Neural Network-Directed Detection and Localization of Faults in Railway Track Circuits: An Application of Dempster-Shafer Theory

The challenge of flexible recognition is frequently addressed using Nearest Neighbor Classifiers. Despite a limited number of reference symbols per class, this categorization principle offers substantial utility. The primary difficulty lies in incorporating a statistical understanding into the recognition task, which becomes particularly evident when multiple classifiers are deployed for the same task. To tackle this, a three-tiered process has been developed that converts the distance measurements of various classifiers into confidence values. Initially, each distance is transformed into the evidence function of a "specialist", who is accountable for a single reference pattern. Subsequently, the evidence function of an individual classifier is formed by amalgamating the votes of these specialists using Dempster/Shafer's theory of evidence. Finally, the outputs of all classifiers are integrated using the same theory. The efficacy of this method is assessed within the context of online script recognition, with the categorization outcomes compared against a previous method. The proposed approach generates an output vector for each distance classifier, making it suitable for statistically tailored classifiers across diverse pattern-matching contexts [16].

Railroad safety significantly depends on the effective inspection of freight vehicles' mechanical parts for potential malfunctions. However, traditional manual detection techniques have limited performance due to the minuscule size of the mechanical components. Additionally, conventional computer vision technology struggles with simultaneous detection of multiple classes of objects. Inspired by the proficiency of deep-learning-based one-step object detectors, this study introduces an innovative SSD-based detector, MFF-net. This detector leverages three modules to enhance detection accuracy and it collaborates with TFDS to facilitate efficient real-time fault detection in the small mechanical components of railway freight cars [17].

Over time, railway bridges exposed to harsh environments may experience a loss of effective cross-section at critical points, potentially leading to catastrophic failure. This paper proposes the application of Vibration-based Convolutional Neural Networks (CNNs) as a promising deep learning method for damage classification across varying extents and levels of cross section loss due to damages like corrosion [18].

As urban rail networks continue to expand in capacity and speed, concerns about their environmental impact intensify. This study delivers an artificial neural network (ANN)-based analysis of quantitative and qualitative predictors, which are gathered from field data collected during monitoring campaigns of ground vibration induced by light rail traffic in urban areas [19].

This research focuses on utilizing artificial neural networks to predict electric loads for railroad transportation. It investigates methods to predict the electrical demands of trains and other fixed rail infrastructure [20]. Key determinants of rail transportation energy consumption are analyzed, and a mathematical model for power usage, utilizing neural networks, is developed. An F-Fisher-based criterion is also proposed for evaluating neural network models. Moreover, the effective management of differential settlement along traffic lines is crucial. Hence, it's important to establish a rapid prediction model that can consider both vertical and parallel settlement profiles based on fundamental excavation information. A settlement profile along railways due to excavation is demonstrated using an empirical technique and a neural network. A database of 370 case records of field measurements of settlements was consulted to visualize the settlement profile features in three dimensions. The empirical technique employed a Rayleigh function distribution for settlement profiles perpendicular to the excavation, and a Gauss function distribution for vertical profiles. Observed data was also utilized to test and refine back-propagation neural networks. The results indicate that the model can accurately forecast the settlement along railways caused by an excavation [20].

In another study, accurate estimation of recoverable train delay can assist railway dispatchers in rescheduling trains and improving service reliability [21]. The researchers aimed to develop a model for predicting primary delay recovery (PDR) based on operational data from the Wuhan-Guangzhou (W-G) high-speed railway. Initial steps involved identifying significant delay factors such as dwell buffer time, running buffer time, extent of primary delay time, and impact of specific sections. An RFR model is proposed for predicting delay recovery in HSR trains experiencing primary delay. The RFR model outperforms conventional delay prediction models according to several performance analyses. Although the model's foundation was built using W-G HSR data, it could be easily extended to other HSR systems. The proposed approach could benefit professionals and academics interested in disruption management [21].

Finally, the challenge of detecting and categorizing railway switches is presented as a suitable application for Deep Neural Networks (DNN). The technique can be employed for landmark localization, environment perception, and condition monitoring of moving parts. A suitable DNN architecture, an anchor box ratio optimization strategy, and transfer learning compensate for the lack of large training datasets in the railway industry, as compared to the automotive sector. The study approach involves decomposing the problem of fault detection into smaller, capacitor-specific pattern recognition tasks, and the results are consolidated using the Dempster-Shafer theory.

