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1 INTRODUCTION

In recent decades, many fields of knowledge have

implemented data collection techniques and data

analysis methods. Among many, machine learning is

one of the most widely applied methods for data self-

analysis. Today, machine learning has concrete

examples in many engineering systems and

intuitively merges concepts such as cyber-physical

systems and cognitive computing into complex and

smart platforms. The previous statement is, in some

ways, the result of advances in the levels of data

storage and data acquisition in engineering systems.

In today's industry, it is common to find physical

engineering systems with hundreds of sensors that

collect data on the performance of their operational

functions. Therefore, it is expected to find hundreds of

contributions and applications that use the

information generated to improve the current state of

engineering systems, qualitatively or quantitatively.

As an example of engineering systems, cooperative

overhead cranes are critical devices in many

continuous production industries. They are usually

Degradation Data Self-Analysis Layer for Integrated

Maintenance Activities

J. Szpytko & Y. Salgado Duarte

AGH University of

Krakow, Kraków, Poland

ABSTRACT: Reliability-oriented approach based on Monte Carlo simulations is a well-established methodology

for coordinating maintenance activities of any technical system. Usually, coordination is conducted using

holistic performance indicators, which are obtained from the convolution between the stochastic system

availability and the system service required in a time horizon of t. Specifically, the system stochastic availability

modeling is composed of the degradation process due to the system operation and the planning of the

maintenance activities needed to keep the system operating at the desired standards. In the case of the

degradation modeling process, given its random nature, it is addressed with predictions, which in practice,

consist of generating random samples of the stochastic degradation processes from probability distributions,

and the parameterization is usually estimated by fitting the distributions to historical degradation data for each

technical component considered. Crucial to forecasting accurate performance indicators is the use of up-to-date

information, i.e., the self-update of historical degradation data. In this paper, to address accurate performance

indicators, we propose using the machine learning approach to update the adaptable model layers affected by

changes in the degradation data. The paper's case study is an overhead crane system of a hot rolling mill

process in a steel plant, which operates under hazardous conditions and continuously. We focus on overhead

cranes because they are critical components of production processes. The paper's subject is validating the

performance of a self-analysis layer, which processes the degradation data of the analyzed technical devices.

The engineering solution ensures well-processed inputs for the problem of coordination of maintenance

activities of overhead cranes, which is the object of the study of this research.

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installed in processes with hazardous conditions and

difficult access. Each overhead crane manages a

section of the ongoing process, is fixed on the

warehouse roof, its movements are limited to a

specific working range, and its criticality comes from

the sometimes-unexpected unavailability of an

overhead crane that can stop an entire production

process.

Today, these cranes are changing their design due

to the new needs of the current industry. For example,

the new generation of overhead cranes is equipped

with sensors to collect information. However, the data

generated by sensor systems are often not exploited

adequately. When talking about cooperative overhead

cranes, the two main research fields studied are load

operation control, such as the examples [9], [11], and

[14], and maintenance, such as the examples [18] and

[19]. The reason researchers focus their attention on

these fields is simple: incorrect load control and weak

maintenance are the most common causes of

overhead crane failures in today's industry. Some

examples of studies that analyze the root cause of

crane failures are [3], [10], and [15]. Even when cranes

with different working conditions and functions are

analyzed in the cited contributions, as a connection

between them, we find that weak maintenance cycles

and wrong load control are the causes of crane

failures.

Focusing on maintenance strategies, a well-

established approach to designing suitable

maintenance strategies is reliability analysis, as it

allows us to measure the risk of possible failures in

overhead crane systems and incorporate and evaluate

potential risk scenarios for the system. Examples of

contributions in this field are [2] and [8]. However, a

weakness of the reliability analysis approach is the

reliance on data to predict realistic scenarios. In this

paper, we propose to integrate reliability-based

maintenance coordination and data analysis methods

by offering an engineering solution for a cooperative

overhead crane system operating in a steel plant,

which is embedded in an integrated digital platform,

and somehow manage to address the weakness of

reliability analysis.

The research conducted here arises from a local

request from the maintenance department of a steel

plant with organizational issues. Although it is a local

solution, the achievements provide a practical

example of how a reliability model can be tailored to

address a local solution using data generated in the

daily work of the maintenance department. The idea

presented in this paper aims to propose, through a

practical example, how an existing overhead crane

system can be adapted to the digital era and

contribute to other data-driven applications. Although

the proposed model is its broadest conception, it is an

oriented engineering solution for coordinating

maintenance activities; the paper's subject is the

Degradation Data Self-Analysis (DDSA) layer, which

ensures the accurate and robust treatment of

degradation data.

