This section presents the theoretical referential of reinforced concrete slabs and the theoretical referential of the AHP method, the Delphi research method, the application of machine learning, and Principal Component Analysis (PCA) in decision problems. The method and procedures used in this work to obtain the proposed objective are also presented.
2.1. Slab Selection Is Based on Their Type and Functionality
Slabs are linear plane elements responsible for transmitting loads from external actions to the beams or the columns. These elements are mostly subjected to bending stress and must be designed, detailed, and built to attend to this internal stress.
Although several types of slab already exist, these elements undergo constant changes. Their construction process frequently evolves to generate the most outstanding possible economy of materials and archive the architectural demand. Thus, among the various possibilities, we can highlight the most common ones: the slab on beams, the waffle slab, the flat slab, and flat slabs with drop panels, as shown in
Figure 2. Slabs on beams are concrete blocks supported on the beams and are usually executed in one single step [
13].
The waffle slabs are composed of the cover and the ribbed section, formed by fillers that can be made of bricks, styrofoam blocks, or any form that allows for the execution of these elements with voids. This slab can be divided into a one-way or two-way slab, depending on the direction that it will be supported on the beams, i.e., if the load is being transmitted to the support in only one direction, it is classified as one-way; if it is sent in two directions, it is called two-way. Due to its voids, this type of slab enables large spans with less concrete when compared to the previous slab. However, its productivity can be affected by the assembly characteristics.
Flat slabs are elements directly supported on the columns, and may or may not undergo a widening in the slab–column transition. According to [
14], these slabs can be formed by the two previous types, i.e., solid or waffle, as shown in
Figure 3. Characteristics of this type of slab include its high productivity compared to the constructive processes. Regarding acoustics, this type of slab can be described as advantageous, as presented in the study [
15]. This manuscript investigates the relationship between the structural form of the slab and acoustics, and proves a good acoustic insulation of this type of structural solution.
The construction techniques of the mentioned slabs differ from each other, causing variations in the production rates of each one. Additional differences between the models, such as the quantity of the utilized material and whether a model has better stiffness or less deformation, may also be statistically quantified. There are also subjective differences, such as the aesthetic suitability of the slab; in other words, a visual difference between the slab system designed by the engineer and what the architect visualized as the final product. For instance, there could be a reduction in the ceiling height due to the thickness of a waffle slab or its aesthetic perception in the absence of a ceiling. These differences between the types impact which slab best suits a specific project.
Finally, there are flat slabs with drop panels and bend beams, which are types of slabs with insufficient thickness to resist the shear stress alone, thus requiring the inclusion of drop panels or bend beams to assist in resisting this effort by increasing the final thickness of the element. The difference between these two types is given by the region in which there will be an increase in thickness, with the first occurring only in the region of the pillar, and the second along the longer span.
2.2. Analytic Hierarchy Process (AHP)
Decision-making problems in which a single alternative must be chosen from a finite number of possibilities are part of every professional’s daily routine. The final choice is usually based on previous knowledge or the experience of each manager or designer working with the state-of-the-art. The multicriteria methods include the Analytic Hierarchy Process (AHP), in which a systematic approach is developed to help the decision-maker choose, based on several criteria, the best alternative to the presented problem. As stated by [
16], the AHP method can be divided into three stages, with the first being the structuring of the problem to be solved, the second the evaluation of criteria and decision alternatives, comprising a hierarchical structure or decision tree, and the third stage the categorization, ranking, ordering or prioritization of other options.
For the initial stages, it is possible to adopt the Delphi method, which “can be characterized as a method for structuring a group communication process so that the process is effective in allowing a group of individuals as a whole to deal with complex problems” [
17] (p. 701). This method can be applied in several areas, such as planning and design, cost and schedule, and construction methods and materials [
18].
To apply the Delphi method, it is important to follow a metric for the research, thus avoiding problems in collecting data in interviews. For this purpose, we can cite the metric developed by Saaty and presented in
Table 1. The scale ranges from 0 to 9 and is intended to indicate the degree of importance that one element has compared to another [
19].
With a large volume of data, machine learning can be used to determine weights based on historical values to reduce the subjectivity of the choice of criteria, whether using the Delphi method or any other similar methods. This area, called Large-Scale Decision Making (LSDM), has become quite popular in decision problems, and could follow the taxonomy stated by [
20] and explain the main ideas in
Figure 4.
For the problem discussed in this article, the Delphi method is necessary to start assembling the matrix values used in the AHP method. With these numbers, the criteria comparison matrix (
CCM) must be developed, as presented in Equation (1), where
is the value for each criterion obtained by the Delphi and PCA methods.
To verify whether the judgments obtained and used in the comparison matrix and its normalized form are valid, consistency validation can be performed from what was exposed by [
21]. This methodology considers the division of the consistency index (Equation (2)) and the random index calculated for square matrices of order
n.
where
is the maximum eigenvalue and
is the dimension of the studied vector. The random index values can be obtained by Oak Ridge National Laboratory publication.
