1. Introduction
The ability to make decisions is an inherent and essential characteristic of human beings. Decisions are constantly being made daily, most of them individually and intuitively. However, when an issue of greater importance or complexity arises that requires a rational approach, a decision process must be adopted. Optimal decision-making is one of the most studied fields by scientific researchers. Although its most significant proliferation has occurred in recent years, from the 1970s onwards, some of the main mathematical tools for multi-criteria decision making (MCDM), now considered classical methods, appeared. Since the definition of the Sustainable Development Goals (SDGs) was introduced by the United Nations in 2015, sustainable design has been one of the most recent trends in decision-making. This approach calls for a paradigm shift in classical decision-making practices, orienting design towards creating products and services that consider the three dimensions of sustainability [
1] from the initial stage to the end of their lifespan.
The construction sector has not been immune to this new trend either. The sustainable construction of buildings and urban districts is directly related to 15 of the 17 SDGs [
2]. The construction of buildings requires a large number of natural resources and land. In addition, the production of materials can consume enormous amounts of non-renewable energy and water and generate harmful emissions to the environment. However, the quality and characteristics of a building have an enormous influence on our well-being, and well-managed construction activities contribute to the development of both the private and public economy. Designing buildings and infrastructure that effectively contribute to the sustainable future demanded by society is becoming a priority for architects and engineers, who are now faced with the challenge of finding a balance between the positive and negative impacts generated in the economic, environmental, and social spheres of their designs. More than 700 methods have been counted since the 1970s that attempt to assess the performance of buildings and their impacts [
3] through quantitative economic, environmental, social, or usability indicators. However, there has been an exponential increase in research on the construction sector oriented toward a sustainable approach and the search for a circular economy in the last decade. Lately, several studies have been carried out to evaluate sustainability in construction projects, from bridges [
4,
5], buildings [
6,
7], construction elements such as pavements [
8], or retaining walls [
9,
10], among many other aspects of construction design and management.
However, the balance between social, environmental, and economic impacts among different alternatives does not lead to a trivial and univocal decision on the best option. It involves the criteria of several stakeholders whose optimization objectives may be at odds [
11,
12]. To address selecting a sustainable solution among a set of possible options, one of the most accepted approaches in the scientific community is to pose it as an MCDM problem. Decision-making techniques provide a rational decision based on specific information, experience, and judgment. Most MCDM methods share the same steps: problem structuring, formulation of criteria, method selection and evaluation, and supporting implementation [
13,
14]. In the first step, the weights of each criterion need to be assigned. Different MCDM procedures were developed in the last decades [
15,
16] to determine the most appropriate solution depending on the decision maker’s (DM) understanding of the problem. Although, to date, it is not possible to show the supremacy of any technique or school of thought concerning the multi-criteria decision-making paradigm. The most widely used decision-making theory is the one presented by Saaty [
17], known as the analytical hierarchical process (AHP).
The AHP owes its appeal to the fact that it translates the DM’s vision into numerical values, judging the relative relevance of each criterion using pairwise comparisons and according to a scale of priorities. The method thus makes it possible to assimilate the tangible and the intangible, the objective and the subjective, and even the rational and the emotional. It is an easy-to-use procedure applicable to numerous real-life scenarios that require a choice between a set of alternatives. The model allows individual and group decisions to be combined, although it is sometimes difficult to reach a consensual agreement [
18]. Moreover, AHP is one of the few multi-criteria techniques with its theoretical axiomatic [
19]. However, the use of classical AHP has been the subject of considerable debate since it assumes that the judgments made by the DM are true. One such criticism is that the weights may be distorted if the AHP hierarchy is incomplete. In addition, because of its linearity, the number of criteria at each level conditions the relative weightings, thus drastically reducing the interest of those sub-criteria and indicators that hang hierarchically from the first, less-weighted level.