Focusing on the performance of this layer has a

well-justified reason. For instance, the layer avoids

human intervention in online filtering and estimation

of the degradation-related parameters of the

reliability-oriented optimization model. In addition,

the layer ensures a comprehensive mathematical and

technical connection between the degradation data

due to the system operation and the optimization

model parameters, considerably decreasing errors

while ensuring that up-to-date information is always

used when running the model. In addition, the

selected frequency models, outputs of the layer, allow

one to prolong in time the degradation due to the

operation of the system, which in turn provides for

evaluation with scenarios of how the planning process

will work in the maintenance department of the steel

company when unexpected failures are considered.

That said, the criticality of this layer in estimating the

performance indicator, a variable that holistically

measures the quality of the maintenance coordination

process, is evident.

The DDSA layer is the process of filtering,

processing, and storing the degradation data of the

proposed engineering solution. The source of raw

degradation data involved in this process comes from

two systems used in the daily work of the steel

company: SAP (Systems, Applications & Products in

Data Processing) and SCADA (Supervisory Control

and Data Acquisition). The proposed paper is an

extended and more in-depth version of the work

presented by [18]. In this paper, we test and validate

the DDSA layer, which fills the decision gaps

presented in previous work. Moreover, with the

validation, we support the conclusions for a specific

scenario and all scenarios analyzed afterward.

Specifically, in this paper, we extend and expose

the functional connection between the system

components, making clear the relevance of the DDSA

layer in the developed model and the algorithm

implemented to execute the fitting process. In

addition, we isolate, test, and show the process of

fitting simple distributions in practice with actual

data. For illustrative purposes, the selected example

and the described data come from a specific 50-ton

overhead crane with 59,501 hours of continuous

operation (6.8 years) and 355 hours of the replacement

or repair process. Also, during the validation, we

include the analysis of the impacts introduced by the

changes in the data, which consists of changing the

input data in a controlled way and re-evaluating the

fitting selection process. Mainly two experiments are

conducted; the first consists of contaminating the

input data with new values generated from the final

selections, and the second one consists of removing

the last records of the dataset.

Once the scope of the paper has been declared, the

remaining sections of the document are presented as

follows: First, the context of degradation data is

discussed, which highlights the value of the DDSA

layer to ensure accurate and robust treatment of the

degradation data. Then, using a 50 tons overhead

crane as a case study, the DDSA layer is tested in

practice. Finally, the conclusions highlight the main

results of this contribution and the connections to

future work.

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2 MATERIALS AND METHODS

2.1 Maintenance – Degradation Modelling

In continuous-process productions, common

indicators for monitoring the performance of a

technical system, which provides a service or

supports a production line process, are based on the

system's availability or its components over time. The

availability A(t) of a technical system over time can be

impacted by two main reasons: planned maintenance

or unexpected failures. From the point of view of state

diagrams, a technological system can have three

possible states: available, unavailable due to

unexpected failures, and unavailable due to planned

maintenance.

In the case of maintenance activities, imagining a

sequence in time, maintenance schedules can be

represented by the variables M maintenance-start-

time and D maintenance-duration-time, where M =

1, m2, …, mk}, D = {d1, d2, …, dk} and k number of

maintenances. Both variables M and D are planned

and can be selected according to maintenance

strategies in the analyzed process.

In parallel, degradation can be treated similarly,

knowing that degradation is inherent in a technical

system. The degradation over time can be represented

by the variables F time-to-failure and R time-to-repair,

where F = {f

1, f2, …, fn}, R = {r1, r2, …, rn} and n is the

number of failures. In particular, variables F and R are

considered random variables. In the case of F,

unexpected failures are related to component

degradation due to system operation and are

unpredictable in almost all cases. In the case of R, the

time depends on the magnitude of the failure, the

expertise of the workers who repair the failure, and

the logistics behind it. In conclusion, it is also defined

as a random phenomenon with certain thresholds.

Since the F and R variables are defined as random, the

system's availability A(t) over time is defined as a

stochastic process.

All the variables M, D, F, and R presented above

are times and can be defined as non-negative

variables. Maintenance scheduling is usually a

planning process, i.e., depending on the planning

window (monthly, quarterly, or annual), planners

propose the sequence to be executed in the next

window. Coordination of the maintenance process is

crucial to ensure the life cycle of any system, and its

optimization is a daily task in any technical system. It

is not surprising that availability A(t) plays an

essential role in maintenance coordination and

appears in several approaches used to coordinate this

process, such as the examples of [6], [12], [7], [4], and

[1]. In approaches in which the modeling of the

availability A(t) is included, it is also necessary to

make decisions related to the random phenomenon,

that is, the random variables F and R. The usual

approach is to assume the progression of the

degradation process in the planned window based on

the historical degradation data of the analyzed

system. The way to include degradation is to fit some

model to the historical degradation data, resulting in

one model representing F and another for R, and then,

based on the fitted model, ~F potential failures and ~R

repair times are simulated. The fitting and simulating

processes are performed for each component of the

analyzed system. The simulated values are

convoluted with the proposed sequence of planned

maintenance, which allows for modeling the system

availability A(t) and assessing the maintenance

scheduled impact.

