Class Thresholds Pre-Definition by Clustering Techniques for Applications of ELECTRE TRI Method

Abstract

The sorting problem in the Multi-criteria Decision Analysis (MCDA) has been used to address issues whose solutions involve the allocation of alternatives in classes. Traditional multi-criteria methods are commonly used for this task, such as ELECTRE TRI, AHP-Sort, UTADIS, PROMETHEE, GAYA, etc. While using these approaches to perform the sorting procedure, the decision-makers define profiles (thresholds) for classes to compare the alternatives within these profiles. However, most such applications are based on subjective tasks, i.e., decision-makers’ expertise, which sometimes might be imprecise. To fill that gap, in this paper, a comparative analysis using the multi-criteria method ELECTRE TRI and clustering algorithms is performed to obtain an auxiliary procedure to define initial thresholds for the ELECTRE TRI method. In this proposed methodology, K-Means, K-Medoids, Fuzzy C-Means algorithms, and Bio-Inspired metaheuristics such as PSO, Differential Evolution, and Genetic algorithm for clustering are tested considering a dataset from a fundamental problem of sorting in Water Distribution Networks. The computational performances indicate that Fuzzy C-Means was more suitable for achieving the desired response. The practical contributions show a relevant procedure to provide an initial view of boundaries in multi-criteria sorting methods based on the datasets from specific applications. Theoretically, it is a new development to pre-define the initial limits of classes for the sorting problem in multi-criteria approach.

1. Introduction

The motivation of this work is based on the fact that in multi-criteria approaches, decision-makers are often faced with subjective analysis. Several methods have been developed to support these subjective tasks on Multi-criteria Decision Analysis (MCDA) problems. These problems present alternatives that can be evaluated under a multiple-criteria view, and the selection of experts for the definition of limits for preferences is crucial in each application [1,2]. In addition, each context demands assertiveness in decision-making, which depends on input information quality, decision-maker experience, and the efficiency of the applied methods. However, input information quality and decision-makers’ know-how are not easy to measure numerically.

The MCDM (Multi-criteria Decision Making) methods have been widely applied to real problems and can guide decision-making to achieve the intended objectives, considering preferences. Nevertheless, sometimes the decision-maker must provide inputs to the method, like weights for criteria and bounds for classes, which depend on the user’s knowledge and assertiveness [3].

Unfortunately, in some cases, the lack of knowledge or experience may harm the efficiency of the method and the entire process. So, the selection of methods sometimes needs to be correctly conducted. A potential alternative to overcome this problem is using methods that do not require a priori subjective information, like clustering approaches. Thus, a quantitative dataset might be an alternative to guarantee the quality of the inputs and after the adjusting of the personal information to attend to the preferences [4,5]. The advance of technology leads to an increase in the information available. Some examples can explain this statement, such as the development of social networks and search websites, the creation of the internet of things, and Industry 4.0. In these scenarios, using Data Mining techniques, it is necessary to create mechanisms to group these datasets [6,7].

In this sense, clustering algorithms are the core of Data Mining. They can divide some databases into groups according to their similarities, usually through an unsupervised approach [8]. This task can be modeled as an NP-hard problem and solved using optimization tools [9].

The clustering methods can be classified into four groups: 1. partitioned, 2. overlapping; 3. hierarchical; and 4. graph-based [10,11]. The first approach divides the data into groups according to their similarities using the Euclidean Distance in most cases. The overlapping methods consider that some samples may belong to different groups according to a membership grid. The hierarchical approaches create a dendrogram (a structured tree representing the clustering levels). The last one, graph-based, creates a weighted graph where the connections are the edges [7,12]. This paper addresses the partitional and overlapping approaches integrated into a comparative analysis with a multi-criteria outranking method to allocate alternatives in classes.

Undoubtedly, the most known partitional clustering algorithm is the K-Means, proposed by [13]. Its fame comes from its simplicity and computational efficiency [6]. The application of the K-Means involves the creation of centroids, artificial samples representing each center of the group. However, this presents drawbacks, such as convergence to a local minimum. once it strongly depends on the initialization of the centroids. In this direction, the K-Medoids were created to overcome the initialization dependency. The main difference between them is using samples’ positions as initial centers in the last case [7].

Many authors proposed the use of bio-inspired meta-heuristics for partitional clustering [14]. In these cases, clusters are again determined by the position of the centroids, as well as modeled as an optimization task, which must be minimized. In this paper, we apply Particle Swarm Optimization (PSO) for clustering [7,15], Genetic Algorithm [16,17], and Differential Evolution [12]. These methods are well known, can be easily implemented, and can escape local optima [18].

The last method investigated is the Fuzzy C-Means, which is an overlapping procedure. It can overcome the problems with the K-Means since the complexity of the relations considering the samples can be challenging to identify. Therefore, this method does not define a sharp partition but a membership matrix once a selection may belong to different groups [4,6,19].

In this context, the objective of this work is to present a methodology to use clustering approaches to support a pre-definition of boundaries for classes in sorting problems for multi-criteria methods and to solve real problems related to this definition.

We compared the clustering results with the ELECTRE TRI outranking method with a dataset extracted from a real application performed in the work published by [20].

This paper is organized as follows: Section 2 presents a Multi-Criteria problem; Section 3 shows the Theoretical basis of the ELECTRE-TRI; Section 4 discusses the methodology used and highlights the application of clustering approaches; Section 5 brings the empirical results and develops a critical analysis. The conclusions and future directions are in Section 6.

2. Problem Statement

Several methodologies have been developed in multi-criteria decision analysis to support decision-makers in expressing their preferences in ordinal, intervallic, or cardinal scales related to real problems. Despite the experience and acquired expertise, sometimes decision-makers feel uncomfortable defining the parameters in a subjective way [21].

The important multi-criteria question in this work is, “how does it assist decision-makers in classifying alternatives in classes?” [22]. In the multi-criteria sorting problem, they need to define weights for criteria, number of classes, boundaries for categories, and thresholds for indifference and preference, which might be, in some cases, a challenging task. For the criteria weight definitions, there are several appropriate methods in which the decision-maker is conducted to reflect on these parameters. Most of them present reliable results in their reports. Among such techniques, some might be quoted: Trade-off analysis [23]; the FITradeoff method [24]; the Swing weights [25]; the Macbeth method [26], and the procedure available in the AHP method [27] that also allows defining weights for criteria.

Regarding the definition of the boundaries, especially for the ELECTRE TRI method [28], it depends totally on the expertise of the decision-makers or specialists. In the AHPsort [29], another sorting procedure is adopted. The decision-maker is invited to compare alternatives in a pairwise combination in order to define the most appropriate class boundaries. It represents a significant effort related to understanding, application, and necessary time to do this; i.e., the decision-makers must be confident in order to define coherent profile limits and indifference and preference thresholds because the alternatives commonly present variations among classes when the multiple criteria are considered. Figure 1 illustrates how the ELECTRE TRI method compares the alternatives with profiles of classes and the indifference and preference thresholds.