In exchange for simplicity, AHP does consider the uncertainty associated with the numerical quantification of opinion, resulting in highly subjective weightings. This means that decision-making can be heavily biased by the so-called non-probabilistic uncertainties associated with the expert’s ability to consistently reflect their view of the problem when performing pairwise comparisons. Moreover, the greater the complexity of the problem to be evaluated, the more the subject’s ability to make judgments decreases so that certainty and precision are mutually exclusive [
20]. This is the case for decision-making problems related to sustainability, particularly construction. Situations of conflict arise between a wide variety of criteria that combine a greater or lesser opposition in the preferences of the DMs who, in addition, individually depend on those taken by the rest in pursuing their interests.
Consequently, research has been carried out over the last few decades to effectively reflect the decision maker’s view of the problem, focusing on minimizing subjectivity by extracting as much information and accuracy as possible from their judgments to obtain meaningful criteria weights. Researchers have begun to use fuzzy [
21] and intuitionistic [
22] perspectives to incorporate non-probabilistic uncertainties coupled with cognitive information derived from complex decision-making problems. More recently, neutrosophic logic has begun to be incorporated into the AHP procedure as a more advanced generalization of fuzzy set theory [
23]. Another existing trend consists of reducing the number of paired comparisons to be solved to simplify the decision-making problem and thus facilitate the consistency of the judgments made by the DM [
24].
To solve the most critical limitations of the AHP method, Saaty [
25] presented the analytic network process (ANP) model. This method emerged as a generalization of the AHP that allows the integration of the interdependence and feedback relationships between the criteria, sub-criteria, or alternatives, generating a genuine network of influences between them when making the final decision. Thus, the ANP has emerged as an appropriate decision-making procedure to address problems related to sustainability [
26,
27,
28]. It allows the complexity of the relationships between decision elements to be accurately captured and the weights of criteria and the local and global priorities of alternatives to be calculated. To consider the influences between the different elements of the system, the calculation of criteria and alternative weights in the ANP requires a specific network structuring instead of the typical linear hierarchical structure of the AHP. The network is formed by nodes or clusters, each comprising a series of elements that can be criteria or alternatives. Feedback is the relationship between elements of the same cluster, while interdependence is the relationship between elements of different clusters.
However, like any other method, ANP also has limitations [
29]. A large number of relationships and criteria complicates calculations. The more relationships between elements, the more questions the experts need to ask to define the influences between all components and elements in the matrices. Therefore, it is necessary to facilitate the methodology by the DM. The usefulness of this methodology lies precisely in the fact that if the decision problem is well formulated, the procedure can be significantly simplified while maintaining the advantages of the network approach. However, the rigor of the method is not lost as it still consists of a structure with groups of networked elements. Consequently, it is a model closer to the complexity of real-world problems.
Here, an adaptive model based on the ANP has been proposed to consider the relationships between the alternatives and the different criteria with quantitative variables, generating an entire network that considers all the influences between them when making the final decision. This method is applied to the sustainability performance of four different design options for the structure of a residential building over its life cycle. Finally, an improved version of the ELECTRE I [
30] method is used in the fuzzy intermediate values environment, combining the weights of the AHP and ANP groups with the so-called ELECTRE IS [
31].
The remainder of the paper is structured as follows:
Section 2 develops the methodology with the different techniques applied in the calculation procedure.
Section 3 presents a sustainable design decision problem as a case study to apply the proposed ANP and ELECTRE IS adaptation.
Section 4 collects the results and their respective analysis; a sensitivity study with other models is also included here. Finally,
Section 5 concludes the research and presents proposals for future work.
2. Materials and Methods
2.1. Fundamentals of the Analytic Hierarchical Process (AHP)
The classical AHP was born out of the need to solve specific decision problems in the U.S. Department of Defense in the late 1970s. This MCDM method, developed by Professor Thomas L. Saaty, ended up extending its application to almost all fields and complex situations that required a decision support tool. The methodology helps to select between alternatives based on a series of selection variables (criteria), usually hierarchical and often in conflict with each other. The structure is hierarchized top-down, starting from the final objective, the criteria, sub-criteria, and indicators (if applicable), and, finally, the alternatives to be compared. A fundamental aspect of the method is the adequate definition of the variables to be considered relevant and mutually exclusive (independent of each other).