Focusing our attention on variables F and R and

the modeling of the fitting process, the problem to be

solved in this case is to find the best model to

represent the historical data, which is then used to

simulate a potential degradation process. The only

known information is that we are dealing with non-

negative variables, and the stochastic process A(t) is

continuous in time. Fitting models can be divided into

non-parametric and parametric.

A contribution supporting this classification is [5],

which tested the performance and covered

convergence issues during the fitting process in both

approaches. The research in the above contribution

concludes an introductory statement that the

convergence of the fitting process depends on the

features of the data. They end that the non-parametric

approach should be used when the data is dense;

otherwise, parametric is the way to go. In our case,

degradation data, the subject of study here, are

usually sparse; even assumptions are sometimes

needed for highly reliable systems. Therefore, given

the features of the data, parametric models are the

tentative choice for our problem. However, the

definition of dense data is not well defined. The

historical degradation data, represented in our case by

F and R, are non-negative random variables. These

recorded values can be assumed as samples of a

generating function, which means the degradation

process, an

d in which the order of the failures is

irrelevant. Given the constraints and features of the

degradation data, the F and R variables are usually

modeled with frequency models. Knowing also that F

and R are used to model, in our case, a continuous

stochastic process, we further restrict the option space

to continuous frequency models.

A comprehensive list of parametric frequency

models is continuous probability distributions. It

should be noted that parametric frequency models

were used in all references cited above, which are

related to the concept of A(t) availability discussed.

On the other hand, non-parametric ones are kernels,

splines, neural networks, a simple frequency

histogram, and the empirical distribution, which have

been applied in [22] and [16]. The selection of the final

approach and model to be used is discretionary and is

guided by the individual viewpoint of the research

conducted. Based on expertise in the field and the

references cited, we consider parametric and

continuous frequency models to be the models that

have the most practical applications, are the most

transparent, and consequently achieve the expected

results.

Deciding on the set of models to be used to model

the variables F and R does not end the discussion. A

parametric model (as well as a non-parametric one) is

estimated based on the available historical data, and it

is well-known that the inference error decreases when

the degradation data are more representative of the

analyzed phenomenon. Consequently, when more

degradation data are available, a good strategy is re-

estimating the model's parameterization, always

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seeking to get closer to the actual phenomenon. Here,

machine learning approaches play an essential role in

the discussion. In current practice, integrating data

monitoring and processing is an important goal. The

main idea is to create smart data processing layers to

update the models' parameterization based on the

newly available data. Several contributions in this

field, such as [17], [21], [13], and [20], have proposed

practical applications, and the research presented in

this paper is another contribution in the same

direction.

Here, we propose a smart data-driven algorithm to

find the most suitable parametric model for the

degradation variables F and R. The algorithm is

encapsulated in the DDSA layer introduced in the

previous section, which runs online and is fully

connected to the SAP-SCADA systems, which makes

it possible to update the data when a scenario is run

on the integrated digital platform. The entire DDSA

layer tested and validated in this paper is

encapsulated in functions implemented in MATrix

LABoratory (MATLAB) that work without interaction

with the model user, and we insist that the validation

of their performance is important in this research.

Finally, before finishing this section, in section 2.2,

we present the description of the algorithm

implemented in MATLAB, which is used in the DDSA

layer, summarizing the standardization efforts of the

fitting process. In section 2.3, we contextualize the

relevance of the fitting process in the integrated

maintenance platform by presenting the functional

modeling at the component and system levels,

discussing, and describing the connection between

them.

2.2 Fitting Single Distributions Algorithm

1. For each i-th overhead crane, the sequence of F

i and

i is filtered from the SCADA-SAP systems by

means of a unique identifier (ID). By construction,

both random number vectors have the same

length.

2. Independently, for each streamed sequence, the

following steps are applied:

− Given a random variable sample ~X = (x

1, x2, …,

n), in our case, either the F or R random

sequence filtered and a set of k-th predefined

and preselected single continuous distributions,

we apply for each k-th case the following steps:

− Fit the data (either F or R) to the k-th single

distribution by maximizing the log-likelihood

function ln 

n(x|θ), i.e., the negative logarithm

value of the product of the probability of the

sample data (X), given the parameters θ of the

distribution. If the fit does not converge for the

given parametric distribution, the process ends;

otherwise, it continues as follows:

− Save the k-th index for ID purposes, the

estimated negative log-likelihood value

when the maximum likelihood estimation

(MLE) method was applied (to record the

process performance), and the parametrized

single distribution structure.