4.1 The Dempster-Shafer Hypothesis

In this study have been created 4256 noise-filled signals at a variety of resistance values for a track circuit that had N=19 trimming capacitors so that we could assess how successful the strategy was. Six hundred and eight of the signals were flawless, while three thousand six hundred forty-eight of them had at least one defective capacitor that had a resistance between and r=1Ω and r=∞ (1). (The capacitor was removed) Estimates of performance were derived using the test set of 1075 signals after the dataset was first randomly segmented into three parts: training, validation, and test. For the purpose of determining whether or not our approach is effective, we compared it to a straightforward technique that makes use of a single multilayer perceptron to make a direct prediction about the location of the defective capacitor (position N+1 is used for fault-free scenarios). The advantages of our approach, which use an extra fusion step to identify and pinpoint a flaw in the system and the construction of as many classifiers as trimming capacitors, are thus quantified.

There was a comparison made between these approaches Here, we use the latter strategy, which offers a malleable framework for dealing with uncertainty and mediating conflicts among many classifiers, as the basis for our investigation. A new method has been revealed in this study to detect the circuit fault in railway systems by prediction of the error occurrence using Dempster-Shafer theory.

The automated train control system would not function without the track circuit. Its primary use is in determining the existence or train tracks in which no cars are allowed to go. The signaling system prevents collisions between trains by having them occupy separate sections of track. On French high-speed lines, the track circuit is an essential component of the track/vehicle transfer system. Coded information, such as the maximum allowed speed along a certain stretch of track due to safety considerations, is sent to the train through a designated carrier frequency.

Several segments of track make up the rail line. Shown in Figure 2 is the track circuit for each of them, which consists of:

– a transmitter at one of the terminals sends out alternating current that varies in frequency;

– the potential transmission line formed by the two railroads;

– there is a receiver at the opposite end of the segment of track that consists mostly of a trap circuit and is used to prevent data from being sent to the next segment of track;

– correction for the track's inductive behavior is accomplished by fine-tuning capacitors linked between the rails at a fixed distance apart. The transmission level may then be increased by performing electrical tuning to reduce the loss of transmitted current. In order to calculate the total number of compensation points, we need to know the length of the track section and the carrier frequency.

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Figure 2. An illustration of an inspection signal and a circuit diagram for a track

The train's wheels and axles create a short circuit with the rails, which are part of the track circuit, allowing for detection of the track. When a train is present in a specific stretch, the signal along the track circuit breaks down because the train's wheels act as a short. To indicate that the area is occupied, the received signal must drop below a certain level. In order to make the transmitted information section-specific and to mitigate the effects of longitudinal interference and transverse crosstalk, four frequencies are used for adjacent and parallel track circuits. Both the sender and the receiver may keep electrical signals separate from those of other circuits on the track by using tuned circuits (capacitance and inductance).

In order to maintain the system's required safety and availability standards, it is crucial that any failures in the track circuit's multiple components (due to, for example, age, environment, or track maintenance operations) be identified as soon as possible. Problems with signaling triggered by the severe attenuation of the sent signal that might result from such failures. Maintainers may be alerted to potential transmission quality issues such track circuit breakdowns via system diagnostics.

An inspection truck fitted with a sensor in front of the front axle used to create a defect diagnostic system based on the vehicle's own data. At each place along the track, the inspection train acts as a "shunt," interrupting the circuit, and allowing the sensor coils to record the short-circuit current (Icc) carrier current level (Figure 2).

Capacitor internal resistance impacted by flaws, which will be the topic of this study. To account for the deviation from the ideal behavior of the capacitor, most of which is due to dielectric aging, we include a serial resistance r that is zero (or weak) while the capacitor is healthy and develops when it is malfunctioning. The system can be realistically simulated, flaws and all, thanks to the development of an electrical model. The lack of fault and the presence of a faulty 10th capacitor are represented by the Icc signals shown in Figure 3. By analyzing the measurement signal, the fault detection and isolation system may ascertain the functioning status of the track circuit.

