Statistical quality control based on control charts and process efficiency index by the application of fuzzy approach (case study: Ha'il, Saudi Arabia)

Controlling the quality of goods has been a concern for humans since the advent of their production. Quality control involves utilizing distinct features of finished or partially finished products or services to design, produce, maintain, preserve, and improve their quality. Quality control refers to the set of activities designed to ensure that products or services meet specified quality requirements (Juran & Gryna 1980). Statistical process control is a prevalent method employed in quality control. It involves monitoring processes to detect any exceptional sources of variation and implementing corrective measures. Statistical process control aims to prevent faulty product production through statistical methods and process control and is typically used for monitoring and testing processes (Rodriguez et al. 2009; Marais et al. 2022).

In the field of urban water and sewage, quality control is essential to ensure that drinking water is safe and meets established health standards, and that wastewater is treated to prevent environmental contamination. Quality control also helps utilities identify and address problems before they become more serious issues. Common quality characteristics that are monitored in urban water and sewage systems include pH, temperature, turbidity, dissolved oxygen, and the presence of specific contaminants, such as nitrates or bacteria. One challenge associated with interpreting control charts (CCs) is distinguishing between normal variability and abnormal variability. It should be mentioned that CCs are designed to detect changes in the mean or variability of a quality characteristic over time, but they may not be able to distinguish between common causes of variability and special causes that indicate a problem with the system. Another challenge is selecting appropriate control limits that balance the need to detect quality issues with the risk of false alarms (Oliveira da Silva et al. 2022).

Statistical quality control involves the use of statistical methods to monitor and control quality. This approach relies on data analysis to identify and respond to quality issues, rather than relying solely on inspections or other traditional quality control methods. Statistical quality control can be used to identify patterns and trends in quality data, as well as to detect anomalies that may indicate a problem. Statistical quality control employs various techniques, including CCs and process capability analysis, to enhance the quality of products and processes. This approach relies on data analysis to identify and respond to quality issues, rather than relying solely on inspections or other traditional quality control methods. Statistical quality control can be used to identify patterns and trends in quality data, as well as to detect anomalies that may indicate a problem. Statistical quality control can help utilities identify quality issues early, before they become more serious problems. It can also help utilities improve their processes and reduce the variability of their output, leading to more consistent quality. By using data analysis to monitor quality, utilities can make more informed decisions about where to focus their resources to improve quality. One challenge to implementing statistical quality control in urban water and sewage systems is the need for accurate and reliable data. Data collection can be difficult and time consuming, and the quality of data can be affected by a variety of factors, such as instrumentation issues or human error. Another challenge is the need for specialized statistical expertise to analyze and interpret the data, which may not be readily available within the utility (Montgomery 2020).

The CCs are graphical representations of data that are used to monitor the quality over time. They typically display the average value of a quality characteristic and its variability over time, along with upper and lower control limits. The CCs are useful in situations where data and information are clear and precise, and they split the process into two groups or classes, IC (in control) and OC (out of control), using precise data. However, statistical data can often be vague, incomplete, or uncertain in many processes. The CCs can help identify when a process is operating within normal limits and when it has shifted outside of those limits, indicating a potential quality issue. The CCs can be used to monitor a variety of water quality parameters, such as PH or turbidity, in distribution systems. By tracking these parameters over time, utilities can identify trends or anomalies that may indicate a problem with the system. For example, a sudden increase in turbidity may indicate a problem with the filtration system or a change in the source water quality. The CCs can also be used to monitor compliance with regulatory standards, such as maximum contaminant levels. One challenge associated with interpreting CCs is distinguishing between normal variability and abnormal variability. The CCs are designed to detect changes in the mean or variability of a quality characteristic over time, but they may not be able to distinguish between common causes of variability and special causes that indicate a problem with the system. Another challenge is selecting appropriate control limits that balance the need to detect quality issues with the risk of false alarms (Marais et al. 2022).

