	0.1137
$X_7$	0.7420	1	0.9944	0.9382
$X_8$	0	0.0039	0.0648	0.0228
$X_9$	0.7240	0.3859	0.0679	0.6687
$X_{10}$	0.0237	0.1005	0.0490	0

,

Table A14. Modified matrix of choice criteria estimates for “Economic value of bare land”.

	${\mathit{\mu }}_{{\mathit{A}}_2}^{\mathit{W}}({\mathit{X}}_{\mathit{k}})$	${\mathit{\mu }}_{{\mathit{A}}_2}^{\mathit{L}}({\mathit{X}}_{\mathit{k}})$	${\mathit{\mu }}_{{\mathit{A}}_2}^{\mathit{S}}({\mathit{X}}_{\mathit{k}})$	${\mathit{\mu }}_{{\mathit{A}}_2}^{\mathit{H}}({\mathit{X}}_{\mathit{k}})$
$X_1$	0.9220	0.9062	0.8928	0.9139
$X_2$	0.6038	0.7523	0.6118	0.6735
$X_3$	0.0109	0	0	0
$X_4$	0.3365	0.4458	0.3498	0.3857
$X_5$	0.1803	0.1685	0.1660	0.1699
$X_6$	1	1	1	1
$X_7$	0.8966	0.8809	0.8679	0.8884
$X_8$	0.7138	0.8211	0.7196	0.7642
$X_9$	0.5723	0.6397	0.5809	0.6026
$X_{10}$	0	0.2819	0.0201	0.1310

,

Table A15. Modified matrix of choice criteria estimates for “Low-slope areas”.

	$\begin{array}{l}{\mathit{\mu }}_{{\mathit{A}}_3}^{\mathit{W}}({\mathit{X}}_{\mathit{k}}),{\mathit{\mu }}_{{\mathit{A}}_3}^{\mathit{L}}({\mathit{X}}_{\mathit{k}}),\\ {\mathit{\mu }}_{{\mathit{A}}_3}^{\mathit{S}}({\mathit{X}}_{\mathit{k}}),{\mathit{\mu }}_{{\mathit{A}}_3}^{\mathit{H}}({\mathit{X}}_{\mathit{k}})\end{array}$
$X_1$	0.7542
$X_2$	1
$X_3$	0.9139
$X_4$	0.4951
$X_5$	0.6263
$X_6$	0.9139
$X_7$	0
$X_8$	0.9510
$X_9$	0.5770
$X_{10}$	0.9573

,

Table A16. Modified matrix of choice criteria estimates for “Distance from water bodies”.

	$\begin{array}{l}{\mathit{\mu }}_{{\mathit{A}}_4}^{\mathit{W}}({\mathit{X}}_{\mathit{k}}),{\mathit{\mu }}_{{\mathit{A}}_4}^{\mathit{L}}({\mathit{X}}_{\mathit{k}}),\\ {\mathit{\mu }}_{{\mathit{A}}_4}^{\mathit{S}}({\mathit{X}}_{\mathit{k}}),{\mathit{\mu }}_{{\mathit{A}}_4}^{\mathit{H}}({\mathit{X}}_{\mathit{k}})\end{array}$
$X_1$	0.1337
$X_2$	0.5290
$X_3$	1
$X_4$	0.4339
$X_5$	0.7968
$X_6$	0.4235
$X_7$	0
$X_8$	0.4464
$X_9$	0.8020
$X_{10}$	0.0388

,

Table A17. Modified matrix of choice criteria estimates for “Connection to transmission system”.