The two findings shown in Figure 3 form the basis of the suggested technique:

– the Examining: The characteristic shape of an Icc signal is a series of arcs, or catenary curves;

– in the case of a problem, only the segment of the signal that is further from the transmitter will be impacted, while the segment closer to the fault will be untouched.

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Figure 3. An example of a simulated Icc signal with and without a faulty 10th capacitor

Similar to a trim cap: a quadratic polynomial used to approximatively describe an arch.

The suggested technique involves parsing the Icc signal for relevant characteristics and then developing a separate classifier for each trimmer. The results of these individual elementary classifiers are then integrated using the language of belief functions as a representation. After much deliberation, a conclusion is reached as to whether or not a defect exists and where it found.

The core ideas of the Dempster-Shafer theory of belief functions (also known as the Evidence theory) will be elucidated. Shafer is the one who came up with the idea and built this uncertain reasoning framework. The Bayesian probability theory considered as having been extended by this. The transferable belief model is a specific interpretation of the Dempster–Shafer theory that has been offered (TBM). The TBM is built on, point of credibility when ideas are considered plausible and swinish echelon at which choices made.

Only the terms needed for the diagnostic approach described below will be defined here.

The collection of feasible solutions to an issue is the discerning framework; designate it with the symbol; ($\Theta$) - Here, it is assumed that has a finite value:

$\Theta=\left\{\theta_1, \theta_2, \ldots \theta_n\right\}$                   (1)

When we assumed that we have a basic belief assignment (BBA), we mean that we have a function m from;(2$\Theta$) to [0,1]- that allocates a "mass of belief" to each subset A of the discerning frame;($\Theta$), such that

$\sum_{A \subseteq \Theta} m(A)=1$                       (2)

Because it is hard to settle on a specific subset of A, we may use the; (m(A)), represent the basic belief mass to quantify the weight we give to the evidence that supports attributing the belief that we have to set; (A⊆ $\Theta$), represent the quantify the weight. A focused set is any two-element set; (A⊆ $\Theta$), such that; (m(A)≻0), It is claimed that a BBA is normal if and only if; m(ϕ)=0, where; (ϕ), - is the empty set. The amount; (m(ϕ)) might be understood to represent the share of faith that is staked on the possibility that none of the hypotheses in are correct (open-world assumption). In the case of; (m($\Theta$)), m is considered a vacuous BBA since it shows a complete lack of understanding of the issue at hand.

Let's say that we're studying a system whose states are denoted by; ($\Theta$={θ₁,θ₂,θ₃}), with (θ₁),representing the normal state. Let's assume we have evidence that suggests the system is in a deficient state with a confidence level of (0.9), These findings are summed up by the following BBA:

$\begin{gathered}m_1\left(\left\{\theta_2, \theta_3\right\}\right)=0.9 \\ m_1(\Theta)=0.3\end{gathered}$

As the data simply suggests that the system is in a defective condition without directing attention to either; (θ₂ or θ₃)- we make note that the mass; (0.9) is associated with; (θ₂, θ₃)- Due to a lack of supporting data, we do not attribute the remaining (0.3) of mass to (θ₁), It clings to the whole framework of perception while remaining uncommitted.

To combine several BBAs stated on the same frame of understanding, Smets developed the conjunctive rule of combination, commonly known as the unnormalized Dempster's rule. In order to apply this rule, the several BBAs must be grounded on distinct pieces of evidence. Consider two BBAs ((m₁)) and ((m₂))- This conjunctive conjunction, represented by; (m₁∩m₂)-yields a BBA, which is defined for all (A⊆ m₂); as

$m_1 \cap m_2(A) \sum_{B C \Theta: B \cap C=A}\quad m_1(B) m_2(C)$                      (3)

Associative and commutative, this rule is quite useful. As a result, we have two options for directly merging n BBAs m₁, …, m_n: (a) using (3) iteratively, or (b) using the method below.

