Optimizing Customer Retention in the Telecom Industry: A Fuzzy-Based Churn Modeling with Usage Data

: Churn is a serious challenge for the telecommunications industry because of the much higher costs of gaining new customers than maintaining existing ones. Therefore, efforts to increase loyalty and decrease customer churn are the focus of telecom’s retention departments. In order to direct antichurn activities, profitable clients who have the highest probability of churning need to be identified. The data used to identify churners are often inaccurate and vague. In this paper, a fuzzy approach to modeling churn intent based on usage data in mobile telecommunications is presented. It appreciates the uncertainty of the data and provides insights into churn modeling. The goal of the study was to evaluate the applicability of the Mamdani and Sugeno models for building a churn model based on a limited but real-world dataset enriched with feature engineering. The additional goal was to find features most usable for churn modeling. Four metrics—accuracy, recall, precision, and F1-score—were used to estimate the performance of the models. The developed fuzzy rule-based systems show that to generalize possible churn identification factors with fuzzy rules, it is advisable to begin with features such as the change in the total amount of the invoice in the last period before the churning compared to the previous one, the total amount of the invoice in the period preceding the churning, the total amount of subscription in two months before the churning, the time of cooperation with the operator, and the number of calls out of the last quarter before leaving.


Introduction
The phenomenon of churn poses a genuine challenge for many customer-centered industries, especially mobile telecommunications, where acquiring new customers costs a lot.In recent years, the mobile telecommunications market has been growing rapidly in many countries.Due to the growing competitiveness of this market-the increasing number of telecom operators, the ease of porting clients' numbers between operators, the introduction of more sophisticated products and services, special offers, etc. [1]-customer churn is becoming an increasingly important challenge facing operators.
Customer retention is particularly important when it concerns those who bring high profits to the company or have the potential to do so, that is, customers with a relatively high lifetime value (LTV).Moreover, long-term customers are usually less sensitive to price changes.In addition, many people are willing to pay a higher price for products from a company they know and trust.They are not willing to take the risk of switching to a cheaper but possibly unreliable competitor [2].
Customer churn is widely classified into two types: voluntary and compulsory.The first type of churn is usually caused by unexpected incidents beyond the control of the telecoms, for example, as a result of customers' financial problems or migration to competitor relocations because of more attractive conditions.On the other side, providers can usually address the reasons for voluntary churn, for example, disappointment with the quality of mobile service, poor customer service such as rejected complaints, poor treatment of a customer with temporary overdue payments, etc.
In order to prevent customers from churning, operators should care for the customers and foster individual long-term relationships with them by providing offers tailored to individual customers' preferences, rewarding loyalty, measuring satisfaction levels, etc.This loyalty-oriented approach is imposed by the fact that the cost of acquiring a customer is generally much higher than the cost of retaining an existing one.
By effectively identifying at-risk customers, the operator can look for ways to increase retention [3].For customers categorized as being at high risk of churn, the operator can take direct measures to prevent churn, such as providing the customer with a personalized offer or addressing the customer's needs in other possible ways, particularly for high LTV customers.
The aim of this paper is to describe the preliminary findings and results of churn fuzzy modeling conducted on real-world data obtained from one of the largest mobile operators in Poland.There is a lack of such analyses in the literature.Based on a limited set of features, we aim to provide ways of engineering and extracting new features that may be usable for fuzzy churn modeling with two different models, utilizing real operator data.
The use of fuzzy theory for churn modeling can be noticed in several papers and implementations.Most of them concern fuzzy clustering [4][5][6] for building customer profiles.The k-means-HF method has been used by Qiu et al. [7] for background segmentation of customers in order to identify groups of customers with the highest churn probability.A hybrid fuzzy unordered rule induction algorithm for predicting customer churn was introduced in [8], incorporating fuzzy c-means clustering.Various classifiers, including a fuzzy one for churn prediction, were compared in [9].Among the most recent papers, Papa et al. [10] proposed a combination of fuzzy rules and neural networks for predicting customer churn in the telecommunications industry.Neural network algorithms were used for searching the parameters of the membership function.Limited research was conducted on churn modeling in the telecommunications sector employing fuzzy classifiers [4,9,11].In [9,11], the authors presented a model designed for precise customer churn prediction and retention, which was validated using specific datasets.In [4], B. Al-Shboul et al. explored the application of a churn prediction approach that combines Fast Fuzzy C-Means and Genetic Programming.While the first algorithm was used to remove outliers, genetic programming was applied to generate a churn prediction model.There is a much bigger selection of churn modeling papers based on other than fuzzy modeling techniques, which are out of the scope of this study, but there is some research worth mentioning.In one of the most recent studies, Bojanowska and Kulisz [12] prepared a Mamdani model with data from CRM systems for churn customer prediction in the banking sector.In [13], Toor and Usman proposed the churn detector (OTCCD), which handles the problems of class imbalance and concept drift.Nabahirwa et al. [14] performed a comparative study of different ML techniques for churn modeling and discovered that the random forest algorithm optimized by grid search outperformed other models.
Rough set theory is yet another approach that can be used for building rule-based systems for customer churn.Amin et al. studied the quality of classification for such systems [15,16].In this approach, all numerical attributes were discretized first, and then rules were generated.
A significant issue in customer churn is data imbalance.Two primary strategies are available to address this issue: data-level solutions, which contain a variety of techniques for balancing the data [17], and algorithm-level solutions, involving the implementation of specialized models dedicated to imbalanced data [18].Notably, focusing more on the minority class often enhances classification performance and will be considered in future research.
We have not come across studies concerning the use of various features generated from usage and payment data for churn modeling with fuzzy systems.The major contributions of this paper are fuzzy rule-based systems (FRBS) for churn modeling in the mobile telecommunications industry.The novel methodology is illustrated by applying fuzzy modeling to various usage and payment features derived from source data, aiming to unveil connections between user behavior and the intent to churn.
The rest of the article is structured as follows: methodology and related work are presented in Section 2. The structure of the experiments and empirical results are provided in Section 3, and conclusions and future research are presented in Section 4.