The mechanics of the method are based on the realization of pairwise comparison matrices at each hierarchical level, where the DM expresses the relative priority of one concept over another and the intensity of this preference. These comparisons are scored according to the Saaty Fundamental Scale [
32], which uses the principle of the Weber–Fechner. As the relationship between stimulus and perception corresponds to a logarithmic scale, the perception evolves as an arithmetic progression if a stimulus grows in geometric progression. Thus, Saaty’s scale makes it possible to transform a set of nine definitions of qualitative relevance into quantitative values between 1 and 9. This semantic scale expresses a gradation of how important a criterion or alternative “
i” is considered to another “
j”, with 1 being “equally important” and 9 equivalent to “
i is extremely more important than
j”. As a result, the so-called decision matrix
A = {
aij} is obtained, which is a square matrix satisfying the properties of: reciprocity (if
aij = x, then
aji = 1/x ∀
i,
j∈{1, …,
n}, where
n is the number of criteria or alternatives to be compared)
; homogeneity (if
i and
j are of equal importance,
aij =
aji = 1, and
aii = 1 ∀
i∈{1, …,
n}).
One of the virtues of the method is to evaluate the coherence of the decision, for which matrix
A must not contain contradictions in the judgments expressed. Consistency can be measured through the consistency index (
CI), defined as:
where
λmax is the greatest eigenvalue and
n is the dimension of the decision matrix. From the
CI value obtained, a consistency ratio (
CR) can be obtained as:
where
RI is the random index, which indicates the consistency of a given random matrix according to
Table 1. If
CI is close to
RI, the matrix has been completed randomly, thus expressing an absolute inconsistency in evaluating the problem to be solved. Conversely, a low consistency ratio means that the DM has a clear knowledge of the problem to be solved, being for
CI = 0 a complete consistency. Inconsistency will be acceptable if the
CR does not exceed the values indicated in
Table 2, in which case the subjective weightings would have to be revised.
Once the consistency has been verified, the weights, which represent the relative importance of each criterion or the priorities of the different alternatives concerning a given criterion, are obtained by solving the following equation:
where
A is the comparison matrix,
w is the eigenvector or preference vector and
λmax is the eigenvalue.
2.2. Fundamentals of the Analytical Network Process (ANP)
In an AHP-based decision model, the criteria (and sub-criteria and indicators, if applicable) and alternatives are structured hierarchically through a linear and unidirectional relationship between levels. According to Saaty [
25], the ANP method allows a much broader representation of the decision problem by structuring it in the form of a network. In this model, interdependencies between all system components are possible. The independence of the elements of a higher level concerning those of a lower level is not assumed, nor is independence between elements of the same level. This allows for a non-linear structure that prioritizes elements and groups or clusters of elements better adapted to the complexity of the real world.
A network model consists of elements or nodes (mainly decision criteria and alternatives) grouped into components, groups, or clusters. The clusters are denoted by Ck (where m = 1, 2, …, m), and it is established that each cluster contains enk elements denoted by e1k, e2k, …, enk. An element of a cluster in the network can have one or bidirectional influence on some or all of the elements of that cluster or a different cluster belonging to the network. These relationships are feedback (between elements of the same cluster) and interdependent (between components of different clusters). In general terms, the ANP consists of two fundamental stages: the first is the structuring of the problem (construction of the network), and the second is the calculation of the priorities of the elements. However, the specific six steps for implementing the ANP are listed below:
Step 1: Model the decision problem as a network. The quality of the network depends mainly on the degree of knowledge of the problem on the part of the DM. The process begins with identifying network elements (criteria and alternatives).
Step 2: Grouping of the elements into components. The DM needs to properly define which of the above elements will be part of each cluster based on sharing some common characteristics.