The literature researched did not show this problem of class limits exactly like a gap. This gap is commonly ignored in studies that involve limit definitions because it is just performed based on the experts’ experience. Thus, the problems are just perceived in real cases, in which the procedures must be re-performed until more coherent results are found, causing delays and occasionally unreliable results.

Additionally, the literature presents several methods, such as clustering techniques and neural networks, applied to drive the same problems using mathematical modeling [10]. On the other side, only the numerical solutions cannot cover the multi-criteria aspect related to decision-makers’ preferences. Consequently, it does not present a solution compromised with the choices and objectives. To fill this gap, it is necessary to promote the integration of these aspects since the numeric methods can aid this procedure with an initial pre-definition of boundaries, and the fine adjustments could be made in the sequence by the traditional subjective ways.

3. Theoretical Foundations and Related Works

The ELECTRE TRI is a multi-criteria classification method that places the existing alternatives into predefined categories (classes) according to their upper and lower limits [30]. The technique presents several uses for decision support in real problems, such as the selection of suppliers explored in the works of [31,32]. Additionally, there are new energy sources and criteria to improve its use in [33] and the research of [34], with applications for assessing the energy performance of school buildings.

In maintenance management, Certa et al. [35] proposed a classification of equipment failure models and [36] developed a procedure to prioritize alternatives for maintenance on water distribution networks. In health, decisions with multiple criteria approaches are assisted by triage models applied to assisted reproduction. In the financial area, the decision-making is helped to select clients who will receive a joint research loan, described in the work of [37]. In addition, using the ELECTRE TRI method in real problems is related to solving issues jointly with other ways to generate more precise decisions.

The fuzzy methodology is used in conjunction with ELECTRE TRI in the work of [32]. However, other vocations also join the fuzzy logic and DEA (data envelopment analysis) with the ELECTRE III method in a ranking problem in [38,39,40,41]. Another method in conjunction with ELECTRE TRI is the Delphi method in [10] to classify intelligent grid policies.

The AHP was another method found to collaborate with ELECTRE III/IV methods in the work of [42,43], with the objective being to rank urban transport projects. The study in [44] proposed a new decision support system for product classification problems that integrate multi-criteria decision-making and feeling analysis to classify products.

The research group from Rivero Gutiérrez, De Vicente Oliva, and Romero-Ania developed studies on Urban Public Transport Systems, with the necessity of a limits definition for classes and weights. Some pieces of the investigation have been developed considering multi-criteria methods, such as AHP, Delphi, ELECTRE TRI, and ELECTRE III, to manage sustainable decisions and the economic efficiency of the operation [2,5,40,41]. Similarly, the VIKOR method was applied to evaluate the degree of satisfaction in Turkish cities [3]. Pala [3] developed a new hybrid decision-making model combining different MCDM methods, considering a mixed-integer linear programming model to prioritize them. It is perceived that all of these studies used some correlated subjective procedure to define limits, weights, or thresholds for preference or indifference.

Thus, in order to solve quantitative aspects in this context, clustering research is geared toward numerous applications. Among these, we highlight classic problems such as maintenance costs in [45]. As presented in the paper from [46], research uses clustering with a genetic algorithm to perform the sequencing of an aircraft manufacturing industry with a flow shop environment with multiple operations. Other applications that use the clustering techniques are water and energy distributions according to consumer demand. Some works like [47] present this subject through the K-Means method, and [48] uses Fuzzy C-means clustering.

The study of [49] proposed a decision support system using the t-SNE algorithm and K-Means clustering to improve security using multiple matching analyses. Applications in health are also presented, as the research of [50] used clustering and the genetic algorithm to group characteristics of individuals with cancer. As noted, clustering can solve several problems to find common elements and form an auxiliary grouping in decision-making.

3.1. Clustering Approach on MCDA Methods

Over many years, the literature has published several clustering approaches to different MCDA methods attempting to improve results. Zhou et al. [43] and Berbel et al. [51] developed a technique based on clustering and goal programming to analyze the decision-making process in irrigated farms.

Goodwin et al. [52] combined Clustering analysis and Multi-Attribute Utility Theory (MAUT) to verify the impact of water pricing on farms. In the study from Azadnia et al. [53], the Fuzzy C-means and ELECTRE II were applied to supplier selection problems in the automotive industry. The K-Means were combined with MULTIMOORA to improve the MCDA analyses [54,55].

3.2. MCDA Methods and Subjectivity

Most MCDA problems rely on decision-maker expertise to identify the best choice. Unfortunately, defining weights and parameters for a multi-criteria problem is difficult, especially in complex situations. The decision-makers’ judgments may directly affect the common choice [20,56]. Thus, it is essential to have a more systematic way to decide, avoiding the negative impacts and personal influences.

As the assignment of criteria consequences is critical to MCDA methods, several ways to define weights were proposed, from subjective to completely objective, besides combining both [57]. Many works show this concern. Ma et al. [58] proposed a two-objective programming model to consider both decision-makers’ emotional and analytical objective weights [46,59].

Entropy Modified Digital Logic (MDL), Criteria Importance through Inter-criteria Correlation (CRITIC), and two new methods were used to define the TOPSIS input weights. A hybrid fuzzy goal programming and Monte Carlo simulation were used to find Pareto-optimal solutions without depending on decision-maker subjective weights [60]. The simulation allows the fuzzy goal programming to find different solutions. Multi-attribute group decision-making that integrates objective and subjective weighting employs statistical variance, TOPSIS, Simple Additive Weighting (SAW), and Delphi-AHP [23,43,61]. The method incorporated weights of the attributes and decision-makers aiming for more accurate results.

Table 1. Summarized theoretical foundations.