Methodology
The primary objective of this study is to introduce a fuzzy approach for modeling churn in mobile telecommunications in order to improve retention activities by aiming them at customers with a high probability of leaving.Our aim is to validate and present the use of fuzzy rules to provide a high-accuracy, explainable churn modeling system.Since the relations between customers' behavior expressed as usage data and churn intent are not instinctively crisp, we decided to use fuzzy theory in order to model those relations and present them as a set of explainable rules.
Fuzzy methodologies exhibit two crucial characteristics: addressing uncertainty through the representation of knowledge using fuzzy sets and rules and achieving excellent interpretability through straightforward linguistic rules.In the proposed fuzzy approach for churn modeling based on actual telecommunications data, we will adhere to the following steps outlined for fuzzy modeling.
In this paper, we have generated two fuzzy rule-based systems.One was based on Mamdani fuzzy inference (the Mamdani model) to explain the relationship between the usage data and the risk of churn.The other was employing Takagi-Sugeno inference (the Sugeno Model) in order to test the prediction capabilities.We are not providing a detailed mathematical model of the system, as we follow the original one presented in [19].
For the Mamdani model, as a first step, an induction of fuzzy partitions was performed.This task allowed for defining linguistic variables, which are meaningful, with memberships of fuzzy sets that will be used later in rule generation.To ensure good interpretability, we have used strong fuzzy partitions [20].In the next step, we have generated fuzzy rules with fuzzy decision trees (FDT) [21,22].
For fuzzy inference, we used the Expert Mode of the GUAJE software version vb4.0 (Generating Understandable and Accurate Fuzzy Rule-based Systems in a Java Environment) [23][24][25].GUAJE allows for the step-by-step building of a knowledge base by directly introducing fuzzy rules from experts or inducing them from data.It also allows for the manual definition of fuzzy partitions or their generation with several methods.It implements the Highly Interpretable Linguistic Knowledge fuzzy modeling methodology (HILK).GUAJE complies with JFML (Java Fuzzy Markup Language) [26,27].
The knowledge base obtained with GUAJE was evaluated using measures such as accuracy and interpretability.While interpretability assesses the ease with which a system can be comprehended by analyzing the complexity of its knowledge base, accuracy gauges the system's predictive capability.Typically, there exists a trade-off relationship between these two metrics, where a simpler system tends to be more interpretable but less accurate [23].
The Sugeno model was generated in order to verify churn prediction capabilities.For this fuzzy inference system, first we divided the data into training and testing sets.For performing validation, we incorporated the 10-fold cross-validation technique.Next, we used fuzzy c-means clustering for initial rules.The parameters of the rules were tuned with ANFIS [25].A Matlab environment with Fuzzy Logic Toolbox [28] was used to create the Takagi-Sugeno model.As the final step, the obtained fuzzy model was evaluated using measures such as Kappa and F1 score [29].