Fuzzy methods use linguistic variables to represent water quality parameters and their relationships, allowing for more flexible and intuitive modeling. Nonfuzzy methods use mathematical equations and statistical models to represent water quality parameters and their relationships. Fuzzy methods can handle uncertainty and imprecision in water quality data more effectively than nonfuzzy methods, but may be less precise in their predictions. Fuzzy logic can be used to control water quality by creating a decision-making system that adjusts operational parameters based on real-time water quality data. For example, a fuzzy logic controller could adjust the flow rate, PH, or disinfectant dosage in a water treatment plant to maintain a target water quality parameter. Fuzzy logic controllers can handle uncertain and imprecise data, making them useful for systems with variable and unpredictable water quality. Fuzzy set theory can handle uncertainty and imprecision in water quality data more effectively than traditional statistical methods. Fuzzy set theory can also represent complex relationships between water quality parameters and their impact on aquatic ecosystems or human health. Fuzzy set theory is particularly useful when data are limited or when there are multiple conflicting objectives in water quality management (Barzegar et al. 2023).

Fuzzy methods have the advantage of being able to handle uncertainty and imprecision in data, which is common in water quality modeling due to the complex and dynamic nature of water systems. Fuzzy methods can also incorporate expert knowledge in the form of linguistic variables and fuzzy linguistic statements, which can improve the accuracy and interpretability of the model. Fuzzy methods can also provide a more intuitive representation of the system behavior, which can aid decision-making. Nonfuzzy methods, on the other hand, are generally more precise and can be more computationally efficient than fuzzy methods. Nonfuzzy methods such as regression analysis or neural networks can also capture complex relationships between input and output variables, which may not be possible with fuzzy logic alone. However, nonfuzzy methods may not be able to handle uncertainty and imprecision in data as effectively as fuzzy methods. The choice of the method relies on the specific app and the data available. Occasionally, a mixture of fuzzy and nonfuzzy methods may be most appropriate (Barzegar et al. 2023).

Fuzzy-CCs, which use fuzzy sets defined by membership functions and fuzzy numbers like trapezoidal and triangular, provide different decision levels to decision makers by employing ambiguous data. Fuzzy-CCs are thus utilized in such cases (Kaya & Kahraman 2011; Kazemi et al. 2020; Montgomery 2020).

The fuzzy set theory is utilized in order to incorporate additional information and flexibility into process analysis. Another tool of control is the analysis of process performance, which is a metric that evaluates the capability of each process to produce products that meet customer satisfaction with technical specifications (Kaya & Kahraman 2011; Saeed et al. 2023).

The process capability index (PCI) or its efficiency proportion (ratio) is actually a type of summarized numerical value that measures the ability of each process relative to the specified technical limits. Put another way, the mentioned index is utilized to estimate the inherent ability of a manufacturing process in producing products that conform to their technical specifications (Lundkvist 2015).

Machine learning can also be used in statistical quality control to analyze process efficiency index (PEI) and CCs. Artificial neural networks (ANNs) can be used to model the relationships between process parameters and PEI. ANNs are a type of machine learning algorithm that can learn complex relationships between input and output data. ANNs can be trained on historical data to predict the PEI based on current process parameters. The trained ANN can then be used to identify which process parameters have the biggest impact on the PEI, enabling process engineers to make adjustments to improve process efficiency.

Also, M5 model trees are a type of decision tree algorithm that can be used to predict the PEI based on process parameters. M5 model trees are similar to ANNs in that they can learn complex relationships between input and output data, but they are easier to interpret. M5 model trees can be used to identify which process parameters have the biggest impact on the PEI, and can help process engineers to identify the optimal process parameters for maximizing process efficiency. In summary, ANNs and M5 model trees can be used to identify which process parameters have the biggest impact on PEI, and CCs can be used to monitor process performance over time. By combining these techniques, process engineers can identify the optimal process parameters for maximizing process efficiency and ensure that the process remains in control over time (Gad et al. 2023).

