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)
that associates each object of interest belonging to the universe of discourse
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
where
and
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
.
Hence, in order to create fuzzy sets
with MFs
reflecting a degree of achieving the decision criteria, the correlation
can be employed for minimized objective functions; alternatively, the correlation
can be used for maximized ones [
5,
12]. Other correlations (for example, [
37]) can also be employed to construct the fuzzy sets. For instance,
can be used for maximized objective functions, while
can be applied for minimized objective functions. The parameters
and
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
are constructed, and an aggregation function
can be applied to generate a fuzzy set named fuzzy decision, which can be presented in the following form [
12]:
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
and
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
where
is the
jth highest element of the
.
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
where
and
. The parameter
is the degree of optimism to be considered and
is the degree of pessimism.
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:
To evaluate the maximum risk level
Rmax(
Xk), it is necessary to use the minimum value of the criterion
(for a case of minimization) or
(for a case of maximization) for each column (scenario) of the payoff matrix [
42,
43,
44]:
for minimized criteria and
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:
In (20),
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. 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:
where
,
,
, and
.
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
are modeled by fuzzy sets
where
is the MF of
. A fuzzy solution
D is defined [
49,
50] as
with the MF
The following problem can be generated:
whose solution is
To obtain (24), it is necessary to build
to reflect the degree of achieving own optima by
. 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
,
,
, and
are replaced, respectively, by the membership values
,
,
, and
. 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
,
,
, and
are replaced, respectively, by the aggregated membership values
,
,
, and
, 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 .
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
, where
X is a set of alternatives
and
is a set of Nonstrict Fuzzy Preference Relations (NSFPRs), which can be presented as
where
is an MF of the
bth NSFPR [
51] for the
bth additional criterion.
Using a square matrix with MF
, an NSFPR
can be modeled that associates a number within a unit interval
for each pair of alternatives
. Thus,
indicates the degree that the alternative
is at least as good as
when the criterion
is considered [
5,
12].
While dealing with
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 have a fuzzy estimate regarding criterion , then the quantity is the degree of preference , while the quantity is the degree of preference .
Based on the conception of an MF of a generalized preference relation [
52] and considering the minimization of
, the quantities
and
, respectively, can be written as [
12,
52]
and
For the maximization of
, (26) and (27) should be written, respectively, as
and
.
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
models requires the reduction of the corresponding quantitative estimates to FPRs.
In particular, since for Additive Reciprocal Fuzzy Preference Relations (ARFPRs), the conditions
and
are met, it is possible to use the [
5,
11,
12] results:
for minimizing
and
for maximizing
.
The ARFPR can be converted to NRFPR [
5,
12] as follows:
The information given in the form of can be processed and applied to narrow the set , leaving as alternatives only the other ones do not dominate from .
Six techniques of analyzing
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
Since
is the MF of the fuzzy set of all alternatives
that are strictly dominated by
, its complement
provides the fuzzy set of the alternatives that are not dominated by
. The MF of this set can be stated as
The alternatives which can be chosen as the problem solution, according to the highest non-dominance level, are the following:
The expressions (31)–(33) are also applicable when
is a vector of FPRs. For example, they serve for the first method of [
12], considering
with the MF [
52,
53]:
When applying (34), the set
[
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
where
is the
th largest value among
. Again, the set of weights
in (35) is to satisfy the conditions
and
.
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
that reflect the level
of the criteria portion that satisfies a concept represented by
Q. The weights can be obtained, after choosing the appropriate FLQ, by
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:
reflects the dominance level of
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
To achieve this result, one has to apply the global nonstrict preference matrix to compute the global strict preference relation
[
55].
Considering that choosing alternatives according to the highest level of non-dominance is reasonable, the alternatives 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.