Feature Construction and Selection
For our analysis, we utilized a dataset supplied by one of the largest mobile telecommunications operators in Poland, encompassing information from 15,000 randomly chosen clients.The data provider had previously conducted an initial segmentation, resulting in a relatively homogeneous group.This dataset exclusively comprised individual (noncorporate) post-paid customers, each with only one active SIM card at the commencement of the studied period.Furthermore, it was assumed that the residential address of the customers remained constant throughout the study, and any changes in the number of active SIM cards on a subscriber's account occurred at most once.
The main data set consists of three tables.The BASE table has information about customers and contracts, helping us figure out if a customer left during the analyzed time.The STATUS table shows the history of changes in customer activity, like being active, temporarily inactive, or permanently inactive.Everyone starts as active at the beginning of the analyzed period.The last table, USAGE, provides a detailed record of how customers used services like SMS and voice calls each month [30].It is worth noting that the operator did not provide usage data about other services, for example, data transfer.
To build a fuzzy rule-based model for churn modeling using customers' usage data, the history of payments, and interactions with the provider, we have performed the following steps: In the first step, a set of all possible usable features for churn modeling has been prepared.In total, 148 features have been generated.A part of them, exactly 10 variables, have been directly extracted from the source dataset, that is, birth date, first activation date, etc.The rest of the 138 features have been constructed from the input dataset on the basis of the authors' domain knowledge and consulting with experts in the field.The authors' own script in a popular programming language was used to preprocess source tables and generate the abovementioned new features.The calculations consisted of, for example, the aggregation of feature values over a period of a month, a quarter, and a half-year, as well as relative changes between periods.
For the purpose of fuzzy modeling, non-numerical features have been removed.As a result, a dataset containing 138 features has been generated.This set of features can be divided into three main groups, which are listed in shortened form in Table 1.The first one consists of five features describing periods of time, mainly between events, for example, cooperation_days_with_operator (customer seniority).The second group contains variables describing service usage, for example, SMS and MMS numbers, outgoing call numbers, and total minutes.For each variable in this group, separate features expressing the total value/number in every subperiod (month, quarter, and half of the year) have been calculated.There were a total of 12 months, 4 quarters, and 2 half-year periods in the analyzed period of time.The time period numbers were appended together with a dash to the names of the created features, with number 0 meaning the last period of analysis, number 1 meaning the period preceding the last period of analysis, and so on, with the highest number giving the first period of analysis.We have also calculated features expressing the dynamics of usage changes between the following periods, for example, sms_number_out_change_quarter_0-change of outgoing SMS number between the last analyzed quarter and the previous one.As domain experts suggest, changes in customers' service usage behavior are usually the first sign of incoming churn intent.The third group of features consisted of variables describing customer payments, subscription amounts (resulting from contracts), and total invoice amounts (including additional services).For this group, additional features for every period and changes between the periods have been generated.

Feature Group Variables
Features describing periods of time between events cooperation_days_with_operator number_of_days_between_deactivation_inquiry_and_first_activation number_of_days_between_deactivation_inquiry_and_date_of_last_promotion number_of_days_between_start_of_analysis_and_resignation user_age_at_the_end_of_analyzed_period Features describing usage of services sms_number_out mms_number_out outgoing_calls_number outgoing_calls_minutes (Total in a period and changes between periods of: months, quarters, and half-years) Features describing contract payments subscription_total_amount invoice_total_amount (Total in a period and changes between periods of: months, quarters, and half-years) In the next step of data preprocessing, we dealt with the missing values for input variables (approximately 2% of all records) by employing a model-based imputer with a simple decision tree [31].
At this stage, there were 138 features in total, which was too many for generating explainable fuzzy rules.In the next step, we have performed feature selection based on the popular Gini ranking algorithm [32].As many features have concerned a similar subject, there was a strong correlation observed between some of them.From those pairs/groups of features, we have selected the one with the highest value of the Gini decrease.As a result, the final set containing the following eight features has been constructed: invoice_total_amount_change_quarter_1, invoice_total_amount_month_1, in-voice_total_amount_month_0, cooperation_days_with_operator, invoice_total_amount_ change_quarter_0, outgoing_calls_minutes_quarter_0, subscription_total_amount_change_ quarter_1, and subscription_total_amount_quarter_1.
Most of the selected features are connected with not only the financial side of the customer's relationship with the operator but also customer seniority and the number of outgoing calls.The target variable was called churned and consisted of information about whether the customer churned (value 1) or remained with the operator (value 0).
As the provided dataset consisted of clients with different histories of cooperation ranging from one to twelve months, the ones with histories shorter than nine months (three quarters) have been removed in order to provide comparable periods of full three quarters.The resulting final dataset for fuzzy modeling contained 8577 records, 278 of which were related to the churning customers.
All the chosen input features for fuzzy modeling were represented on a numeric scale, spanning from fractional values to thousands.Developing a churn model based on customers' usage data and behavioral features, expressed numerically, can yield a precise model.However, this approach does not assure that the resulting model will be easily comprehensible and practical.Transforming numerical features into fuzzy, rough concept features allows for a more meaningful and interpretable representation of customers' behavior and usage data (e.g., low, average, and high).