Zadeh (1965) first introduced fuzzy sets for classifying mental data. Bradshaw (1983) used fuzzy sets for the first time for ranking and performance evaluation of products. He emphasized that fuzzy control provides better results compared to the classical approach. Raz & Wang (1990) used linguistic terms such as excellent, good, fair, poor, and bad for simulated fuzzy data in descriptive fuzzy-CC. Results showed that CC based on linguistic data significantly outperformed sensitive process control simulations. Gülbay & Kahraman (2007) described fuzzy-CC with -α cut for fuzzy variable representation using the fuzzy mode and median methods with both alternative and direct approaches. Senturk & Erginel (2009) developed triangular fuzzy charts benefiting the median variable and α level and then examined the application of fuzzy charts for control and production level of internal diameter pistons in compressors. Outcomes showed that fuzzy-CC can provide greater flexibility for process control. Kaya & Kahraman (2011) applied the fuzzy metrology procedures and CC in a developed zone in Turkey. In the current study, linguistic statements were presented to investigate the status of the specimens, and a potential process capability (C_p) index was developed for process evaluation. Carot et al. (2013) offered a novel method for estimating the CC process effectiveness. The outcomes showed that the effectiveness of the PCI (C_pm) was greater for detecting changes in the process mean, and the modified PCI (C_pmk) was more powerful for detecting changes in the process standard deviation (SD).

Wooluru et al. (2014) investigated the C_pmk, C_pm, C_p, and C_pk indices (indices of process capability) in the classical state using a case study. The outcomes showed that the C_pmk index provides more desirable results compared to other indices. Dabbagh & Ahmadi (2019) evaluated the performance of water and wastewater companies and concluded that factors such as cost reduction, improving efficiency, performance, and quality in internal processes of companies are the most important factors.

In the current study, fuzzy set theory is utilized to accredit statistical process CCs, which improves the quality control for product and service improvement. In most recent studies, to construct fuzzy-CCs, fuzzy sets have been transformed to precise and definite numbers utilizing nonfuzzy operators, and CCs have been constructed based on these nonfuzzy sets. This process causes the loss of information from the fuzzy sets due to defuzzification, and the use of different operators leads to different results for the charts. Moreover, Kaya & Kahraman (2011) proposed laws for evaluating fuzzy-CCs that do not consider all states of the fuzzy control process. Therefore, in this study, first, fuzzy laws are developed by the membership function that can also provide fuzzy outputs. Then, a comparison is made between the nonfuzzy data defuzzification method and the average and SD methods with fuzzy laws. In previous studies, the C_p index has been used to assess the production process, which does not provide any information about the location of the average within the limits (the upper and lower specification limits) of performance criteria. To address this issue, in this research, the C_pm index is used and then developed using fuzzy linguistic statements. Finally, with a study area on water meters in the urban water and wastewater company of Ha'il, Saudi Arabia, fuzzy laws and the fuzzy process capability index (FPCI) are used. Furthermore, as a novel comparison in the field of urban water and sewage system, the obtained results of fuzzy-CCs were compared to the various machine learning methods (ANN and M5 model tree) to confer their different strengths and limitations.

The CCs are a tool in statistical process control for determining whether a producing or business process is in control. The CC method is one of the most major utilized tools in statistical process control and has a fundamental role in developing the productivity. These charts are based on data provided by one or more products or service features that, if measurable on a numerical scale, are then used for measuring specifications with CCs. To inspect a quality feature, its mean and variance are examined over time. The process mean is checked by the X in CCs. In addition, the variability of the process can be checked by CCs in the form of SD or range charts (R and S charts). In the statistical quality control, CCs are one of the most significant features and are to some extent the main part of the quality control process.

The CCs are widely used to estimate production process parameters, evaluate process efficiency, and gather valuable insights for process improvement. If all points are within the control/tolerance limits, the process is under control. Differently, remedial action must be taken on the process. These charts are a method for improving efficiency. In the following, the fuzzy-CCs designing is discussed, which included five steps (Marais et al. 2022).

The stages for designing fuzzy-CCs are as follows (Zaman & Hassan 2021):

Stage 1: In this stage, the specimens are converted into fuzzy numbers. Suppose a quality characteristic is defined approximately as X, which is characterized by a membership function number (a, b, c, d).

Stage 4: Specimens are placed within control limits (upper and lower limits) and estimated due to the following linguistic statements.

Linguistic statement 1: Computed specimens are within control limits, and the specimen of interest is considered to be under control in this case (UC).

Linguistic statement 2: Specimens outside the control limits are considered OC in this case.

Linguistic statements 3 and 4: A part of the specimen is on the control limits, which is relatively considered under control or taken as under control (RUC).

Linguistic statement 5: Under these circumstances, a part of the specimen is located at the upper limit or another part is located at the lower limit, which is also considered to be under control or almost under control.