	$\begin{array}{l}{\mathit{\mu }}_{{\mathit{A}}_6}^{\mathit{W}}({\mathit{X}}_{\mathit{k}}),{\mathit{\mu }}_{{\mathit{A}}_6}^{\mathit{L}}({\mathit{X}}_{\mathit{k}}),\\ {\mathit{\mu }}_{{\mathit{A}}_6}^{\mathit{S}}({\mathit{X}}_{\mathit{k}}),{\mathit{\mu }}_{{\mathit{A}}_6}^{\mathit{H}}({\mathit{X}}_{\mathit{k}})\end{array}$
$X_1$	0.8622
$X_2$	0.4618
$X_3$	0.9597
$X_4$	1
$X_5$	0.6619
$X_6$	0.3712
$X_7$	0.4590
$X_8$	0.1064
$X_9$	0
$X_{10}$	0.2930

and

Table A18. Modified matrix of choice criteria estimates for “Site accessibility”.

	$\begin{array}{l}{\mathit{\mu }}_{{\mathit{A}}_7}^{\mathit{W}}({\mathit{X}}_{\mathit{k}}),{\mathit{\mu }}_{{\mathit{A}}_7}^{\mathit{L}}({\mathit{X}}_{\mathit{k}}),\\ {\mathit{\mu }}_{{\mathit{A}}_7}^{\mathit{S}}({\mathit{X}}_{\mathit{k}}),{\mathit{\mu }}_{{\mathit{A}}_7}^{\mathit{H}}({\mathit{X}}_{\mathit{k}})\end{array}$
$X_1$	0.3333
$X_2$	1
$X_3$	0.8333
$X_4$	0
$X_5$	0.8333
$X_6$	0.7643
$X_7$	0.8333
$X_8$	0.1667
$X_9$	0.6667
$X_{10}$	0.3128

. Finally, the aggregated payoff matrix of the choice criteria estimates is given in

Table A19. Aggregated matrix of choice criteria estimates.

	${\mathit{\mu }}_{\mathit{D}}^{\mathit{W}}({\mathit{X}}_{\mathit{k}})$	${\mathit{\mu }}_{\mathit{D}}^{\mathit{L}}({\mathit{X}}_{\mathit{k}})$	${\mathit{\mu }}_{\mathit{D}}^{\mathit{S}}({\mathit{X}}_{\mathit{k}})$	${\mathit{\mu }}_{\mathit{D}}^{\mathit{H}}({\mathit{X}}_{\mathit{k}})$
$X_1$	0.1337	0.1337	0.1337	0.1337
$X_2$	0.0959	0	0.0104	0.0870
$X_3$	0.0109	0	0	0
$X_4$	0	0	0	0
$X_5$	0.1803	0.1685	0.0069	0.1699
$X_6$	0.1745	0.2278	0	0.1137
$X_7$	0	0	0	0
$X_8$	0	0.0039	0.0648	0.0228
$X_9$	0	0	0	0
$X_{10}$	0	0.0388	0.0201	0

. This matrix indicates that the solution alternatives $X_1,X_5, \mathrm{and} X_6$ are indistinguishable.

In this way, the application of the $<X,R>$ models are needed for the subsequent ordering of the solution alternatives $X=\left\{X_1,X_5,X_6\right\}$, which are the following: $X_1$—Central region of the Buritis municipality (solution $X_1$ so far); $X_2$—North of the Formoso municipality (solution $X_5$ so far); and $X_3$—Northwest of the Espinosa municipality (solution $X_6$ so far).

The alternatives have been analyzed by applying additional criteria: (1) Distance from planned TLs; (2) Social potential; and (3) Natural vulnerability.

Evaluating the solution alternatives, e.g., considering the criterion “Distance from planned TLs”, is based on quantitative information. At the same time, the estimates of alternatives from the point of view of the criteria “Social potential” and “Natural vulnerability” were obtained in a qualitative form.

Table 4. Evaluations of solution alternatives.