Step 3: Analysis of the network of influences. Each element mij in the ANP matrix is filled in with values 0 or 1, where 1 means that element i is influenced by element j. It should be noted that this is not a reciprocal matrix, i.e., element i can be influenced by element j, but element j does not necessarily have to be influenced by element i. Thus, a correlation matrix is obtained, which is called the influential supermatrix.
Step 4: Calculation of priorities between elements. For each cluster, only the non-zero components of the matrix will be considered. There are as many pairwise comparison matrices between elements associated with a network element as groups of elements belonging to the same cluster that influence that element. This influence is obtained by the conventional AHP method, using Saaty’s fundamental scale to complete the entries of the paired comparison matrices. Let us assume as an example that elements
e2,
e3, and
e4, both belonging to cluster
C1, have an influence on element
e1 (
Figure 1). A simple AHP model will be constructed to determine how much influence each of the three elements has on
e1.
By performing this process with each element of the influential supermatrix, the so-called unweighted supermatrix is constructed. The elements that indicated the existence of influence with a “1” are now replaced by the quantification of such influence. The inputs collect the weights of the relative influence of the elements located in the rows of the matrix on the elements located in the columns, as shown in
Figure 1.
Step 5: Calculation of priorities between clusters. It should be noted that the matrix of the previous step is not stochastic, i.e., its columns do not add up to 1. For the unweighted supermatrix to be stochastic, the elements of each cluster will be multiplied by the weight of each cluster (considering both criteria and alternative clusters). A pairwise inter-cluster comparison matrix associated with a given network cluster is one whose rows and columns consist of all the clusters in the network that influence that given component. There are as many paired comparison matrices between clusters in the model as groups of clusters influencing any given cluster in the network. These weights are again obtained using a conventional AHP procedure. The resulting stochastic supermatrix is then called the weighted supermatrix.
Step 6: Determine the criteria weights and the preferred alternatives. The stochastic weighted supermatrix is raised to successive powers until its entries converge and remain stable. Such matrix is called the limiting supermatrix, and all its columns are equal. If you want to know the final ranking of the alternatives, look at the entries in any column of the limiting supermatrix corresponding to the rows associated with the alternatives. These values will not sum to one but can be normalized by dividing each value by the sum of the column.
2.3. Group Aggregation Technique
When several experts participate in the decision-making problem, the question arises of how to include in the process the preferences of each expert based on their relevance within the group. The calculation of expert voting power adopted in this study is based on the recent paper by Sodenkamp et al. [
33], which proposes determining each expert’s relevance based on the neutrosophic triad (truth, indeterminacy, and falsity). A simplified version of the fuzzy function [
34] is employed here. Two parameters are set to determine the voting power of each expert, namely, their competence through self-assessment and their consistency in completing the evaluation matrix. Thus, voting power (
Φi) of expert
i is calculated as the Euclidean distance from each point to the ideal point of maximum credibility 〈1, 0〉, formulated as:
The Delphi method evaluation technique is followed to characterize the expert panelists, which usually considers aspects such as years of professional experience, presentations at conferences, authorship of articles in peer-reviewed journals, qualifications, committee membership, etc. [
35]. In this case, the degree of knowledge in specific evaluation fields is incorporated to calculate the coefficient of the voting power of each expert, following the methodology applied by Sierra et al. [
36].