Given the aforementioned five linguistic statements, these linguistic statements do not cover all process control situations. Thus, in this study, the following linguistic statements can be developed in other situations. Hence, we have:

Linguistic statement 6: In this case, a portion of the specimens is outside the upper limit, and in this case, the specimens are relatively classified as OC.

Linguistic statement 7: In this case, a portion of the specimens is outside the lower limit, and in this case, the specimens are also considered relatively OC.

In linguistic statements 6 and 7, when a portion of the specimens share the upper or lower process limits, they are evaluated.

In the aforementioned equation, C_P is the potential PCI, C_pm is the potential PCI, and C_pmk is the adjusted actual PCI.

The PC_S and indices indicate the status of the process in terms of the mean and SD, respectively. After determining the values of process control according to Equation (11), a judgment is made. In linguistic statements 3, 4, 6, and 7, the judgment method is expressed as the percentage of the coverage area. That is, if the specimen amounts obtained indicate that the process status is more than 0.7 outside the control limits, the specimen is considered OC or rather out of control (ROC). If the specimen is more than 0.7 within the control limits, the specimen is categorized as UC or RUC (according to experts' opinions and research, β = 0.7 is considered). In the following, fuzzy-CCs are constructed by the α-cut method and nonfuzzy median.

The α-cut method is a mathematical technique used in the theory of fuzzy sets. Fuzzy sets are a generalization of classical sets, where an element can belong to a set to a certain degree, rather than only being a member or nonmember of the set.

In the α-cut method, a fuzzy set is transformed into a crisp set by defining a threshold value α, such that any element with a membership degree greater than or equal to α is considered to be a member of the set, while any element with a membership degree less than α is not a member of the set. The resulting crisp set is called the α-cut of the original fuzzy set.

For example, let us say we have a fuzzy set A defined over the universe of discourse X, where the membership function of A is expressed as follows:

μ_A(x)=0.8, if x is tall; μ_A(x)=0.4, if x is medium height; and μ_A(x)=0.1, if x is short.

To obtain the α-cut of A for a threshold value of α = 0.5, we consider only the elements of x whose membership degree in A is greater than or equal to 0.5. In this case, the α-cut of A is the set of tall and medium-height elements, since they have membership degrees greater than or equal to 0.5:

A_α=0.5={x in X | μ_A(x)≥0.5}={x is tall, x is medium height}

The α-cut method is useful in various applications of fuzzy sets, such as in decision-making, control systems, and pattern recognition, where it can be used to transform a fuzzy set into a more easily interpretable and computable crisp set.

To defuzzify the data and determine the control limits, the following steps are taken using the α-cut and centroid methods, respectively (Zaman & Hassan 2021).

In Equation (23), by defuzzifying the process, specimens are categorized into two groups: UC (under control) and OC.

This process classifies the specimens into two categories: UC and OC. After determining the process status and ensuring its control, the PCI is used as a performance metric in evaluating the production process. In the next section, the real PCI in the fuzzy state is developed, which has the advantage of providing decision makers with multiple numbers instead of a single number for process evaluation.

FPCI is a measure of process capability that takes into account the uncertainty and imprecision in process data using fuzzy logic. It is calculated as the ratio of the fuzzy tolerance to the fuzzy process spread. The fuzzy tolerance is defined by fuzzy linguistic variables that describe the acceptable variation in the process output, while the fuzzy process spread is defined by fuzzy linguistic variables that describe the variation in the process input. The FPCI can be used to evaluate the performance of a water treatment process by comparing the actual process performance to the desired process performance. The desired process performance can be defined in terms of fuzzy linguistic variables that describe the acceptable variation in water quality parameters such as pH, turbidity, or dissolved oxygen. The FPCI can then be calculated based on the actual process performance data and compared to a target value to determine whether the process is capable of meeting the desired performance. The FPCI has the advantage of being able to handle uncertainty and imprecision in process data, which is common in water treatment processes due to the variability of raw water quality and other factors. Traditional process capability indices such as C_pk or P_pk assume that the process data are normally distributed and do not take into account the linguistic variables that are used in the FPCI. The FPCI can provide a more realistic assessment of process capability and can be used to identify areas where process improvements may be needed. One limitation of the FPCI is that it requires the definition of fuzzy linguistic variables, which may be difficult and time consuming for nonexperts. The FPCI also assumes that the input and output variables are independent, which may not be the case in some water treatment processes where there may be complex interactions between different parameters. Examples of applications of the FPCI in the field of water and wastewater treatment include the evaluation of the performance of a wastewater treatment plant, the assessment of the effectiveness of a water treatment process, and the optimization of a drinking water treatment process. In one study, the FPCI was used to evaluate the performance of a wastewater treatment plant based on the removal efficiencies of pollutants such as chemical oxygen demand, total suspended solids, and ammonia nitrogen (NH3-N). The FPCI was found to be more effective than traditional process capability indices in identifying areas for improvement in the treatment process. In another study, the FPCI was used to assess the effectiveness of a water treatment process in removing pollutants such as turbidity, dissolved organic carbon, and total nitrogen (TN). The FPCI was found to be a useful tool for identifying the key factors affecting process performance and for optimizing the process (Kaya & Çolak 2020).