Reference	Year	Approach
Multi-criteria concepts and methods
[35]	2017	Classification of equipment failure models
[36]	2012	Procedure to prioritize alternatives for maintenance on water distribution networks
[37]	2015	Selection of clients who will receive a joint research loan
[32]	2018	Fuzzy methodology is used in conjunction with ELECTRE TRI
[38]	2016	Decisions based on interval-valued intuitionistic fuzzy information
[39]	2016	Power station site selection under intuitionistic fuzzy environment
[42]	2015	Evaluation of urban transportation projects.
[43]	2010	Evidential reasoning-based nonlinear programming model for MCDA
[44]	2019	A new decision support system for product classification problems that integrate multi-criteria decision-making
[56]	2020	Supply chain management with AHP, DEMATEL, and TOPSIS
[57]	2016	Subjective, objective, and combinative weighting in multiple criteria decision making
[58]	1999	Determination of attribute weights
[59]	2009	Fuzzy TOPSIS based on subjective and objective weights
[60]	2015	Supplier selection and order allocation in reverse logistics systems
[61]	2014	TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets
[2]	2021	Managing Sustainable Urban Public Transport Systems: an AHP Multicriteria Decision Model
[5]	2021	Multiple Criteria Decision Analysis of Sustainable Urban Public Transport Systems
[40]	2022	Economic, Ecological, and Social Analysis Based on DEA and MCDA for the Management of the Madrid Urban Public Transportation System
[41]	2022	Economic Evaluation of the Urban Road Public Transport System Efficiency Based on Data Envelopment Analysis
[3]	2022	A mixed-integer linear programming model for aggregating multi–criteria decision-making methods
[62]	2021	Evaluating the satisfaction level of citizens in municipality services by using picture fuzzy VIKOR method: 2014 2019 period analysis
Clustering
[10]	2019	Swarm intelligence for clustering
[45]	2018	An investigation that used genetic algorithms and Fuzzy C-means for evaluations of maintenance costs
[46]	2014	Clustering with a genetic algorithm to perform the sequencing of an aircraft manufacturing industry with a flow shop environment with multiple operations
[47]	2018	K-Means method for energy recovery in water distribution networks
[48]	2017	Fuzzy C-Means clustering for evaluation of household monthly electricity consumption
[49]	2014	Decision support system using the t-SNE algorithm and K-Means clustering to improve security using multiple matching analyses
[50]	2019	Clustering and genetic algorithm to group characteristics of individuals with cancer
Clustering and MCDA combined
[51]	1998	Developed a technique based on clustering and goal programming to analyze the decision-making process in irrigated farms
[52]	2004	Combined Clustering analysis and Multi-Attribute Utility Theory (MAUT) to verify the impact of water pricing on farms
[53]	2011	Fuzzy C-Means and ELECTRE II were combined to supplier selection problems in the automotive industry
[54,55]	2019	K-Means was combined with MULTIMOORA to improve the MCDA analyses

summarizes the main theoretical foundations related to this work. These works were selected by their relevance to this theme, the number of citations, and their connection to the developed approach.

However, we did not find efforts using clustering approaches in conjunction with ELECTRE TRI to define class boundaries. Therefore, in this paper, we propose using the clustering methods to help the ELECTRE TRI boundaries definition procedure support the decision process and reduce the doubtful subjectivity inherent to the decision-making process.

4. Methodology

This section describes the methods applied in this work to compare clustering techniques and the ELECTRE TRI method. Firstly, the results of an application involving water distribution networks are evaluated. A sorting experiment is conducted, prioritizing areas in categorized classes. The numerical application is detailed in Section 5. The primary information about the utilized methods is presented in Section 4.1 and Section 4.2. Figure 2 illustrates the methodological context giving an overview of the involved parameters for the sorting approach.

4.1. Multi-Criteria Outranking Sorting Method ELECTRE TRI

The ELECTRE TRI method is a multi-criteria outranking technique developed that allocates alternatives in predefined categories, called the sorting procedure. The allocation occurs when an option “x” is compared with defined profiles of the limits from the categories or boundaries b_h. In this work, the multi-criteria outranking method ELECTRE TRI is applied as a base of comparison among clustering algorithms to search for an initial definition of boundaries for category profiles [28,63]. It uses the concordance and discordance indexes to evaluate the statement xSb_h (“x outranks b_h”). The partial concordance c_j (x, b), concordance c (x, b), and partial discordance d_j (x, b) are calculated by the expressions (1), (2), and (3) below:

$$c_j(x,b_h)=\{\begin{array}{}\\ 0 if& g_j(b_h)-g_j(x)\ge p_j(b_h)\\ 1 if& g_j(b_h)-g_j(x)\le q_j(b_h)\\ & \frac{p_j(b_h)+g_j(x)-g_j(b_h)}{p_j(b_h)-q_j(b_h)}\begin{array}{cc},& \mathrm{otherwise}\end{array}\end{array}$$

(1)

where

$$c(x,b)=\begin{array}{ccc}\frac{{\displaystyle \sum _{j\in F}{k}_j{c}_j(x,b_h)}}{{\displaystyle \sum _{j\in F}{k}_j}}& & \end{array}\begin{array}{ccc}& & \end{array}\begin{array}{ccc}& & \end{array}$$

(2)

where

$$d_j(x,b_h)=\{\begin{array}{c}0\\ 1\\ \in \end{array}\begin{array}{c}if\\ if\\ [0,1]\end{array}\begin{array}{c}{g}_j(x)\le g_j(b_h)+p_j(b_h)\\ g_j(x)>g_j(b_h)+v_j(b_h)\\ ,\mathrm{otherwise}\end{array}$$

(3)

Still, an index σ is calculated, being σ(x, b_h) ∈ [0, 1] σ(b_h, x), respectively, which represents a credibility degree of the assertion in which xSb_h, x ∈ A, h ∈ B, as shown in Equation (4).

$$\sigma (x,b_h)=c(x,b_h).{\displaystyle \prod _{j\in F}\frac{1-d_j(x,b_h)}{1-c(x,b_h)}}$$

(4)

where: $\stackrel{\_\_}{F}=\{j\in F:d_j(x,b_h)>c_j(x,b_h)\}$

Mousseau et al. [64] presented two assignment procedures: one pessimistic, which compares x with b_i, to i = p, p − 1, …, 0, b_h, starting with the first profile in which xSb_h is the category CL_h+1(x → CL_h+1); and the optimist one, which compares x with bi, to i = 1, 2, …, p, b_h, starting with the first profile, such that “b_h is preferable to x” states that CL_h for category (x → CL_h).

4.2. Clustering Techniques

Section 1 mentions that the literature divides the central clustering approaches into partitional, overlapping, hierarchical, and graph-based [7,9,10,11]. The partitional process is the most used, and it is also the focus of the investigation conducted in this paper, together with the Fuzzy C-Means, as an overlapping method [9,58].

The partitional approach creates groups according to their similarities. It performs the clustering by inserting artificial points named centroids or centers, presenting the exact dimensions of the samples (features). Each centroid is the representative point of a respective group so that the samples belonging to the same group are represented by the nearest centroid [13,15]. Then, the task to be solved by a partitional algorithm is determined by using an iterative process to find the best location of the centroids, as shown in Figure 3.

Mathematically, let x_i = [x_i,₁, x_i,₂, …, x_i,_D] the i-th object (samples), i = 1, …, N, each presenting D features. The clustering algorithms aim to allocate these N samples into K < N clusters, represented by the c_k = [c_k,1, c_k,2, …, c_k,D] centroids, k = 1, …, K. The most used metric to determine the distance (dist) of an object i to a centroid k is the Euclidean Distance, defined in Equation (5) [9]:

$$dist\left(x_i-c_k\right)={‖x_i-c_k‖}^2=\sqrt{\sum _{d=1}^D{\left(x_{i,d}-c_{k,d}\right)}^2}$$

(5)

The following subsections present in detail the algorithms addressed in this investigation.