$m_1 \cap \ldots m_n(A)=\sum_{B_1 \cap \ldots \cap B_n=A}\quad m_1\left(B_1\right), \ldots, m_n\left(B_n\right)$                   (4)

The value of ($m(\phi)$), which is applied to the empty set, seen as a quantitative representation of the tension between the two inputs.

Case 2 Let's say that more information indicates the system is not in state (θ₂), with a confidence level of (0.7), continuing the scenario established in Case 1. The following BBA might be used to portray this evidence:

$\begin{gathered}m_2\left(\left\{\theta_1, \theta_3\right\}\right)=0.7 \\ m_2(\Theta)=0.4\end{gathered}$

The BBA (m=m₁∩m₂), is defined as the sum of (m₁) and (m₂) which is obtained by using (3).

$\begin{gathered}\left(\left\{\theta_3\right\}\right)=0.8 \times 0.6=0.48 \\ m\left(\left\{\theta_2, \theta_3\right\}\right)=0.8 \times 0.4=0.32 \\ m\left(\left\{\theta_1, \theta_3\right\}\right)=0.2 \times 0.6=0.12 \\ m(\Theta)=0.2 \times 0.4=0.08\end{gathered}$

The multitudes are still adding up to one, as we can see.

One way to account for the credibility of a source is to apply a discount rate; (1-α) -on the original BBA m, where; (α)-is a coefficient between zero and one. The discount rate increases as the level of trustworthiness decreases. With this BBA in hand ^αm

$\begin{gathered}{ }^\alpha m(A)=\alpha(A) \forall A \subset \Theta \\ { }^\alpha m(\Theta)=1-\alpha[1-m(\Theta)]\end{gathered}$                   (5)

By shifting some of the weight of belief from; (α) to ($\Theta$)- the information content of the BBA suffers. The trustworthiness of the source is represented by the coefficient; (α), When (α=1) holds, the belief function remains unaltered, and we have complete faith in the credibility of the source. As approaches 0, the resultant BBA is more like the empty belief function; (m($\Theta$)=1), -which means that the source's information is ignored. There are a number of approaches that have been presented to use data to get the best possible answer for α.

Case 3 the following scenario: we are going to flip a coin, and our expectation of the result is shown on the frame; ($\Theta$={Head, Tails})- If the coin is honest, we may express our faith in; ($\Theta$)-using the BBA below.

$\begin{aligned} & m(\{\text { Head }\})=0.5 \\ & m(\{\text { Tails }\})=0.5\end{aligned}$

A Bayesian BBA is a kind of BBA in which the focus sets are all unique. This type of BBA is comparable to a probability distribution. Let's pretend for a moment that we have just a; (0.8) -level of confidence in the fairness of the coin. In such case, a discount off the aforementioned BBA is in order; (1-α=1-0.8=0.4) -The BBA after discounting is

$\begin{gathered}{ }^\alpha m(\{\text { Head }\}) 0.8 \times 0.5=0.4 \\ { }^\alpha m(\{\text { Tails }\}) 0.8 \times 0.5=0.4 \\ { }^\alpha m(\Theta)=0.4\end{gathered}$

Some of the weight seems to have been moved to the discernment framework, which a result of a misunderstanding of the odds.

The credal level is where one's beliefs are entertained, whereas the pianistic level is where one's BBAs are converted into probability distributions and used to form decisions in line with the idea of maximal expected benefit.

When a decision has to be made, the BBAs are transformed into probabilities. The BBA m is transformed using the pianistic procedure defined above, and the resulting pianistic probability function, BetP is then constructed.

$\operatorname{BetP}(\theta)=\sum_{\{B \subseteq \Theta, \theta \in B\}} \frac{1 m(B)}\quad{|B| 1-m(\phi)}\quad \forall \theta \in \Theta$               (6)

where, the cardinality of (B) - is denoted by |B|, - it is taken for granted that (m(ϕ)≺1)

For all practical purposes, this definition assumes that; (m(B)) -is uniformly distributed among; (B) - constituent parts in the absence of any other information (B⊆$\Theta$) - For decision making in accordance with traditional Bayesian decision; theory, the pianistic probability is a useful classical measure of probability.