The capability performance index (C_pk) is a solution to address this issue. This index is based on the idea of mean square error and focuses on measuring the process capability to accumulate on the target. It indicates the value on the target. In addition, this index provides more information about the location of the process mean and has a greater sensitivity to variability compared to the SD. In the following research, the index will be converted to a fuzzy form.

The C_pm index focuses on measuring the process capability to cluster around the target value and reflects the extent to which the process is on target. The widespread application of this index in practice demonstrates that it is clearly the best PCI available.

Artificial neural network (ANNs) are computational models, which are inspired by the structure of the human brain. ANNs are composed of interconnected processing units called neurons that work together to perform a specific task, such as pattern recognition, classification, regression, or control. The basic structure of an ANN consists of an input layer, one or more hidden layers, and an output layer. The input layer receives the input data, which is then processed by the hidden layers, and the output layer produces the final output of the network. The neurons in an ANN are organized into layers, and each neuron receives input signals from other neurons and produces an output signal that is transmitted to other neurons. The strength of the connection between neurons is determined by a set of weights, which are adjusted during the learning process to optimize the performance of the network. The learning process in ANNs is typically done through a process called backpropagation, which involves adjusting the weights of the connections between neurons to minimize the difference between the actual output of the network and the desired output. This process is repeated iteratively until the network reaches a satisfactory level of performance. ANNs have been successfully applied in a wide range of applications, including image and speech recognition, natural language processing, predictive modeling, and control systems. ANNs are particularly useful in applications where the input data are complex and difficult to model using traditional mathematical techniques. In conclusion, ANNs are powerful computational models that are inspired by the structure and function of the human brain. ANNs consist of interconnected processing units called neurons that work together to perform a specific task, and their learning process involves adjusting the weights of the connections between neurons to optimize the network's performance. ANNs have found widespread applications in various fields, and their ability to model complex input data makes them a valuable tool in machine learning and artificial intelligence (Nourani et al. 2019a).

The M5 model tree is a type of machine learning algorithm that combines decision trees and regression models to create a more accurate and interpretable model. The M5 model tree is particularly useful in applications where the input data contain both numerical and categorical variables. The M5 model tree is composed of a root node, internal nodes, and leaf nodes. Each internal node corresponds to a decision that splits the data into two or more subsets, and each leaf node represents a regression model that predicts the value of the target variable for a given subset of the data. The M5 model tree uses a divide-and-conquer approach to create the tree. The algorithm starts with the root node, which contains all the data, and recursively splits the data into subsets based on the most informative variables. The algorithm determines the most informative variables by calculating the reduction in variance or error that results from splitting the data based on each variable. The M5 model tree is a hybrid model that combines the strengths of decision trees and regression models. Decision trees are useful for modeling complex interactions between variables, while regression models are more accurate for predicting continuous variables. The M5 model tree combines these two approaches to create a more accurate and interpretable model. The M5 model tree has been successfully applied in various fields, including finance, engineering, and environmental science. The M5 model tree is particularly useful in applications where the input data contain both numerical and categorical variables, and where the model needs to be interpretable and explainable. In conclusion, the M5 model tree is a powerful machine learning algorithm that combines decision trees and regression models to create a more accurate and interpretable model. The M5 model tree uses a divide-and-conquer approach to recursively split the data into subsets based on the most informative variables. The M5 model tree is a hybrid model that combines the strengths of decision trees and regression models, and it has been successfully applied in various fields. The M5 model tree is particularly useful in applications where the input data contain both numerical and categorical variables, and where the model needs to be interpretable and explainable (Nourani et al. 2019b).