Inducing Fuzzy Partitions for Input Features for the Mamdani Model
As a preliminary step for fuzzy rule generation, we have fuzzified input variables with automatic induction of partitions.There is a great choice of validity indices for fuzzy clustering, although there is no universally agreed-upon index that can be used to find out the optimal cluster number and shape [33][34][35].
Hierarchical Fuzzy Partitioning (HFP) creates standardized, uniformly strong fuzzy partitions via an ascending procedure, where at each step, two fuzzy sets are merged for each variable.This algorithm discovers highly concentrated data areas using the agglomeration method of hierarchical clustering, analyzes and combines the data areas, and then uses the evaluation function to find the optimal clustering scheme [36].
K-means aims to partition observations into k clusters, where each observation belongs to the cluster with the nearest mean (cluster centroid).It is a well-known method used for clustering objects in many applications.
The decision of how to fuzzify each variable with which automated method (HFP or k-means) into how many clusters (fuzzy sets) is usually one of the most difficult tasks in fuzzy sets induction from real data, as there are not generally available, commonly accepted rules.In order to make the best possible choice of cluster shape and number, we have performed clustering several times with different numbers of clusters, ranging from two to five, and both clustering methods.The results of every iteration were compared with the following metrics: Chen Index, Partition Coefficient, and Partition Entropy.
Dunn's Partition Coefficient (PC) [37,38] measures the closeness between the fuzzy solution and the corresponding hard solution.The hard solution is built by classifying each object into the cluster that has the largest membership.According to Formula (1), the coefficient ranges from 1/k to 1, where k is the number of clusters.When all memberships are equal to 1/k, their value is 1/k.The value of one results when, for each object, the value of one membership is unity and the others are zero.
Bezdek's Partition Entropy (PE) [39] is a simple index based on fuzzy membership values of fuzzy partitions.Its formula is defined in Equation ( 2).
Chen index (Ch) [36] is an effective fuzzy partition measure as a cluster validity criterion associated with the fuzzy c-means algorithm and is defined in Equation (3).
A good partition should minimize PE while maximizing PC and Ch metrics.Table 2 presents the results of partition induction for every input variable.Results of fuzzification, including histograms and fuzzy partitions, for three of the variables are presented in Figures 1-3.The K-means partitioning method was applied for the majority of variables.