4.3. K-Means and K-Medoids

The K-Means is the most usual method for partitional clustering due to its simplicity and fast convergence [6,11,13,15]. The samples are separated into a predefined number of groups according to the final positions of the centroids.

The initialization randomly generates the centroids. Then, distances between them and objects are calculated by Equation (5). Each point is allocated to the cluster with the smallest space. After, the positions of the centroids are recalculated using Equation (6) [6]:

$$c_k=\frac{1}{n_k}\sum _{i=1}^{n_k}{x}_i^k$$

(6)

where the number of patterns within a cluster k is n_k.

This process is repeated until the stop criterion is reached as the number of epochs or a slight change in centroid position. The K-Means minimizes the Sum of Squared Error (SSW) given by Equation (7) [65]:

$$SSW=\sum _{k=1}^K\sum _{i=1}^{n_k}{dist}^2\left(x_i-c_k\right)$$

(7)

Remarkably, the method is highly dependent on the initialization. As this process is random, it can lead to poor performance. Additionally, the K-Means need to work better with overlapping databases [9].

A way to minimize this disadvantage is to address the K-Medoids algorithm, similar to the K-Means. The difference between them is just the initialization: K-Means randomly chooses K patterns (samples) of the data set as the initial positions of the centers. It can contribute to a better initialization of the algorithm.

4.4. Bio-Inspired Metaheuristics

Bio-Inspired Metaheuristics have gained prominence in optimization in the last two decades due to their search capability [66]. As the name indicates, these methods were inspired by some aspects of the natural world, like Darwin’s Evolution Theory or the collective behavior of groups of animals [16,67].

Clustering optimization is the process in which the algorithms reduce the distance between the centroids and the data inside the cluster. These algorithms are populational since solutions are simultaneously maintained during the iterative search. In this case, each key is called an agent. Therefore, the approaches above are effective candidates to deal with this task [68,69].

In the most used codification scheme, an agent y_p is a centroids vector. Therefore, it is a vector with K × D elements, which concatenates the spatial positions of the K centroids, as seen in Expression (8) [10,14]:

$${\mathit{y}}_{\mathit{p}}={\mathit{C}}_{\mathit{p}}=\left({\mathit{c}}_{\mathrm{1,1}},{\mathit{c}}_{\mathrm{1,2}},{\dots ,\mathit{c}}_{1,\mathit{D}},\dots ,{\mathit{c}}_{\mathit{K},1},{\mathit{c}}_{\mathit{K},2},{\dots ,\mathit{c}}_{\mathit{K},\mathit{D}}\right)$$

(8)

This work addresses two leading evolutionary algorithms, Genetic Algorithm and Differential Evolution, and the essential swarm-based proposal, Particle Swarm Optimization (PSO).

4.5. PSO for Clustering

Kennedy and Eberhart developed Particle Swarm Optimization (PSO) in 1995 [70]. It is the first proposal and the most prominent swarm-inspired algorithm, with many applications in various fields, such as optimization and clustering [7,10].

The biological inspiration came from the social interactions of animals, like flocks of birds and schools of fish. Each candidate solution is named a particle, which moves in the search space according to its self-experience, pbest^t_p, and the collective experience gbest^t.

Indeed, the first is the best position achieved by particle p during the iterative process, and the second is the best position of the entire swarm until iteration t. The candidate solutions present a performance index named fitness. These collective interactions create a complex behavior that allows the algorithm to optimize a cost function [15].

Assume a swarm a population of p = 1, 2, …, P particles. The agents move in the D dimensional space (cost function) according to Equation (9):

$$y_p^{t+1}=x_p^t+v_p^{t+1}$$

(9)

in which t denotes the current iteration, y_p = [y_p,1, y_i,2, …, y_p,D] is the position of an agent in the search space and v_p = [v_p,1, v_p,2, …, v_p,D] is the velocity, updated using Equation (10) [65]:

$$v_p^{t+1}=\omega v_p^t+c_1{r}_1\left({\mathbf{p}\mathbf{b}\mathbf{e}\mathbf{s}\mathbf{t}}_{\mathit{p}}^{\mathit{t}}-x_p^t\right){+c}_2{r}_2\left({\mathbf{g}\mathbf{b}\mathbf{e}\mathbf{s}\mathbf{t}}^{\mathit{t}}-x_p^t\right)$$

(10)

being ω the inertia weight, r₁ and r₂ are random vector values generated according to a uniform probability in the interval [0,1], and c₁ and c₂ are the cognitive and social coefficients, respectively.

Initialization of the swarm is performed by spreading the agents over the space according to a uniform random generation. The velocity is initiated with zero. Note that ω is below 1, and the speed is limited in the interval [−Vmax;+Vmin]. Finally, in clustering tasks, the fitness assignment is given by Equation (6): the closer the data are to the respective centroid, the better the fitness, or the smaller the SSW.

4.6. Genetic Algorithm for Clustering

The Genetic Algorithm (GA) is the primary evolutionary algorithm for optimization tasks. Darwin’s Theory of Evolution by natural selection is the inspiration for the method, which influences the agents (individuals or chromosomes) and the entire population [16].

The GA is a probabilistic and populational algorithm like the PSO. The initial individuals, candidate solutions, are randomly generated subjects to the search space constraints. In this case, each agent is represented according to Equation (8), and the vector elements are called genes. The genes compose the genotype of the individual [9,67].

As usual, a fitness value is assigned to them. However, unlike the PSO, the search is performed by creating new generations, which substitute the old ones. In this sense, the agents with higher fitness tend to survive and belong to the next generation or maintain their genetic load in the population via the offspring. The next step is selecting the individuals to pass through the genetic operators. This paper uses the roulette wheel scheme [16,65,67], and better fitness tends to be chosen.

Once individuals are chosen, the final steps apply the genetic operators: crossover, which performs a local search (exploitation), and mutation for global tracking (exploration). In this work, we perform the classic one-point crossover [65]. The first two individuals randomly selected by the roulette wheel (parents) change their genes, creating two new ones according to a probability PC. These will be in the next generation. This process is repeated until the number of individuals in the next generation equals the original. Note that some agents may be selected more than once. Finally, the mutation is applied to Pm genes of the population. We add a small value drawn using a normal distribution to the selected genes.

Again, in clustering problems, the agents are represented according to Equation (8), and the fitness is calculated by Equation (7) [65].

4.7. Differential Evolution for Clustering

Differential Evolution (DE) is an optimization method that follows similar principles of the Genetic Algorithm, inspiration, and genetic operators. The agents proposed by Storn and Prince in 1995 have named vectors generated similarly to the last two presented approaches [12]. Again, it is used in a population method that maintains a set of possible solutions at each iteration. However, the crossover and mutation are different from the GA: here, the vector difference is used to perform the search [12,65].