$\begin{gathered}{BetP}\left(\left\{\theta_1\right\}\right)=\frac{0.12}{2}+\frac{0.09}{3} \simeq 0.08 \\ Bet P\left(\left\{\theta_2\right\}\right)=\frac{0.33}{2}+\frac{0.09}{3} \simeq 0.2 \\ BetP\left(\left\{\theta_3\right\}\right)=0.49+\frac{0.33}{2}+\frac{0.13}{2}+\frac{0.09}{3} \simeq 0.75\end{gathered}$

Case 4 recalling the BBA m from equation above, represented the pianistic probability function.

4.2 Methods for diagnosis

A high-level schematic of the proposed diagnostic system's whole architecture is shown in Figure 4. The track circuit is shown as the system Σ, and the N subsystems; (S₁, …, S_N) -are the trimmers. The application's stated goal is to identify the location of the faulty capacitor should a problem be found in any of these capacitors. One faulty capacitor is thought to be the maximum. The measured information for capacitor; S_i, -is shown as I_i, in Figure 4. The vector of information; (I₁, …, I_N) - consists of the points along the curve in Figure 2. As demonstrated by the arrows connecting; S₃ and (I₃, …, I_N), a failure in capacitor; (S₃)- would alter the measurement data (i. e., the shape of catenary curves) pertaining to all capacitors downstream.

A diagram of the proposed system's ; (N)- local classifiers is shown in Figure 4; ((D₁, …, D_N) -), - A priori, each local classifier ;(D_i)- tasked with learning to make a prediction about the health of either capacitor; (S_i) or the health of the capacitors upstream from it. The terms "local coding" and "distributed coding" will be used to describe these two methods from now on. Although both of these strategies make sense on the surface, we'll go into them more below.

4.jpg

Figure 4. An overview of the fault isolation and localization methodologies guiding principles

To do this, feature values are taken from inspection data and fed into a classifier (D_i), which then calculates a probability based on the data; (P_i). After that, the preferred encoding modifies the output into a Dempster-Shafer mass function m_i. In order to apply pignistic probability to identify whether or not a defect exists and, if so, where it is located, we must first combine the N mass functions into a single mass function m using Eqns. (3), (4), (6).

4.3 Determining important characteristics

Figure 3 depicts an Icc curve, which is the shape the inspection data take, The space for measurements is thus defined. Segments of the Icc curve have an arched shape. These arcs stand in for individual capacitors that serve as fine-tuners. In order to construct a sparse representation of the data, each segment was approximated by a quadratic polynomial, and its three coefficients; (a_i, b_i and c_i) - were used as features in the process (Figure 5). Hence, a total of (3N)- characteristics were used to characterize the whole Icc curve.

Together with the i th capacitor for fine-tuning. Hence, classifiers' feature spaces are nesting.

In this study, we focus on probabilistic classifiers such as those based on neural networks or decision trees. As was mentioned before, every local classifier D_i has the potential to be educated to recognize defects in either capacitor S_i or any other capacitor that lies between S₁ and S_i As can be shown in Figure 6, the encoding of neural network outputs is affected because of the nature of the learning job that neural networks are assigned when they are used as classifiers. Let's pretend that capacitor S_i (at i=3 in Figure 6) has developed a defect. We shall examine two scenarios:

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Figure 5. Taking out features

– in order to put into practice, the local coding that is shown in Figure 6, we would begin by setting the intended output for classifier D_i, to the value 1, while leaving the desired outputs for the other classifiers at the value 0.

– with dispersed coding (Figure 6 coding 2), classifiers D_j with j ≥ i have 1 set as their intended output and 0 set as the outcome of every other classifier.

As a matter of first principles, distributed coding seems to be the best option for taking into consideration the physical proximity of individual systems. Yet, this research will investigate and test both approaches.

Due to the fact that classifiers D_i= (i=1, ..., N) only analyze regional data, their results must be averaged before a definitive conclusion can be drawn about whether or not a faulty capacitor is present and where it found. Classifier results are represented in Dempster-Shafer form, and then merged using Dempster's rule (3), (4).