This approach is inspired by the technique used by the Russian State Committee for Science and Technology [
37], which considers two types of parameters to determine the expert profile of the panelists, namely knowledge-oriented parameters and argumentation-oriented parameters. The knowledge-oriented parameters are based on the general knowledge of the expert, namely, years of professional experience, authorship of JCR articles, and papers presented at conferences. The higher the score on these parameters, the more critical thinking ability is revealed. The other set of parameters, those oriented to argumentation, is related to expertise in the specific fields to be evaluated, in our case, sustainability and its dimensions, construction, and multi-criteria analysis. The resulting indicator reflecting the voting power of each expert is then obtained as the average of each parameter. Based on the above, the credibility of each expert is determined as follows:
where
PAi indicates the years as an active professional of
i-th expert;
SEi counts the number of years of experience in sustainable issues;
RJi and
RPi quantify the scientific production as primary author in articles for journals with JCR impact factor and papers in international congresses, respectively;
max{
PAk},
max{
SEk},
max{
RJk} and
max{
RPk} are the maximum of these attributes among the
k-experts. The
KFm,i parameters integrate the expert’s knowledge in several disciplines associated with the decision-making problem. In this case,
n = 5 fields have been chosen, representing the level of competence in construction and civil engineering, economic appraisals, environmental assessment, social analysis, and MCDM methods.
At last, the inconsistency (
εi) is evaluated from the inconsistencies derived from each of their pairwise comparisons performed by each expert in the AHP group and is calculated on a single matrix as:
In addition, for each expert of the ANP group, it is calculated on the total number of matrices to be completed as follows:
where
CRi is the consistency ratio of the
i-th expert on the comparison matrix filled in a classical AHP decision process for the set of criteria;
CRij is the consistency ratio of the i-th expert regarding the j-th comparison matrix filled along the ANP decision process;
CRlim and
CRlim,j are the limiting consistency ratios in the AHP and ANP matrices, respectively, depending on the number of elements to compare according to
Table 2;
Mi represents the total number of matrices filled in by expert
i belonging to the ANP group.
Once the weights (
wij) for each criterion
j have been determined for each
i-th expert along with their voting power (
Φi), the final AHP/ANP group weights of the
k-experts are obtained for each criterion as follows:
2.4. Outranking Methods
The last part of the methodology consists of aggregating the one-dimensional life cycle performance results to evaluate each alternative from a three-dimensional approach to sustainability that allows ranking them in order of preference. In addition to the ranking obtained with the pairwise comparison MCDM methods, the criteria weights obtained by AHP and ANP are combined with an outranking MCDM method to compare the results.
These methods establish a ranking order among a discrete set of alternatives in which each solution shows a degree of dominance over the others about a criterion. Any preference structure can be defined using an outranking relation, establishing the conditions for alternative
A to overcome alternative
B. Thus, alternative
A surpasses (
S) alternative
B if the DM prefers it to
B or shows indifference (
I) between the two. Among the methods that strictly apply this definition of outranking relation, those of the ELECTRE family stand out, considering ELECTRE I was historically the first outranking method [
30]. They can deal with incomplete and fuzzy information and allow alternatives to be ranked according to the preference relation between them.
ELECTRE IS
The modeling of the decision maker’s preferences in the ELECTRE IS version [
31] is less rigid than in ELECTRE I since the following argument is supported: if the difference between the valuations of the alternatives
A and
B is minimal, will the DM continue to prefer one of them?
Therefore, this method improves the previous version by incorporating fuzzy overclassification logic through pseudo-criteria, allowing the DM to choose decision parameters as intervals instead of fixed (true) values. The steps of ELECTRE IS and calculations are presented below.
The first step is to obtain the optimum value for each criterion among all the alternatives to be evaluated. It will correspond to the highest or lowest score depending on whether the variable is to be maximized or minimized, respectively.
The pseudo-criterion is a function in which the preference between two alternatives is characterized by two non-zero thresholds: one of indifference qj and one of strict preference pj (pj ≥ qj) for a specific criterion j.