Fuzzy Rule Generation for the Mamdani Model
We used the Fuzzy Decision Trees (FDT) algorithm for generating fuzzy rules based on fuzzy variables obtained in the previous step.The FDT algorithm sorts inputs according to their importance, with the goal of minimizing entropy.Then, a tree is translated into a quite general incomplete rules set, as only a subset of input variables is considered.
The following parameters of the FDT algorithm have been set: maximum tree depth: 8; leaf minimum cardinality: 10; minimum significant level that is the minimum matching degree for an item to be considered to belong to the node: 0.1; tolerance threshold that represents the tolerance on the matching degree to the node majority class: 0.1; minimum entropy/deviance gain: 0.001; coverage threshold: 0.9; relative entropy: enabled; prune: enabled; and split whole subtree/leaf: enabled.
As a result, FDT generated 158 rules.In the next step, pruning was performed, which resulted in a model with only eight fuzzy rules.In the last rule base, simplification was performed with a strict limit imposed in order not to lose coverage or introduce any error cases.As a result of simplification, we received only five rules, which constitute a very concise fuzzy rule set.The resulting rule set can be characterized with the following metrics: coverage = 1.0, accuracy = 0.983, average confidence firing degree = 0.887, mean square classification error = 0.014, error cases (EC) = 116, ambiguity cases (AC total) = 36, ambiguity cases error (AC error) = 30, and zero unclassified cases (UC).
The confusion matrix (Table 3) reveals that the model has noticeably better prediction for non-churners (precision = 0.988 and sensitivity = 0.995) than for clients who left the operator (precision = 0.803 and sensitivity = 0.629).This characteristic of the model is probably caused by an imbalanced dataset.Still, it offers good predictions for real churners.All five rules are presented in Table 4, and they reflect a focus on usage data.The interpretability index (0.67) is high, meaning that the rules are easily understandable and explainable owing to the granular approach.The total rule length [40] is equal to 14, while the average rule length is only 2.8.The accumulated rule complexity [41] is equal to only 5.1.Those metrics, in particular the short average rule length, reflect the simple and understandable nature of the discovered fuzzy rules.Of the eight input variables, only four have been implemented in fuzzy rules.For example, rule no. 1 shows that if invoice_total_amount_change_quarter_0 is low, invoice_total_amount_month_0 is low, cooperation_days_with_operator is low, and outgoing_calls_minutes_quarter_0 is low, then we may assume that a client will churn.The second rule differs only in one condition, where cooperation_days_with_operator is high.This seemingly small change means that a more senior client may rather stay with the operator.This conclusion is consistent with the opinions of domain experts, who say that customers with a long relationship are less inclined to churn, whereas clients who have a short relationship are going to leave the operator, for example, in pursuit of better welcome offers among other providers.It can be noticed that the attribute invoice_total_amount_change_quarter_0 appears in all rules.This indicates the very high importance of this attribute.It can also be observed that the last rule contains a condition only for this attribute, which leads to a decision.
We need to emphasize that the fuzzy rule-based system generated can only serve as an evaluation of usage and financial data for modeling the churn phenomenon in the telecommunications industry for the analyzed environment and customer group.Due to the highly imbalanced dataset, a split between training and testing was not performed.Therefore, the resulting fuzzy rules cannot be used as a churn prediction tool.We are aware that this is just a preliminary verification of the presented approach to modeling churn in the mobile telecommunications industry, similarly to other applications [42].

Fuzzy Rules Generation for the Sugeno Model
We have also generated the Sugeno model with five rules based on the eight input features described above.Every input variable and output churn variable have been fuzzified into five clusters.By fuzzying churn into five clusters, the resulting rules indicate the probability of churning in fuzzy linguistics, which seems more useful than binary information.
The average accuracy on the test dataset based on the 10-fold stratified cross-validation is 0.98, while Kappa statistics is 0.502, which means that the obtained model has extremely good churn prediction capabilities, even taking into account that the studied customer group was quite homogenous.The average F1 score for churned customers is 0.98, while for customers not leaving the provider, it equals 0.51.
Below, we present the best model (for one partition), with an accuracy of 0.99 and a Kappa of 0.60.The five rules of the model are shown below.In Figure 4, we also show one membership value for the variable outgoing_calls_minutes_quarter_0.The rules of the Sugeno model require some effort to be interpreted, but the predictive quality is very high.Figure 5 shows the surface visualization for the Sugeno model for two variables: invoice_total_amount_change_quarter_0 and invoice_to-tal_amount_month_0.In this figure, one can observe that the decrease in the total amount paid by the customer in the last quarter before churning is a sign of a possible churn.If invoice_total_amount_change_quarter_0 is in cl 1,1 , invoice_total_amount_month_0 is in cl 1,2 , invoice_total_amount_month_1 is in cl 1,3 , cooperation_days_with_operator is in cl 1,4 , invoice_total_amount_change_quarter_1 is in cl 1,5 , outgoing_calls_minutes_quarter_0 is in cl 1,6 , subscription_total_amount_change_quarter_1 is in cl 1,7 , and subscription_total_ amount_quarter_1 is in cl 1,8 , then churn is in cl 1,1 .
The rules of the Sugeno model require some effort to be interpreted, but the predictive quality is very high.Figure 5 shows the surface visualization for the Sugeno model for two variables: invoice_total_amount_change_quarter_0 and invoice_total_amount_month_0.In this figure, one can observe that the decrease in the total amount paid by the customer in the last quarter before churning is a sign of a possible churn.The rules of the Sugeno model require some effort to be interpreted, but the predic tive quality is very high.Figure 5 shows the surface visualization for the Sugeno mode for two variables: invoice_total_amount_change_quarter_0 and invoice_to tal_amount_month_0.In this figure, one can observe that the decrease in the total amoun paid by the customer in the last quarter before churning is a sign of a possible churn.