At the initialization process, some population agent is randomly selected and named target vector v_i. Next, the first operation is the mutation generating a new vector y_i^new. This process follows Equation (11) [28]:

$$y_i^{new}=y_i^{r1}+F\left(y_i^{r2}-y_i^{r3}\right)$$

(11)

where r1, r2, r3 ∈ 1, 2, …, NP are the index of the other three agents randomly chosen, and F is a real number drawn in the interval [0, 2]. Note that Equation (11) performs a vector subtraction [65].

The next step is the crossover between the target vector and y_i^new. A new agent called trial vector u_i is created according to Equation (12) [8]:

$$u_{id}=\left\{\begin{array}{cc}{y}_{id}^{new}& if{r}_d\le CR\vee i=I_i\\ v_{id}& if{r}_d>CR\vee i\ne I_i\end{array}\right.$$

(12)

where d = 1, …, D is the current dimension (gene); r_d is sorted using a uniform distribution in the interval [0, 1]; CR is defined by the user; and I_i ∈ 1, …, D is a randomly chosen index, which guarantees that the new vector will receive at least one gene from v_id. Finally, the vector that remains in the population is selected by a greedy criterion (best fitness) between v_i and u_i.

4.8. Fuzzy C-Means

Fuzzy C-Means (FCM) is an overlapping clustering method, fast to converge and simple to implement [4,6,71]. The FCM is an excellent alternative to clustering tasks since it overcomes the disadvantages of K-Means and some partitioned methods.

Some good aspects of the method are that it works better with overlapping data and is less sensitive to initialization and noise. Therefore, it has been successfully applied in many tasks.

The main difference between the methods previously described is that the FCM does not generate a hard boundary for the clusters. Instead, the algorithm creates a membership matrix $={\left\{{\mu }_{ij}\right\}}_{i,j}^{N,K}$. In this case, the inputs of this matrix are μ_ij of each object i for the cluster j. Note that ${\mu }_{ij}\in \left[\mathrm{0,1}\right]$, ${\sum }_{j=1}^K{\mu }_{ij}=1$, and $0<{\sum }_{j=1}^K{\mu }_{ij}<N$ are given by Equation (13):

$${\mu }_{ij}^m=\frac{1}{{\sum }_{k=1}^C\frac{{‖x_i-c_j‖}^{\frac{2}{m-1}}}{‖x_i-c_k‖}}$$

(13)

where ∥.∥ is the norm, often the Euclidean distance from Equation (4), and m is the fuzziness parameter. Then, the cluster centers are updated according to Equation (14):

$$c_j=\frac{{\sum }_{i=1}^N{\mu }_{ij}^m}{{\sum }_{i=1}^N{\mu }_{ij}^m}$$

(14)

The FCM firstly updates the grade of membership μ_ij for each object and center and then calculates the centers employing Equation (9) in each iteration. At the end of the process, the goal is to optimize the cost function given by Equation (15) [6]:

$$J_m=\sum _{i=1}^N\sum _{j=1}^K{{\mu }_{ij}^m‖x_i-c_j‖}^2$$

(15)

4.9. Numerical Application

An application was performed to present an empirical analysis using data collected from a Brazilian city (Ponta Grossa—Parana state) located in the south of Brazil with 120 thousand water connections and approximately 320 thousand inhabitants. The data were based on the study published by [20], in which the ELECTRE TRI method was used in a real case to classify maintenance priorities in water distribution networks. This case study was chosen because it was conducted using threshold definitions from experts’ perceptions and datasets from an automated system with quantitative characteristics necessary to provide the integration and comparisons with clustering techniques. The study [20] was applied in the sanitation company of the cited city and provided real results, mainly because of the support from the automated system, without only subjective evaluations.

Urban growth and the emergence of new housing are conditions in urban water distribution systems and lead to mobilizing investment in infrastructure, necessary to accommodate it. Therefore, the level of investment allocated to each sectorial area will depend on the characteristics and priorities for these areas [72,73,74]. This infrastructure includes water pipes; fittings; reservoirs (placed strategically to ensure supply at critical moments); and measuring devices, such as control valves, flow meters, and pressure gauges. It needs an organized structure to be configured. One technique that can be used to manage development areas is ‘sectorization’. This is a categorization of flow measurements from macro systems into smaller domains and presents an alternative to the modeling problem. As some sanitation companies use automation systems in their water distribution systems, this contributes directly to water loss reduction and area measurement information that can be digitally collected to make new models for solving problems in this sector [20,75,76]. Trojan et al. [36] show strategies or means that could be adopted to achieve goals individually, and some alternatives should be considered that can achieve the expected results for the goals.

4.10. Sorting Areas into Three Categories

The first step to be considered is defining criteria close to the problem.

Table 2. Criteria.

Criteria
g1	Number of connections (CN)
g2	Measured Volume (MV)
g3	Water Losses (WL)
g4	Meters per connection (MC)
g5	Population (POP)
g6	Public Economies (PE)

presents the adopted standards from the original application [20].

In multi-criteria applications, the definitions from decision-makers (experts) are typically performed for criteria, weightings, indifference and preference thresholds, and the definition of regional class boundaries. In this work, these parameters were adopted from [20] to compare the results with the clustering techniques used in experiments to define started values of boundaries for class profiles.

This was based on the dataset consolidated by the application, as shown in

Table 3. Parameters for the boundaries between classes (original application with criteria 3 and 4 changed to benefit).

Categories /Classes	Maintenance	Borders	Criteria
			(CN) g1	(MV) g2	(WL) g3	(MC) g4	(POP) g5	(PE) g6
CL1	Proactive	b1	3500	37,000	93.00	27.00	12,000	15
CL2	Preventive
		b2	1900	18,500	86.00	25.00	8000	5
CL3	Corrective
Weights			23%	10%	20%	15%	12%	20%
Direction of Preferences			↑ up	↑up	↑up	↑up	↑up	↑up

L_n → Classes; g_n → Criteria; b_n → Class Border.

. Initially, Indifference and Preference thresholds equal to ‘zero’ were considered to compare the clustering results. After these proposed start values, the decision-makers could be invited to promote adjustments on these thresholds [77], as preconized in multi-criteria analysis.

Table 4. Original dataset.