Once the priority vector of the criteria comparison matrix has been calculated, and its consistency validated, the criteria stage is concluded and the alternatives stage starts. This phase follows the same process, i.e., the formation of the comparison matrices and their normalized form and the calculation of their respective priority vectors.
In the third and final stage, local priorities are grouped, and the alternatives are classified.
2.3. Principal Component Analysis
Datasets sometimes have many features that make it difficult to analyze and obtain information contributing to decision-making processes. One way to minimize this is using dimensional reduction techniques such as Principal Component Analysis (PCA). The purpose of PCA is to represent the multidimensional dataset in a set of coordinates in a new dimensional space.
The central idea is to adapt the coordinate systems into sets that are easy to analyze and discard unnecessary ones (those that do not present an advantage for data analysis). One of the main goals of PCA is dimensionality reduction using the first principal components that reflect the structure of the data [
22].
The decomposition of the original data into subspaces (master and residual) can be performed according to Equation (3).
where
and
are the
ith principal component and the corresponding load vector, respectively;
P and
T are the load matrix and score matrix of the master subspace, respectively;
represents the residual matrix; K represents the number of retained principal components [
23] (p. 4). The statistics used to monitor the two subspaces can be calculated according to Equation (4).
where
is the covariance matrix of
T.
The control limits (Equation (5)) are utilized to assess if the observed process operates within the expected parameters, thus indicating that the process is functioning normally.
“Among them, is the F distribution of degrees of freedom and with the confidence . and are the mean and variance estimated from the Q statistic.”
The use of PCA in the context of the Architecture, Engineering, and Construction (AEC) industry can be found in the most diverse areas, from application in demolition waste generation to material characteristics, as presented by [
24], who used this method to design modern concrete mixes. This work was based on a sample of 550 types of mixture, which were later categorized and divided according to the compressive strength of the concrete. The main objective is to verify the possibility of using PCA to create the amount of each ingredient needed to reach a determined compressive strength. The author concluded that its use for this purpose was unsatisfactory. Reference [
25] studied the application of PCA for the concrete mix design process, using 38 different recipes to achieve a refined one.
For [
26], the PCA was used to transform the correlated properties of the concrete to obtain combinations of parameters for multiple performance characteristic indexes and consequently optimize the properties of the fiber-reinforced geopolymer composites. In this paper, PCA was combined with the Taguchi Method.
In addition to the use of the method in the material subject, works such as [
27] present the possibility of using PCA for purposes of material strength. In the case of [
27], this method was used to reduce the features of the original dataset and, from this reduction, apply them to Artificial Neutral Network (ANN) to predict the shear strength of reinforced concrete beams.
The combination of ANN with PCA can also be seen in the work of [
28], who used the dimensionality reduction of the data for application in the machine learning algorithm to obtain a predictive model of the slump and compressive strength of concrete with mineral additives.
Other applications can be found in [
29], which develops a prediction model based on machine learning algorithms combined with PCA for demolition waste; of the algorithms combined with PCA, the k-nearest neighbors proved to be the ones with the best results. The authors of [
30] studied the assessment of the fire-induced concrete spalling of columns using the k-nearest neighbors, but PCA was only used to compress the number of features.
2.4. Methodology
The development of this research began with a literature review aiming to analyze and define the criteria used in the decision-making process of the studied problem. This survey was carried out using the following databases: Compendex, ScienceDirect, Google Scholar, Wiley, and SpringerLink, searched in the English and Portuguese languages, with the emphasis on scientific articles published between the years 2001 and 2022. The following research questions can be asked at this stage: Which criteria do professionals in theArchitecture, Engineering, and Construction (AEC) industry believe impact the decision of which slab to adopt in each project? This survey highlighted the following criteria:
Structural performance: Capacity of slabs to support the applied loads, considering the displacement of the slab as a function of the span (serviceability), the internal forces as bending moment (resistant capacity), and the durability of the element;
Productivity/constructability: Related to the simplicity of the construction process, directly proportional to the execution time and the allocated labor resources;
Cost: Monetary value associated with the cost of the materials used in the slab’s construction (concrete, steel, and formwork), as well as the value of the execution;
Appearance: Related to the aesthetic when integrating the structural element and the architectural concept;
Waste generation: Environmental impact generated by construction waste;
Thermal/acoustic comfort: Related to the performance of the slab concerning the thermal comfort provided by the heat transmission of the slab and the acoustic comfort offered by the insulation of noise between floors.
Other criteria, such as the type of structural system, can be applied in research that covers more types of construction, such as warehouses and large buildings, among others. However, because this research is limited to residential and commercial buildings, these factors do not have as significant an impact as the others that are listed. This choice is made as this type of building is more used and common in several cultures [
31].
With the selected criteria, and based on the five slab alternatives (slab on beams, one-way waffle slab, two-way waffle slab, solid flat, and waffle flat), it is possible to develop a criteria tree for the presented problem, as shown in
Figure 5.