A Dempster-Shafer model cannot be constructed without first defining the context in which judgments are made, in this particular instance, we will utilize set $\Theta$= {1, ..., N, N+1}, where N refers to the total number of capacitors included inside the rail circuit. Each unique pair {i}, i =1, ..., N+1 represents a distinct potential fault location. The lack of a problem corresponds to the virtual position N+1 In this section, we will begin by discussing the issue of local coding, and then we will go on to discuss the scenario of distributed coding, detailing how the outputs of classifiers are translated into mass functions and integrated.

4.4 Combining several classifiers using neighborhood codes

Using local encoding, if classifier D_i, returns a 1, then the failure is assumed to be in capacitor S_i Figure 6. And if there isn't, then something's wrong.

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Figure 6. The N classifiers are trained using one of two output coding techniques

If capacitor S₃ is faulty, the N classifiers are trained using one of two output coding techniques. Just the intended result of classifier D₃ is coded as a 1 using local coding (code 1).

All classifiers D_j with j ≥ 3 have their targets set to 1 using dispersed coding (coding 2) in a capacitor of value S_j whereby either j ≠ i or {i} holds true. It follows that BBA mi, with the singleton {i} and its complement {i}/ Θ as its focal sets, used to represent D_i,'s classification results.

$m_i(\{i\})=P_i, \quad m_i(\Theta /\{i\})=1-P_i$                  (9)

Characterized by the fact that the output of classifier D_i, is denoted by P_i $\in$[0,1].

When applying the unnormalized version of Dempster's rule (4) to the combination of the N BBAs m₁, …m_N the result is a BBA m that has N+1 focal sets. These sets are referred to as singletons and the empty set respectively. The following is a transcription of the phrase that may be found on this BBA:

$\begin{gathered}m(\{i\})=P_i \prod_{J \neq i}\left(i-P_j\right) \forall i=1, \ldots, N \\ m(\{N+1\})=\prod_{i=1}^N\left(1-P_i\right)\end{gathered}$

$m(\phi)=1-\sum_{i=1}^{N+1} m(\{i\})$                  (10)

Let us suppose for the moment that we are going to reduce the output of each classifier by a rate of 1-α_i with 0≤ α ≤1. As will be shown in Section 5, the discount rate figured by using the classifier error as a determining factor. After this, the value of BBA m_i, which represents the output of classifier D_i, is

$m_i(\{i\})=\alpha_i P_i, \quad m_i(\Theta /\{i\})=\alpha_i\left(1-P_i\right), \quad m_i(\Theta)=1-\alpha_i$                   (11)

An unnormalized variant of Dempster's rule is used to combine these N BBAs into a BBA m, which described as follows.

$\begin{gathered}m(\{i\})=\alpha_i P_i \prod_{j \neq i}\left[\alpha_j\left(1-P_j\right)+\left(1-\alpha_j\right)\right] \forall i=1, \ldots, N, \\ m(\{N+1\})=\prod_{i=1}^N \alpha_i\left(1-P_i\right)\end{gathered}$

$m(A)=\prod_{i \notin A} \alpha_i\left(1-P_i\right) \prod_{i \in A}\left(1-\alpha_i\right) \forall A \subset \Theta, N+\in A, 1 \prec N$                 (12)

The remaining portion of the mass being allotted to the set that is empty. If the coefficient indexes are negative or 0, as is often believed to be the case, then the products in the preceding phrases disappear.

Let's assume that the classifier D_i 's output is 0, and that distributed coding was employed. Thus, either there is an issue between S_i+1 and S_N or there isn't a problem there. On the other hand, if classifier D_i 's output is 1, then it is clear that there is an issue with the link between S₁ and S_i. Hence, the following BBA used to describe the data provided by the classifier D_i.

$m_i([1, i])=P_i, m_i([i+1, N+1])=1-P_i$                 (13)

where, [1, i] represents the range of numbers from 1 to i and P_i $\in$[0,1] represents the output of the classifier D_i, as in the previous illustration.

When Dempster's rule (4) is applied to the combination of m₁, …m_N the following BBA is produced:

$\begin{gathered}m(\{i\})=\prod_{j=1}^{i-1}\left(1-P_j\right) \prod_{k=i}^N P_K \forall i=1, \ldots, N, \\ m(\{N+1\})=\prod_{i=1}^N\left(1-P_i\right),\end{gathered}$

$m(\phi)=1-\sum_{i=1}^{N+1} m(\{i\})$               (14)

The BBA produced by the ith classifier, m_i, changes to reflect the discount rate of 1-α_i applied to all BBAs reflecting classifier D_i's output, as previously.