	Benefit	Benefit	Cost	Cost	Benefit	Benefit
Weights	23%	10%	20%	15%	12%	20%
Areas (flow sectors) an	Number of Connections (index)	Measured Volume (index)	Water Losses (index)	Meters of Network Per Connections (index)	Population (index)	Public Economies (index)
a1	883	12,404	41.50	14.49	3033	10
a2	3255	57,729	39.18	15.81	14,960	5
a3	1850	19,130	29.69	10.19	6470	5
a4	1310	16,810	53.55	13.31	4267	2
a5	1192	11,425	37.24	8.91	4059	2
a6	2783	30,220	36.94	11.98	10,220	5
a7	14,375	180,585	55.25	12.62	48,444	20
a8	3397	32,938	51.11	10.01	11,574	6
a9	2622	33,182	65.26	13.05	8837	5
a10	2779	35,797	51.77	11.34	10,160	5
a11	3286	45,784	39.63	11.70	11,750	7
a12	2208	20,382	44.10	9.93	7647	3
a13	3333	35,474	41.50	10.52	11,521	20
a14	2685	26,499	38.90	10.17	9259	20
a15	23,474	302,947	34.13	12.34	83,563	10
a16	1830	22,472	66.49	12.24	6226	3
a17	8667	92,686	46.94	10.76	29,887	10
a18	5124	53,416	32.72	10.76	17,268	6
a19	1705	19,250	61.68	11.29	5881	5
a20	865	10,864	46.32	11.53	3077	5
a21	974	9158	64.80	9.99	3289	5
a22	727	7483	66.76	10.35	2520	3
a23	1844	20,141	54.08	11.09	6362	5
a24	2961	34,724	65.70	11.81	10,604	3
a25	4586	51,197	39.39	10.44	15,729	4
a26	2156	36,343	27.52	16.74	9074	4
a27	4527	50,234	74.81	11.45	15,545	5
a28	1876	23,153	37.07	12.04	6623	4
a29	4651	56,296	52.43	11.98	16,290	9
a30	2774	33,143	59.60	11.94	9668	8

presents the original dataset extracted from [20] with the six criteria, four of which are defined as ‘benefit target’ and two as ‘cost target’. The two criteria of cost target were normalized and transformed to benefit the cluster application and to provide equal scales for comparisons with ELECTRE TRI results.

Then, three procedures were proposed to perform the cluster experiments, and all had the benefit target defined for every criterion [78,79].

The first one, shown in

Table 5. Normalized weighted dataset—Procedure 1.

	Benefit	Benefit	Benefit	Benefit	Benefit	Benefit
Weights	23%	10%	20%	15%	12%	20%
Areas (flow sectors) an	Number of Connections (index)	Measured Volume (index)	Water Losses (index)	Meters of Network Per Connections (index)	Population (index)	Public Economies (index)
a1	0.16	0.17	11.70	4.31	0.08	8.89
a2	2.56	1.70	12.16	1.78	1.84	3.33
a3	1.14	0.39	14.06	12.55	0.58	3.33
a4	0.59	0.32	9.29	6.57	0.26	0.00
a5	0.47	0.13	12.55	15.00	0.23	0.00
a6	2.08	0.77	12.61	9.12	1.14	3.33
a7	13.80	5.86	8.95	7.89	6.80	20.00
a8	2.70	0.86	9.78	12.89	1.34	4.44
a9	1.92	0.87	6.95	7.07	0.94	3.33
a10	2.07	0.96	9.65	10.34	1.13	3.33
a11	2.59	1.30	12.07	9.66	1.37	5.56
a12	1.50	0.44	11.18	13.05	0.76	1.11
a13	2.63	0.95	11.70	11.92	1.33	20.00
a14	1.98	0.64	12.22	12.59	1.00	20.00
a15	23.00	10.00	13.17	8.43	12.00	8.89
a16	1.12	0.51	6.70	8.62	0.55	1.11
a17	8.03	2.88	10.61	11.46	4.05	8.89
a18	4.45	1.55	13.46	11.46	2.18	4.44
a19	0.99	0.40	7.66	10.44	0.50	3.33
a20	0.14	0.11	10.74	9.98	0.08	3.33
a21	0.25	0.06	7.04	12.93	0.11	3.33
a22	0.00	0.00	6.65	12.24	0.00	1.11
a23	1.13	0.43	9.18	10.82	0.57	3.33
a24	2.26	0.92	6.86	9.44	1.20	1.11
a25	3.90	1.48	12.12	12.07	1.96	2.22
a26	1.44	0.98	14.50	0.00	0.97	2.22
a27	3.84	1.45	5.04	10.13	1.93	3.33
a28	1.16	0.53	12.59	9.00	0.61	2.22
a29	3.97	1.65	9.51	9.12	2.04	7.78
a30	2.07	0.87	8.08	9.20	1.06	6.67

, considered a Normalized Weighted Dataset (Procedure 1). The original data were normalized on a scale of (0 to 100) and multiplied by their respective weights. Thus, the better value for the alternative is the exact value of the criterion weight, and the worst is equal to zero [80]. The procedure for normalizing criteria 1, 2, 4, 5, and 6 was the Linear Max–Min normalization method presented in Equations (16) and (17) [72].

$$n_{ij}=\frac{r_{ij}-r_j^{min}}{r_j^{max}-r_j^{min}}\hspace{1em}\hspace{1em}\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{b}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{t}$$

(16)

$$n_{ij}=\frac{{r_j^{max}-r}_{ij}}{r_j^{max}-r_j^{min}}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{t}$$

(17)

where n_ij is the normalized value; r_ij the alternatives i criteria j values; and r_j^max and r_j^min are the maximum and minimum criteria values, respectively. For criterion 3 [Water Losses (index %)], the normalization for the percentage was adopted as (100—r_ij).

The second procedure shown in

Table 6. Non-Normalized weighted dataset—Procedure 2.

	Benefit	Benefit	Benefit	Benefit	Benefit	Benefit
Weights	23%	10%	20%	15%	12%	20%
Areas (flow sectors) an	Number of Connections	Measured Volume (m3/month)	Water Losses (%)	Meters of Network Per Connections (Index)	Population (Inhabitants)	Public Economies (Number)
a1	203	1240.4	58.5	0.069	363.9	2.0
a2	748	5772.9	60.8	0.063	1795.2	1.0
a3	425	1913.0	70.3	0.098	776.4	1.0
a4	301	1681.0	46.4	0.075	512.0	0.4
a5	274	1142.5	62.7	0.112	487.0	0.4
a6	640	3022.0	63.0	0.083	1226.4	1.0
a7	3306	18,058.5	44.7	0.079	5813.2	4.0
a8	781	3293.8	48.8	0.100	1388.8	1.2
a9	603	3318.2	34.7	0.077	1060.4	1.0
a10	639	3579.7	48.2	0.088	1219.2	1.0
a11	755	4578.4	60.3	0.085	1410.0	1.4
a12	507	2038.2	55.9	0.101	917.6	0.6
a13	766	3547.4	58.5	0.095	1382.5	4.0
a14	617	2649.9	61.1	0.098	1111.0	4.0
a15	5399	30,294.7	65.8	0.081	10,027.5	2.0
a16	420	2247.2	33.5	0.082	747.1	0.6
a17	1993	9268.6	53.0	0.093	3586.4	2.0
a18	1178	5341.6	67.2	0.093	2072.1	1.2
a19	392	1925.0	38.3	0.089	705.7	1.0
a20	198	1086.4	53.6	0.087	369.2	1.0
a21	224	915.8	35.2	0.100	394.6	1.0
a22	167	748.3	33.2	0.097	302.4	0.6
a23	424	2014.1	45.9	0.090	763.4	1.0
a24	681	3472.4	34.3	0.085	1272.4	0.6
a25	1054	5119.7	60.6	0.096	1887.4	0.8
a26	495	3634.3	72.4	0.060	1088.8	0.8
a27	1041	5023.4	25.1	0.087	1865.4	1.0
a28	431	2315.3	62.9	0.083	794.7	0.8
a29	1069	5629.6	47.5	0.083	1954.8	1.8
a30	638	3314.3	40.4	0.084	1160.1	1.6

has adopted a Non-Normalized Weighted dataset (Procedure 2), in which the original data was multiplied by the weight in percentage. The cost targets were also normalized considering the formula $n_{ij}=\frac{1}{r_{ij}}$ for numerical data and (100—r_ij) for loss index percentages.