The indifference threshold may reflect the minimum uncertainty limit in the data, while the preference threshold may report the maximum uncertainty limit. Note that when
pj =
qj, a pseudo-criterion becomes a true criterion. The concordance index
Cj (
A,
B), which states to what extent alternative
A is at least as good as alternative
B for criterion
Zj, will be a value between 0 and 1 which is defined as:
The concordance index values are incorporated into the concordance matrix by aggregating the DM weights as follows:
Correlatively to the preference and indifference thresholds, it also introduces an additional threshold, called the veto threshold. It reinforces support when it diminishes the importance of the coalition with which it agrees, making it possible to build authentic ex æquo classes (ties). The no veto condition can be formulated as:
where
vj is the veto threshold concerning the
j criterion (
vj ≥
pj ≥
qj);
Zj is the normalized score for each criterion as a function of each alternative;
ηj is the importance coefficient;
Cij corresponds to the concordance index of each pair of alternatives; and
c* is the concordance threshold defined as the next value greater than or equal to the average in the scores of the concordance matrix.
Finally, the Ai alternative outperforms Ak provided the following conditions are met:
Concordance criteria: Cj (A,B) ≥ c*;
Discrepancy criterion: There is no criterion j such that Zj (B) − Zj (A) > vj;
that is, Zj (A) − Zj (B) ≥ −vj for all the criteria.
5. Conclusions
Building a better world means aligning with the SDGs set for 2030. In this sense, the construction sector has a fundamental role to play, as it can be responsible for a large number of effects, both positive and negative, on the environment, the economy, and society. The sustainable design of buildings and urban districts has focused the efforts of a significant part of the scientific community. Mitigating their considerable negative impacts on the environment and boosting economic growth and social welfare are essential to achieving the sustainable future to which our society aspires. However, sustainability and construction management are complex issues involving multiple competing criteria. Moreover, the quantification of sustainability is difficult to objectify since it depends on the subjective perception of each DM and the relevance assigned to each criterion. These issues require techniques that make it possible to model the decision-making problem as closely as possible to reality, considering the interdependence and feedback relationships between the different criteria that are limited by methods such as AHP. With the ANP method, more reliable results can be obtained, but at the cost of many more questions and comparisons that further complicate the calculations by increasing the number of relationships and criteria. This means that the experts have to intervene much longer, diluting the concentration and introducing more significant uncertainties in their judgments. An ANP model adapted to quantitative units, specific to the criteria for sustainability assessments in building structures, has been calibrated to simplify this process.
This paper evaluates sustainability performance among four different structural design alternatives for a single-family dwelling over its life cycle based on various combinations between other MCDM methods of pairwise comparison and outperformance. In most cases, the preferred design option for sustainability performance is based on lightweight slabs with recycled plastic hollow corps. The results show the advantages of using ANP when the problem, such as the one at hand, can be formulated from a quantitative definition of the criteria used in the decision-making process. The model has significant advantages since it continues to identify the feedback and interdependencies in the network but greatly simplifies the computation of the network by reducing the number of questions to be answered by the experts involved in the decision process. For these cases, the ANP methodology makes it possible to automatically complete part of the clusters from the data input, which reduces the experts’ judgments and thus increases their consistency.
Finally, based on the weights obtained by the AHP and ANP groups, the MCDM method of outranking ELECTRE IS was applied to aggregate the seven impact categories to compare each design option through a sustainability ranking. The ELECTRE technique, specifically the IS variant, simultaneously considers the heterogeneity of the criteria ranges and the imperfect knowledge inherent in decision-making situations more similar to those of the real world. Introducing pseudo-criteria through imprecision thresholds in the data allows taking advantage of the fuzzy type overcoming relationship, which implies tuning less rigid modeling in terms of the decision maker’s preferences to obtain the best compromise solution. A sensitivity study has also been included with other techniques for obtaining subjective weights (BWM and FUCOM) and other MCDM methods (SAW, COPRAS, TOPSIS, and VIKOR).
Future lines of research will focus on two objectives. Firstly, to make the methodology as easy as possible for the DM and to further reduce the complexity of the intervention of each expert, an attempt will be made not only to simplify the process but also to avoid the size of the pairwise comparison matrices being a constraint by considering all the model elements as part of a single cluster. Secondly, minimization through neutrosophic logic of the effect of non-probabilistic uncertainties is associated with the decision maker’s ability to consistently reflect his view of the problem when making a judgment.