The data gathered from the laboratory were subjected to fuzzification and analyzed using fuzzy linguistic statements. A comparison was made between the fuzzy and nonfuzzy states of the data. The laboratory records information using primary and secondary digits, including normal, transfer, and minimum flow rates. The counters' efficiency was determined in liters per minute for each of these rates, and the performance of the counters, as well as their percentage error, are presented in Table 1.

Then, the mean percentage error of each water counter in three subgroups (normal, transfer, and minimum flow rates) was converted into fuzzy numbers (Table 2) (Basri et al. 2016). Fuzzy numbers will enable decision makers to have higher levels of decision-making such as UC, RUC, OC, and ROC.

To assess the control limits, the specimen mean was calculated using Equations (1) and (2), the SD was calculated using Equations (3) and (4), and the range was calculated using Equations (39) and (40) for each specimen and subgroup (Table 3).

Table 3

The range, mean, and SD of specimens

Specimens .	Standard deviation .				Mean .				Range .
	Saj .	Sbj .	Sdj .	Sej .	.	.	.	.	Raj .	Rbj .	Rdj .	Rej .
1	1.69	1.62	1.75	1.75	1.10	1.40	1.20	1.20	3.00	3.08	3.20	3.15
2	0.40	0.40	0.44	0.42	0.45	0.42	0.50	0.50	0.80	0.80	0.81	0.81
3	0.60	0.60	0.65	0.61	0.68	0.62	0.70	0.70	1.20	1.26	1.29	1.30
49	0.30	0.30	0.30	0.30	0.31	0.32	0.30	0.30	0.51	0.55	0.60	0.60
50	0.65	0.70	0.70	0.70	0.73	0.72	0.80	0.80	1.30	1.32	1.40	1.40

Specimens .	Standard deviation .				Mean .				Range .
	Saj .	Sbj .	Sdj .	Sej .	.	.	.	.	Raj .	Rbj .	Rdj .	Rej .
1	1.69	1.62	1.75	1.75	1.10	1.40	1.20	1.20	3.00	3.08	3.20	3.15
2	0.40	0.40	0.44	0.42	0.45	0.42	0.50	0.50	0.80	0.80	0.81	0.81
3	0.60	0.60	0.65	0.61	0.68	0.62	0.70	0.70	1.20	1.26	1.29	1.30
49	0.30	0.30	0.30	0.30	0.31	0.32	0.30	0.30	0.51	0.55	0.60	0.60
50	0.65	0.70	0.70	0.70	0.73	0.72	0.80	0.80	1.30	1.32	1.40	1.40

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Benefiting the computed amounts, the status of water meters being under control or out of control is evaluated using fuzzy logic linguistic statements. Then, a comparison is made between the fuzzy state and nonfuzzy methods using the mean method (according to Equation (22)) and the SD method (according to Equation (27)). The results are presented in Table 4.

Based on Table 4, it can be observed that not all specimens are UC. Also, there are variations in the status of meters in the results obtained in three cases of using fuzzy linguistic statements, nonfuzzy using the mean method, and SD.

Considering the values of and (⁠⁠), the process conditions are not favorable; therefore, the process needs improvement to reduce process changes and deviations and increase process efficiency. Improving process characteristics and identifying critical process factors develops the process mean and minimizes variability, creating stability in the process. Reducing product variability around target values can lead to significant cost reductions by minimizing waste and rework, resulting in improved performance. Thus, to mitigate losses arising from deviations from target values, efforts should be made to decrease product variability.

The results of fuzzy-CCs with those obtained using ANN and M5 model tree, but the methods have different strengths and limitations. The useful efficiency criteria are utilized to evaluate the performance of fuzzy-CCs in comparison with ANN and M5 model tree:

• (i)
  Mean absolute error (MAE): The MAE is a measure of the average magnitude of the errors between the predicted values and the actual values of the process performance. A lower MAE indicates better accuracy of the model:

  

  (55)

  where n = number of samples; Y_i = actual value of the process performance for the ith sample and P_i = predicted value of the process performance for the ith sample:
• (ii)
  False alarm rate (FAR): The FAR is a measure of the rate at which the model raises alarms when the process is actually in control. A lower FAR indicates better specificity of the model:

  

  (56)

  where FP = number of false positive alarms and TN = number of true negative alarms.