	${\mathit{F}}_1({\mathit{X}}_{\mathit{k}}):{\mathit{C}}_{1\left[\mathrm{meters}\right]}$	${\mathit{F}}_2({\mathit{X}}_{\mathit{k}}):{\mathit{C}}_2$	${\mathit{F}}_3({\mathit{X}}_{\mathit{k}}):{\mathit{C}}_3$
$X_1$	$500.00$	$\mathrm{Moderate}$	$\mathrm{Very} \mathrm{Large}$
$X_2$	$30,000.00$	$\mathrm{Very} \mathrm{small}$	$\mathrm{Large}$
$X_3$	$25,800.00$	$\mathrm{Very} \mathrm{small}$	$\mathrm{Moderate}$

reflects the evaluations of each alternative following the considered criteria.

For the first criterion, it is possible to apply (29) to construct the corresponding ARFPR. The application of the transformation function (30) permits the conversion of the obtained ARFPR to the following NRFPR:

$${\mu }_{R_1}(X_k,X_l)=\begin{array}{cccc}& X_1& X_2& X_3\\ X_1& 1& 1& 1\\ X_2& 0& 1& 0.86\\ X_3& 0.14& 1& 1\end{array}$$

Let us consider the criterion “Social potential”, which is qualitative. A social potential can be defined as a set of current conditions, measured by the productive, natural, and human dimensions, which determine the starting point of a municipality or region to achieve sustainable development. Regions with high social potential have a well-developed transport infrastructure, economic activities, and social conditions (more discussions can be found in [66]). Thus, the rationality of installing a generation plant may be associated with prioritizing areas with high social potential. However, installing a generation plant can stimulate a region’s social development due, for example, to job creation. Thus, this step has a strategic character regarding a public policy for needy regions. Therefore, this study considers that the discussed criterion should be minimized. Figure 6A shows this criterion’s evaluations (levels of MFs) and its spatial distribution for the Minas Gerais State. Figure 6B also shows the evaluations (levels of MFs) for the criterion “Natural vulnerability”, which also should be minimized and whose spatial character can be observed in the map. Figure 6C shows the MFs (or fuzzy values) of the estimates utilized for these criteria.

From the point of view described, we can observe that the social potential and natural vulnerability criteria assume a conflicting character since the proposed analysis searches areas with low social potential (localized in the northern region mainly) and low natural vulnerability (mainly localized in the east and south). The proposed technique allows the consideration of the problems’ characteristics and the choosing of a more appropriate solution.

Applying (26) and (27) for the estimates presented in Figure 6C, defined for $F_p(X_k)\le F_p(X_l)$ and $F_p(X_l)\le F_p(X_k)$, respectively, we can construct the following NRFPRs:

$${\mu }_{R_2}(X_k,X_l)=\begin{array}{cccc}& X_1& X_2& X_3\\ X_1& 1& 0.60& 0.60\\ X_2& 1& 1& 1\\ X_3& 1& 1& 1\end{array}$$

$${\mu }_{R_3}(X_k,X_l)=\begin{array}{cccc}& X_1& X_2& X_3\\ X_1& 1& 0.60& 0.60\\ X_2& 1& 1& 0.94\\ X_3& 1& 1& 1\end{array}$$

As mentioned above, choosing the techniques for analyzing $<X,R>$ models is a prerogative of DMs. In particular, using (34), we can construct

$${\mu }_R(X_k,X_l)=\begin{array}{cccc}& X_1& X_2& X_3\\ X_1& 1& 0.60& 0.60\\ X_2& 0& 1& 0.86\\ X_3& 0.14& 1& 1\end{array}$$

It is possible to obtain the following SFPR by applying (31):

$${\mu }_R^S(X_k,X_l)=\begin{array}{cccc}& X_1& X_2& X_3\\ X_1& 0& 0.60& \\ X_2& 0& 0& 0\\ X_3& 0& 0.14& 0\end{array}0.46$$

Finally, applying (32) generates the set of non-dominated alternatives with the following MF:${\mu }_R^{ND}(X_k)= [1 0.40 0.54]$. Therefore, the analysis based on applying the three additional criteria and the first discussed technique generates the following ranking of the alternatives: $X_1\succ X_3\succ X_2$.