Conclusions and Future Research
Churn modeling plays an important role in retention processes in the mobile tele communications industry.By identifying potentially leaving customers, many steps can be taken to influence their decision.In this paper, two preliminary fuzzy models were generated for churn modeling for one of the biggest mobile operators in Poland.Eigh important parameters were selected and successfully used to implement the model with very good prediction capabilities, even taking into account the quite homogenous group of customers.

Conclusions and Future Research
Churn modeling plays an important role in retention processes in the mobile telecommunications industry.By identifying potentially leaving customers, many steps can be taken to influence their decision.In this paper, two preliminary fuzzy models were generated for churn modeling for one of the biggest mobile operators in Poland.Eight important parameters were selected and successfully used to implement the model with very good prediction capabilities, even taking into account the quite homogenous group of customers.
We have created two models: the Mamdani model, with the purpose of explaining factors influencing churn behavior, and the Sugeno model for testing prediction capabilities.Input variables for the Mamdani model have been translated into two (low and high) and three fuzzy sets (low, average, and high) for the sake of understandability.As a final result, a simplified five fuzzy rules set was generated with full coverage and accuracy equal to 0.983 on the training set.Although metrics for churn cases are lower than for non-churners, with precision equal to 0.803 and sensitivity equal to 0.629, the model still offers good churn prediction capabilities.The interpretability index is very high.Only one of the rules describes customers who are going to churn.It should also be noted that the generated rules are very concise and short; one of them has only one condition in the premise.This is very convenient for experts and real applications.
The Sugeno model, while tested with 10-fold stratified cross-validation, had an accuracy of 0.98, which was near perfect considering the fact that it was based mostly on usage data (both call/messaging services) and payments (resulting from usage and selected plans).The average F1 score for churned customers was 0.76, proving that the model has good capabilities for identifying potential churners from a group of individual customers.
Here, the rules are more difficult to interpret than for the previous approach.But it should be noted that only two rules allow us to obtain such high accuracy of the model.
Our findings complement other results in the existing literature [4,9,11,12] with two fuzzy systems based on hard-to-reach real-world data from Poland.Based on a limited set of features, we showed ways of engineering and extracting new features, which turned out useful in fuzzy churn modeling with two different models utilizing operator data.
In future research, we are planning to build a hybrid churn model incorporating neural fuzzy networks with an extended set of features, including not only numerical but also categorical ones such as demographic data.Those features should help us our understanding of key elements indicating clients' leaning toward churn and possibly obtain better modeling results to help overcome this phenomenon, also for more heterogeneous customer groups.It is also planned to apply the concept of rough sets, which also use categorical data to generate rule-based systems.
Funding: Financed by the Minister of Science under the "Regional Excellence Initiative" Program.

Figure 1 .
Figure 1.Distribution and fuzzy partitions for the input variable invoice_total_amount_month_0.

Figure 2 .Figure 3 .
Figure 2. Distribution and fuzzy partitions for the input variable cooperation_days_with_operator

Figure 1 . 1 Figure 1 .
Figure 1.Distribution and fuzzy partitions for the input variable invoice_total_amount_month_0.

Figure 2 .Figure 3 .
Figure 2. Distribution and fuzzy partitions for the input variable cooperation_days_with_operator

Figure 2 . 1 Figure 1 .
Figure 2. Distribution and fuzzy partitions for the input variable cooperation_days_with_operator.

Figure 2 .Figure 3 .
Figure 2. Distribution and fuzzy partitions for the input variable cooperation_days_with_operator

Figure 3 .
Figure 3. Distribution and fuzzy partitions for the input variable outgoing_calls_minutes_quarter_0.

Table 1 .
Features groups for churn modeling.

Table 2 .
Fuzzy partitions for variables with values of evaluation indices.

Table 2 .
Fuzzy partitions for variables with values of evaluation indices.Variable Partition Induce Method Partitions Number PC PE Ch cooperation_days_with_operator k-means 2 0.849 0.225 0.772

Table 2 .
Fuzzy partitions for variables with values of evaluation indices.Variable Partition Induce Method Partitions Number PC PE Ch cooperation_days_with_operator k-means 2 0.849 0.225 0.772

Table 2 .
Fuzzy partitions for variables with values of evaluation indices.

Table 3 .
Confusion matrix with accuracy metrics for the Mamdani model.

Table 4 .
Fuzzy rules in the Mamdani model.