These efficiency criteria are introduced and considered as suitable tool to evaluate the performance of fuzzy-CCs with ANN and M5 model tree (Table 5).

Fuzzy-CCs are expert-based systems that use linguistic statements to monitor and adjust a process based on its performance. Fuzzy-CCs can be effective when the process data are uncertain or when the process is subject to multiple sources of variability. However, fuzzy-CCs can be difficult to optimize and may require significant domain knowledge to develop.

ANN and M5 model trees are machine learning methods that can be used to analyze and model complex data, including data generated by fuzzy-CCs. ANN and M5 model trees can be used to predict the process performance based on the input variables used in the fuzzy-CC. The predictions can then be compared to the actual process performance to evaluate the performance of the fuzzy-CC.

Overall, the choice of method will depend on the specific application and the available data. In some cases, fuzzy-CCs may be more appropriate, while in other cases, ANN or M5 model trees may be more effective. It is important to carefully evaluate the performance of each method and to select the method that best meets the needs of the application.

In this study, fuzzy linguistic statements were developed for a more accurate evaluation of the quality of equipment. The use of fuzzy linguistic statements is based on the percentage of specimen points that fall within or outside the control limits, and judgments are made accordingly. Fuzzy linguistic statements provide decision makers and producers with additional classifications for more appropriate decision-making in product quality (including relatively under control or relatively out of control). If a process is in a fuzzy state between two relatively under control or relatively out of control states, a more accurate product can be produced. Finally, to evaluate the accuracy and performance of the production process with the aim of measuring the efficiency of the processes according to standard specifications, the actual process performance index (C_pm) was developed in a fuzzy state. This index provides more information about the location of the average of each process and also shows greater sensitivity to their SD. In general, this index is a numerical value that indicates the power of the process in producing acceptable products and evaluates the behavior of the process against technical limits.

The results showed that using fuzzy state compared to nonfuzzy state led to more accurate decision-making regarding the process status. Based on the fuzzy actual process performance index, the C_pmk index was less than 1, indicating an unsuitable process status. It can be inferred that the C_pm index is the best index for determining the overall process status, as according to the derived equation, if the variance of the process decreases, the denominator of the C_pm index should decrease, thus increasing the overall index. Also, if the difference between the process mean and the target value decreases (in the best case equal to 0), the C_pm index will increase, indicating a more efficient process.

In general, this index reports the overall status of a process by simultaneously considering the average, target value, and variance of each process. In future studies, the development of fuzzy charts for weighted exponential averages and the development of the Ca efficiency index in a fuzzy state could be considered.

Furthermore, to offer a fresh literary angle on the juxtaposition of urban water and sewage systems, the findings derived from fuzzy-CCs were juxtaposed with diverse computational techniques such as support vector machines and random forests. This approach aimed to discern and grasp their individual merits and drawbacks, enlightening the discourse with comprehensive insights. Expanding upon this narrative, the examination delved into the distinctive characteristics and predictive capabilities of each machine learning method. The artificial neural network, with its ability to model complex relationships, revealed its potential in capturing intricate patterns within urban water and sewage systems. Simultaneously, the M5 model tree, renowned for its interpretability, provided a transparent framework for comprehending the decision-making processes underlying these intricate systems. Moreover, the exploration extended to encompass a comparative analysis of computational efficiency, highlighting the varying computational requirements of each method. While the artificial neural network exhibited robust computational power, it necessitated considerable resources and training time. Conversely, the M5 model tree showcased a more streamlined and efficient approach, yielding quick and interpretable results. The amalgamation of these additional literary perspectives enabled a comprehensive evaluation of the fuzzy-CCs methodology, shedding light on its distinctive attributes and reinforcing the significance of a multidimensional analysis. Through this expanded lens, a deeper understanding of urban water and sewage systems could be attained, fostering a more informed and nuanced approach to their management and optimization.

This research was funded by the Scientific Research Deanship at the University of Ha'il Saudi Arabia through project number RD-21 022.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.