Table 7. Original dataset Cost Normalized—Procedure 3.

	Benefit	Benefit	Benefit	Benefit	Benefit	Benefit
Weights	23%	10%	20%	15%	12%	20%
Areas (flow sectors) an	Number of Connections	Measured Volume (m3/month)	Water Losses (%)	Meters of Network Per Connections (Index)	Population (Inhabitants)	Public Economies (Number)
a1	883	12,404	58.50	18.99	3033	10
a2	3255	57,729	60.82	17.67	14,960	5
a3	1850	19,130	70.31	23.29	6470	5
a4	1310	16,810	46.45	20.17	4267	2
a5	1192	11,425	62.76	24.57	4059	2
a6	2783	30,220	63.06	21.50	10,220	5
a7	14,375	180,585	44.75	20.86	48,444	20
a8	3397	32,938	48.89	23.47	11,574	6
a9	2622	33,182	34.74	20.43	8837	5
a10	2779	35,797	48.23	22.14	10,160	5
a11	3286	45,784	60.37	21.78	11,750	7
a12	2208	20,382	55.90	23.55	7647	3
a13	3333	35,474	58.50	22.96	11,521	20
a14	2685	26,499	61.10	23.31	9259	20
a15	23,474	302,947	65.87	21.14	83,563	10
a16	1830	22,472	33.51	21.24	6226	3
a17	8667	92,686	53.06	22.72	29,887	10
a18	5124	53,416	67.28	22.72	17,268	6
a19	1705	19,250	38.32	22.19	5881	5
a20	865	10,864	53.68	21.95	3077	5
a21	974	9158	35.20	23.49	3289	5
a22	727	7483	33.24	23.13	2520	3
a23	1844	20,141	45.92	22.39	6362	5
a24	2961	34,724	34.30	21.67	10,604	3
a25	4586	51,197	60.61	23.04	15,729	4
a26	2156	36,343	72.48	16.74	9074	4
a27	4527	50,234	25.19	22.03	15,545	5
a28	1876	23,153	62.93	21.44	6623	4
a29	4651	56,296	47.57	21.50	16,290	9
a30	2774	33,143	40.40	21.54	9668	8

presented the Original dataset Cost Normalized (Procedure 3), in which just the cost target for criteria 3 and 4 were normalized. The other data were considered equal to the original dataset.

5. Results and Discussion

In the ELECTRE TRI method, it is possible to utilize original data for just normalizing cost targets. So, Procedure 3 was used to make an experiment with a similar application of the ELECTRE TRI and compare it with clustering results. The other procedures were utilized for investigating the behavior between clustering techniques and original ELECTRE TRI border definitions. These procedures were adopted because they can change the experiments if considering the data direction, mainly in the clustering application.

First, the analyses and comparisons were performed based on the results presented in Figure 4, illustrating each technique with its respective procedure (1, 2, and 3). To do so, the following set of parameters was used:

• K-Means and K-Medoids: 100 independent runs;
• PSO: population of 20 agents, 50 iterations, c₁ = c₂ = 2.05, ω with linear decay, 30 independent runs;
• GA: population of 20 agents, 50 iterations, 80% of crossover probability, 30% of mutation probability, 30 independent runs;
• DE: population of 20 agents, 50 iterations, 80% of crossover probability, F = 0.8, 30 independent runs;
• FCM: 100 independent runs.

The algorithms were run several times until the smallest SSW was achieved since the final responses depend on the initialization.

Figure 5 illustrated the final results with original sorting from ELECTRE TRI, the FCM experiment, and the new results of the ELECTRE TRI application using the limits suggested by the results of FCM. It demonstrates that the FCM (Procedure 1) was able to capture the agreements of the data for the proposed categorizations.

In

Table 8. Definition of limits after sorting by Clustering method—FCM—Procedure 1.

		Benefit	Benefit	Benefit		Benefit		Benefit	Benefit
Classes/Limits	an	Number of Connections	Measured Volume (m3/month)	Water Losses (%)		Meters of Network Per Connections (m/connections)		Population (Inhabitants)	Public Economies (Number)
High(Hi1)	a15	23,474	302,947	65.87		12.34		83,563	10
Proactive Class	a7	14,375	180,585	44.75		12.62		48,444	Max20
	a13	3333	35,474	58.50		10.52		11,521		20
Low(Lo1)	a14	Lo1 = 2685	Lo1 = 26,499	38.90		Lo1 = 10.17		9259		20
Limits for Preventive/ Proactive		$Lim=\frac{{\mathrm{H}\mathrm{i}}_2-{\mathrm{L}\mathrm{o}}_1}{2}+{\mathrm{L}\mathrm{o}}_1$	$Lim=\frac{{\mathrm{H}\mathrm{i}}_2-{\mathrm{L}\mathrm{o}}_1}{2}+{\mathrm{L}\mathrm{o}}_1$	$=Max$		$Lim=\frac{{Hi}_2-{Lo}_1}{2}+{Hi}_2$		$Lim=\frac{{\mathrm{H}\mathrm{i}}_2-{\mathrm{L}\mathrm{o}}_1}{2}+{\mathrm{L}\mathrm{o}}_1$	$=Max$
LIMIT VALUES		(8667 − 2685/2) + 2685 = 5676	(92,686 − 26,499/2) + 26,499 = 59,592	= 72.48		(23.29 − 10.17/2) + 23.29 = 29.85		(29,887 − 9259/2) + 9259 = 19,573	= 20
High(Hi2)	a17	Hi2 = 8667	Hi2 = 92,686		53.06	10.76		Hi2 =29,887	10
Preventive Class	a2	3255	57,729		60.82	15.81		14,960	5
	a29	4651	56,296		47.57	11.98		16,290	9
	a18	5124	53,416		67.28	22.72		17,268	6
	a25	4586	51,197		60.61	23.04		15,729	4
	a11	3286	45,784		60.37	21.78		11,750	7
	a26	2156	36,343	Max72.48		16.74		9074	4
	a6	2783	30,220	63.06		21.50		10,220	5
	a28	1876	23,153	62.93		21.44		6623	Lo2 = 4
	a3	1850	19,130	70.31		Hi2 = 23.29		6470	5
Low(Lo2)	a1	Lo2 = 883	Lo2 = 12,404	58.50		18.99		Lo2 = 3033	10
Limits for Corrective/ Preventive		$Lim=\frac{{\mathrm{H}\mathrm{i}}_3-{\mathrm{L}\mathrm{o}}_2}{2}+{\mathrm{L}\mathrm{o}}_2$	$Lim=\frac{{\mathrm{H}\mathrm{i}}_3-{\mathrm{L}\mathrm{o}}_2}{2}+{\mathrm{L}\mathrm{o}}_2$	$Lim=Max$		$Lim={\mathrm{H}\mathrm{i}}_3$		$Lim=\frac{{\mathrm{H}\mathrm{i}}_3-{\mathrm{L}\mathrm{o}}_2}{2}+{\mathrm{L}\mathrm{o}}_2$	$\frac{{\mathrm{H}\mathrm{i}}_3-{\mathrm{L}\mathrm{o}}_2}{2}+{\mathrm{L}\mathrm{o}}_2$
LIMIT VALUES		(4527 − 883/2) + 883 = 2705	(50,234 − 12,404/2) + 12,404 = 31,319	= 62.76		= 24.57		(15,545 − 3033/2) + 3033 = 9289	(8 − 4/2) + 4 = 6
High(Hi3)	a27	Hi3 = 4527	Hi3 = 50,234		25.19		22.03	Hi3 = 15,545	5
Corrective Class	a10	2779	35,797		48.23		22.14	10,160	5
	a24	2961	34,724		34.30		21.67	10,604	3
	a9	2622	33,182		34.74		20.43	8837	5
	a30	2774	33,143		40.40		21.54	9668	Hi3 = 8
	a8	3397	32,938		48.89		23.47	11,574	6
	a16	1830	22,472		33.51		21.24	6226	3
	a12	2208	20,382		55.90		23.55	7647	3
	a23	1844	20,141		45.92		22.39	6362	5
	a19	1705	19,250		38.32		22.19	5881	5
	a4	1310	16,810		46.45		20.17	4267	2
	a5	1192	11,425	Max62.76		Hi3 = 24.57		4059	2
	a20	865	10,864	53.68		21.95		3077	5
	a21	974	9158	35.20		23.49		3289	5
Low(Lo3)	a22	727	7483	33.24		23.13		2.520	3