Let us consider the solution using the OWA operator jointly with fuzzy quantifiers as an aggregation rule for considering the criteria. The utilization of the concept of fuzzy majority for three additional criteria provides the vector of the order weights ${\omega }_1=0.0667;{\omega }_2=0.6667$, and ${\omega }_3=0.2667$, obtained through (36). Their use generates the following global nonstrict preference matrix, obtained based on (35):

$${\mu }_R(X_k,X_l)=\begin{array}{cccc}& X_1& X_2& X_3\\ X_1& 1& 0.83& 0.63\\ X_2& 0.73& 1& 0.92\\ X_3& 0.77& 1& 1\end{array}$$

The concept of fuzzy majority aggregates the elements of the global nonstrict preference matrix obtained above and is applied to obtain the global ranking of the alternatives. Excluding the main diagonal elements, it is possible to obtain the order weights ${\omega }_1=0.4;{\omega }_2=0.6$, based on applying (36). Thus, it is possible to obtain the following global ranking of the alternatives with the use of the QGDD expressed by (37):

$$\mathrm{QGDD}(X_k)=[0.71 0.81 0.86]$$

(39)

Thus, following the application of this approach, the alternatives are ranked as follows: $X_3\succ X_2\succ X_1$.

4. Discussion

This research adapts, improves, and applies decision-making models and methods to determine locational alternatives for HREGS. The analysis starts with the SMA in uncertain conditions based on applying the possibilistic approach and S-OWA operator. First, the spatial relevant criteria are geographically modeled (in the geo-fields format) and normalized through fuzzy sets. The normalization process allows one to express all the values of the criteria on the same scale (the unit interval) and, at the same time, to express the preferences of the DMs regarding the performance of each alternative concerning each criterion. In the analysis, the DM defines the MF for each criterion, which reflects his/her preferences in the best way. In the case study presented in this work, correlations (7)–(10) have been used. Aggregating the criteria for generating a final decision map was performed using the S-OWA operator, with the levels of optimism $\alpha =0.4$ and pessimism $\beta =0.6$ for five analyzed scenarios. These procedures were executed to treat internal uncertainties. Figure 5 presents the SMA results in each scenario. It is possible to observe that, although the decision maps obtained in each scenario are quite different, predominantly more attractive areas for installing HREGSs are in the northwestern, northern, and northeastern regions of the Minas Gerais State.

Analyzing or validating the SMA’s results is challenging since Minas Gerais has no HREGs in operation. In this way, the research considers existing wind and solar photovoltaic plants to evaluate the obtained results. In this sense, Figure 5 shows the percentages of existing photovoltaic solar power plants (Figure 5A) and existing wind power plants (Figure 5B) at different ranges of aptitude values (0–0.2; 0.2–0.4; 0.4–0.6; 0.6–0.8; 0.8–1) of the generated decision maps (Figure 5) in each one of the five analyzed scenarios.

Firstly, it is possible to observe that there are no plants in any analyzed scenario in areas classified with the highest aptitude values (range of 0.8–1). However, this can be considered acceptable since the existing plants are not HREGs. On the other hand, none of the existing plants are located in constrained regions (those with aptitude values equal to zero) or localities with the lowest aptitude values. This fact can indicate that the utilized models can identify the worst locations for installing HREGs, aside from the prohibitive areas.

Most existing solar photovoltaic power plants are located in regions classified in the aptitude range of 0.4–0.6 in all scenarios (Figure 5A). The existing wind power plants are equally distributed (33%) in regions with aptitude values equal to 0.2–0.4, 0.4–0. 6, 0.6–0.8 in the case of Scenario 1. In all other scenarios, most existing wind power plants are located in areas with aptitude values of 0.4–0.6 (Figure 5B). The “optimal areas” for installing the power plants are those whose aptitude values are between 0.6–0.8 and 0.8–1, according to the criteria of [65]. Given that common plants were considered to evaluate the results obtained for HREGs, the range of 0.4–0.6 can also be classified as an “optimal area.” This evidence suggests that decision maps have the desired quality even with limited data.