$m_i([1, i])=\alpha_i P_i, \quad m_i([i+1, N+1])=\alpha_i\left(1-P_i\right)$

$m_i(\Theta)=1-\alpha_i$             (15)

Dempster's rule, in its unnormalized version, was applied to this set of BBAs, and the outcome is what we have right now.

$\begin{gathered}m(\{1\})=\alpha_1 P_1 \prod_{j=2}^N\left[\alpha_j P_j+\left(1+\alpha_j\right)\right], \\ m=(\{2\})=\alpha_2 P_2 \alpha_1\left(1-P_1\right) \prod_{j=3}^N\left[\alpha_j P_j+\left(1-\alpha_j\right)\right], \\ m(\{i\})=\alpha_i P_i \alpha_{i-1}\left(1-P_{i-1}\right) \prod_{j=i+1}^N\left[\alpha_j P_i+\left(1-\alpha_j\right)\right] \prod_{K=1}^{i=2}\left[\alpha_K\left(1-P_K\right)+\left(1-\alpha_K\right)\right] \forall i \in[3 i N], \\ m(\{N+1\}) \alpha_N\left(1-P_N\right) \prod_{j=1}^{N-1}\left[\alpha_j\left(1-P_j\right)+\left(1-\alpha_j\right)\right], \\ m([i, j])=\alpha_{i-1}\left(1-P_{i-1}\right) \alpha_j P_j \prod_{m=i}^{j-1}\left(1-\alpha_m\right) \prod_{K=j+1}^N\left[\left(1-\alpha_j\right)+\alpha_K P_K\right] \prod_{l=1}^{i-2}\left[\alpha_1\left(1-P_l\right)+\left(1-\alpha_i\right)\right] \forall i \prec j[[i, j] \subset \Theta, \end{gathered}$

$m(\Theta)=\prod_{i-1}^N\left(1-\alpha_i\right)$                 (16)

Empty set receives the remaining fraction of the mass allocation.

For example, let us take into consideration a made-up system that is comprised by N=3 different subsystems. The discerning framework is denoted by the equation Example. Let us take into consideration a made-up system that is comprised by N=3 different subsystems. The discerning framework is denoted by the equation $\Theta$={1, 2, 3, 4}.

Classifier $\mathrm{D}_1: m_1(\{2,3,4\})=1-p_1$,

$m_1(\{1\})=p_1 \text {, }$

Classifier $\mathrm{D}_2: m_2(\{3,4\})=1-p_2$,

$m_2(\{1,2\})=p_2,$

Classifier $\mathrm{D}_3: m_3(\{4\})=1-p_3$,

$m_3(\{1,2,3\})=p_3 \text {, }$

Hence, the total BBA is

$\begin{gathered}m(\{1\})=p_1, p_2 p_3, \\ m(\{2\})=\left(1-p_1\right) p_2 p_3, \\ m(\{3\})=\left(1-p_1\right)\left(1-p_2\right) p_3, \\ m(\{4\})=\left(1-p_1\right)\left(1-p_2\right)\left(1-p_3\right), \\ m(\phi)=1-\sum_{i=1}^4 m(\{i\})\end{gathered}$

Without taking into account discounts, the above are the various BBAs that are supplied by the classifiers.

Experts in railway management and a specialist in railway maintenance reviewed the comparative findings. The potential reductions in both the work for maintenance staff and the obstruction for train operators have been welcomed by everybody. The new model produces a favorable timetable, although one that requires more nights overall. Five nights a week, on average, are set aside for scheduled maintenance activities, however this affects only a fraction of the network. This suggests that there is still opportunity for new endeavors in other areas. Since the hindrance points cannot be simply translated into these numbers, a thorough examination of the schedule is required to determine the exact number of trains hampered. When the electricity of the overhead wire is turned off for repair, the number of trains that are impeded is calculated based on the number of trains required to start the schedule on a typical workday.