, the limit definition was calculated based on a middle value between classes in some criteria or based on a maximum value for other criteria, according to the target of the criteria. The calculations are detailed in the blue cells of Table 8.

According to the sorting experiments, the FCM—Procedure was considered to find each classes’ high and low limits. It was the most similar result found compared to the original application of the ELECTRE TRI method. An adjustment was necessary to delimitate a mean point between classes’ high and low limits. Thus, a medium point between these limits was considered to define the “a priori” boundaries. This adjustment was made based on the targets of each criterion. For example, criteria 3 and 4 were initially normalized as benefits, so the maximum reduction for losses and meters per connection is required. Table 8 presents the categorization (sorting) realized through the Fuzzy C-Means technique and the formulation for the adjustments to calculate each value for the limits of classes.

Table 9. Final Boundaries adopted defined by FCM Procedure 1.

Categories /Classes	Maintenance	Borders	Criteria
			(CN) g1	(MV) g2	(WL) g3	(MC) g4	(POP) g5	(PE) g6
CL1	Proactive	b1 b1*	3500 5676	37,000 59,592	93.00 72.48	27.00 29.85	12,000 19,573	15 20
CL2	Preventive
		b2 b2*	1900 2705	18,500 31,319	86.00 62.76	25.00 24.57	8000 9289	5 6
CL3	Corrective

presents the values of comparison between the original application from ELECTRE TRI, and the boundaries found with Fuzzy C-Means highlighted as (b_n*) in Table 9.

In this application, the Fuzzy C-Means (FCM) results provided pre-defined boundaries to a new application of ELECTRE TRI with the limits defined with this technique based on the dataset. It is possible to state that the results after clustering applications were very similar to the original application with ELECTRE TRI. It is essential to highlight that the delimitation limits in the original application were performed by an elicitation (inference) process, which demands time and effort to understand the data behavior. It was performed by an intuitive process [20]. So, it is coherent to state that the results found in this application with cluster algorithms were satisfactory regarding promoting a start view for decision-makers about the class limits and thresholds.

An extra analysis with performance indicators for the utilized cluster algorithms was also realized. In

Table 10. Final Boundaries defined by FCM Procedure 1.

Method	Average SSW	Average SSB	Average Silhouette
K-Means	940.42	106.43	0.65
K-Medoids	945.20	140.81	0.80
FCM	630.53	80.01	0.53
PSO	145.76	89.18	0.41
GA	142.60	102.16	0.58
DE	145.07	108.62	0.64

, it is possible to learn a verification of these indicators. Even though FCM does not figure out the best performance, it has a Fuzzy procedure embedded in its concepts, which is closed with these basic application features.

Still, it is crucial to determine whether decision-makers can rearrange these definitions according to their preferences. The main objective of these experiments is to provide a good understanding and an initial starting point to follow with a more robust and time-reducing multiple-criteria sorting analysis.

6. Conclusions

The methodology and experiments developed in this work aimed to create a new methodology that helps in the pre-definition of boundaries (limits between classes) in multiple-criteria sorting problems using clustering techniques. It was initially performed with the elicitation of decision-makers in a sanitation company to solve the problem of sorting flow areas for investments and priorities for maintenance actions.

The proposed evaluation exploited six clustering techniques: K-Means, K-Medoids, Fuzzy C-Means, PSO for clustering, a Genetic Algorithm for clustering, and Differential Evolution. These algorithms were compared with the original application of ELECTRE TRI and provided a start view for limits of classes in this problem. In ELECTRE TRI or another sorting method, the decision-makers intuitively define these limits. However, commonly, they do not have an initial start or “north” to do this.

This work provided this overview contributing practically to support the decision process. The practical contributions may be highlighted as the opportunity offered to the decision-makers to analyze the limits between classes only by the data behavior when they do not feel comfortable defining it alone. In this sector, the experts had notarial knowledge to define these class limits; most sanitation companies’ cases did not have enough information to correctly define these class limits and thresholds. By applying this methodology, the decision-makers could not have much experience defining the initial limits for analysis.

Thus, this work also can contribute to the theory with a development that aggregates methodologies in the multi-criteria sorting analysis, widening the existing results and methods in the literature.

Some limitations, such as the existence of reliable datasets to support the clustering application and the definition of weights for criteria, still need expert understanding to do it coherently. A correct definition of these parameters can lead to biased results. Future works can be developed to generalize this methodology, like developments of this proposal considering several cases studies related to sanitation or electrical systems and sensitivity analysis to make improvements. In addition, further investigations on other clustering methods, such as self-organized maps and other bio-inspired metaheuristics, should be performed.