It is worth mentioning that the parameters utilized in the aggregation implement a pessimistic decision attitude, which reflects the lower performance for each solution alternative concerning all analyzed criteria. A change in the values of the optimism/pessimism levels leads to the generation of different decision maps, which may reflect, for example, a more compensatory attitude toward decisions.

For the treatment of external uncertainties, the possibilistic approach is applied. For this purpose, payoff matrices have been constructed for each criterion of the factor type. In the case study, only two criteria (“Potential of energy of a hybrid power system” and “Economic value of bare land”) have associated deep uncertainties. However, the payoff matrices were constructed since all criteria were to be considered. This explains the presence of equal values in Table A3, Table A4, Table A5 and Table A6 (Appendix A).

The final result, reflected by the aggregated matrix (Table A19), indicates that, according to the criteria of Wald and Hurwicz, the best solution alternative is $X_5$. At the same time, from the point of view of the criteria of Laplace and Savage, the best solutions are $X_6$ and $X_1$, respectively. Therefore, formally, these solutions cannot be distinguished. It is necessary to apply the corresponding procedures to perform an additional analysis of these alternatives.

In particular, the utilized procedures allow one to use criteria of quantitative and qualitative character (spatial or non-spatial). The corresponding qualitative information can be represented by applying various preference formats, and the DM can choose the best one based on his/her knowledge, experience, and intuition.

Two techniques for analyzing $<X,R>$ models have been utilized to process the preferences. Applying the first technique in the presence of the three additional criteria provides the following ordering of the alternatives: $X_1\succ X_3\succ X_2$.

The second technique reflects different decision attitudes oriented by different FLQs. Using the “Majority” quantifier leads to the following ordering of the alternatives: $X_3\succ X_2\succ X_1$. Other quantifiers may also be utilized, for example, “At least one” and “As many as possible.” In this sense, using the quantifier “All” reduces the OWA operator to the minimum operator, providing the same solution obtained based on the first technique. The quantifiers implement different decision attitudes, mainly related to the degree of risk tolerance for the DMs. The application of different techniques, as well as different FLQs, can lead to different solutions. It is to be considered natural, and the choice of the technique and the quantifier is a prerogative of the DM.

5. Conclusions

The methodologies that compose this analysis have been chosen to adequately consider the internal and external uncertainties inherent to decision-making processes. External uncertainties are treated by applying the possibilistic approach, which permits one to consider characteristic estimates used in constructing the classic approach’s choice criteria to deal with uncertainties as objective functions. The internal uncertainties are considered by applying fuzzy sets to normalize the spatial criteria. The aggregation process for generating decision maps is based on adjusting the S-OWA operator’s optimism and pessimism parameters. This allows one to generate maps (geo-fields) that reflect different decision attitudes related to the degree of risk tolerance for the DMs. At the second moment, the analysis of the internal and external uncertainties is realized through the modeling and processing of FPRs, guided by methodological principles for analyzing $<X,R>$ models. This stage permits one to include additional criteria of quantitative and qualitative character.

The main contribution of this research is the adequate treatment of the uncertainties related to the problem of the determination of locational alternatives for HREGSs through the possibilistic approach and the consideration of a wide range of criteria, as well as the use of strategies incorporating the DM’s knowledge, experience, and intuition in generating the preferences for the decision process.

The study results for the Minas Gerais state show that decision maps are visually meaningful and can indicate feasible and unfeasible areas for setting up HREGs. These results are significant as they consider the large dimension of the territory and the complexity related to the decision-making processes required for the energy planning of areas such as these. Moreover, the results allow the determination of locational options for HREGSs using a wide range of quantitative and qualitative spatial criteria.

Future developments can include evaluating the energy complementarity between three or more sources and models to consider the complementarity between sources in different parts of the territory and to consider the existing electric power transmission system.