For the construction of judgment matrices, because the criteria are divided into two categories, quantitative (structural performance, productivity and cost, and waste generation) and qualitative (appearance and thermal/acoustic comfort), the assembly of the matrices was divided between the machine learning analysis and the Delphi methodology, respectively. For the subjective criteria, to apply the Delphi research methodology, it is necessary to follow a sequence of scientifically validated steps that include an elaboration of the system to be adopted and the flow that will be used, as presented in
Figure 6.
Once the method was chosen, 12 specialists were selected and divided as follows: four builders, four structural engineers, and four architects. The decision was made on the premise that each professional has unique perspectives and priorities that influence the decision-making process. For example, the engineer will prioritize the structural performance, the builders will prioritize the minimum cost, and the architects will prioritize the structure’s appearance in the edification. The heterogeneity of this sample aims to provide a more holistic view of the process.
The experts were selected based on their academic backgrounds and extensive market experience. The study included four architects, all of whom are university professors. Three of them have over 15 years of experience in project development, while the fourth has more than 15 years of experience in project management in renowned construction companies. The study also included four structural engineers, three of whom have more than 10 years of experience in reinforced concrete projects, including one with over 35 years of experience. All the structural engineers are university professors. Two of the four builders are university professors, each with more than 30 years of experience. The other two have over 10 years of experience, with one specializing in project management and the other in the research and development of new products for a large company in the prestressed concrete industry.
After selecting the specialists, the first round of interviews consists of open questions aiming at the exactification of criteria, which are later compared with those raised by a previously executed bibliographical survey. With these data, the second-round initiates a quiz assembled through Google Forms on a scale comparison from 0 to 9, as shown in
Table 1. It should be noted that this phase focuses on subjective criteria.
For the objective criteria, two different procedures were developed due to the different characteristics of the data. The first analysis procedure was carried out for the cost and structural performance and is based on the mean of every value regarding each type of slab. The cost’s unit of measurement is the monetary value per m² and its composition is based on the sum of the values of the materials used for the construction of each, being a function of the area of formwork, volume of concrete, and weight of conventional or prestressed steel. The hierarchy of this item is presented in
Table 2 and is developed by averaging the construction value of each of the slab types.
The second procedure was used to calculate the productivity and the waste generation criteria, and is based on a machine learning algorithm. This ML algorithm learned from 2147 examples of executed slabs throughout the Brazilian territory. The database presents the five types of slabs that were studied, with more examples of two-way waffle slabs (47.1%) due to the regional construction characteriztics, followed by the one-way waffle slab (34.5%), flat waffle slab (8.0%), flat slab (5.6%) and slab on beams (4.8%), respectively.
Figure 7 shows the frequency at which each type of slab appears.
These data come from a Brazilian company that rents and makes formwork for reinforced concrete slabs, thus storing the data from the constructions with which it performs this service. The performance of this company covers the entire national territory, which reduces the impact of regional construction culture; however, the main structural systems adopted in the country and their respective construction processes prevail. The base is composed of slabs with spans ranging from 4 to 22 m, with 8 being the value of the average, mode, and median. The concrete compressive strength range of these elements is from 25 to 50 MPa, with a mode of 35 MPa.
The dataset used for the productivity study has the following features: the proportion between the area of each type of slab and the daily labor performed by each construction worker category; for example the construction manager uses a given number of days to build a given slab area. For expository purposes,
Table 3 presents the average of each ratio calculated for each type of slab,
Table 4 represents the standard deviation, and
Table 5 represents the median.
The dataset used for waste generation consists of the following features: form area and the rates per square meter of concrete, steel, and prestressed steel.
Table 6 displays the average ratio of the materials, calculated for each type of slab for explanatory purposes,
Table 7 represents the standard deviation and
Table 8 represents the median.
With all the data collected in the previous steps, the hierarchy calculation was developed using the AHP method. Finally, the alternatives to the slabs were developed and hierarchized based on the obtained results.
From the authors’ point of view, the choice of a multi-criteria method among those available, when applied to a given context, should be adapted to the characteristics of the problem in question. Important points will include the evaluation of the problem, the decision objects, and the available information. According to [
32], the choice of method should be the result of an evaluation of the chosen parameters, the type and precision of the data, the decision-maker’s way of thinking, and his knowledge of the problem. It is also noteworthy that a direct consequence of the possibility of choosing between different methods is that the results can be discordant and even contradictory. Also, in agreement with [
32], one should not complicate the evaluation, since the observed differences are more related to the diversity of results than to contradictions, and some criteria allow for a validation of the chosen method.
In the problematic “Multicriteria Evaluation of Reinforced Concrete Slabs using Analytical Hierarchy Process”, the application of the AHP method came from the decision-maker’s acceptance of the method, which meant that the questions being presented to the decision-maker made sense to him, and he was confident in answering them. In addition to this point, the need to assess the acceptance of the data, the properties used by the method, and whether the result supported the decision process was highlighted. Secondary issues, such as the existence of tools such as expert choice, were also observed, as they allowed for greater integration with the problem being addressed. Still, these authors agree with [
33], who stressed that the multi-criteria decision support methodology has several methods that can be applied to the most diverse problems. Therefore, the choice of a multi-criteria decision support method is already a multi-criteria problem.