− II. Apply two goodness-of-fit tests to

analyze the results of the fit: first, the one-

sample non-parametric Kolmogorov-

Smirnov (KS) test, defined as

( ) ( )

( )

maxD Fx Fx= −

where

( )

is the empirical cumulative

distribution function of the data and F(x) is

the cumulative distribution function of the

fitted single parametrized distribution; and

then the Anderson-Darling (AD) test,

defined as

( ) ( )

( )

n Fx Fx wxdFx

∞

−∞

−

∫

where n is the number of data points in the

sample,

( )

and F(x) as described above,

and w(x) is a weight function defined as

( ) ( ) ( )

( )

1wx Fx Fx

−



= −



In the case of the AD test, the data is the

ordered sample.

− Apply the Akaike information criterion

(AIC) defined as

(

)

2log 2

−+

where log L(

) denotes the optimal log-

likelihood objective function value, and k is

the number of parameters of the single-fit

distribution.

− Estimate the parametrized fit distribution

structure's theoretical mean µ and variance

− Store the frequency of the data (number of

records) and the sum of the data (in the case

of variable F, we are storing the hours of

operation).

− Add a logical check value following the rule:

if the k-th fitted distribution has finite µ and

, Flagk = 1; otherwise, Flagk = 0.

− Reject distributions with Flag

k = 0 (fits with

infinite mean or variance).

− If the data records are less than ten, the best fit

is the exponential distribution. Otherwise, the

best fit is the k-th distribution with minimum

AIC value.

3. End of the fitting process algorithm.

2.3 Functional Modelling of the Integrated Maintenance

Platform

Looking at contextualizing the integrated digital

platform, we can say that the platform is adapted to

support online maintenance activities coordination. In

the system analyzed, in our case, a set of overhead

cranes in a selected manufacturing industry, the

maintenance department, which uses the digital

platform for planning purposes, has risk

circumstances when unanticipated failures occur

during scheduled maintenance activities.

The platform was created to reduce the

overworking days in the maintenance department by

minimizing the convolution between scheduled

maintenance activities and unanticipated failures. The

impact of risk situations exists because the set of

overhead cranes is critical for moving demanding

loads on the production line of the manufacturing

industry. This undesirable situation stresses workers

because they work under pressure during the

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interaction between scheduled maintenance activities

and unanticipated failures.

The integrated digital platform comprises three

blocks of self-analysis: data filtering and synthesis,

model simulation through scenario evaluation, and

optimization layer to coordinate maintenance

activities. Each block is supported by independent

algorithms implemented in MATLAB. Also, the data

sources are mainly two professional systems: SAP

(Systems, Applications, and Products in Data

Processing) and SCADA (Supervisory Control and

Data Acquisition). Integrating all blocks with a

dynamic window that visualizes the model

performance conforms to the digital platform and

connects all blocks through the integrated digital

platform.

The data processing (filtering and synthesis) block

ensures the existence of all the necessary parameters

to evaluate the modeled scenario. Therefore, as we

can see, each time a scenario is loaded for evaluation,

the model parameterization is calibrated with the

updated information.

In addition, the digital platform provides, for a

given proposed scenario to be evaluated, the option of

using an optimization algorithm (heuristic in our

case) to find the best maintenance activities schedule

for the scenario. In this case, the optimization

algorithm minimizes the interaction between

unexpected failures and the maintenance activities

scheduled for the set of overhead cranes considered.

Having briefly contextualized the integrated

digital platform, we focus on the first block, the data

processing block, specifically the degradation data

processing.

Previously, we introduced how the measurement

of system availability A(t) enables coherent

coordination of the maintenance process. Now, we

intend to describe and deepen the integrated

platform's functional modeling point of view. For this

purpose, we divide the description into two levels,

component, and system, making visually clear the

inner workings of the digital platform. However,

although the definition has been divided, everything

is connected. Furthermore, it is necessary to

emphasize that the platform operates without human

intervention in data processing, and the platform user

only interacts with it through the system-level settings

of the parameters of the assessed scenario.

Fig. 1 shows the model composition of each

component considered on the digital platform, in our

case, overhead cranes.

The stochastic functional capacity of each overhead

crane z

1 = f (t | θ1) is composed of the convolution

between the degradation process C

D = f (t | θ) and the

planned maintenance process C

M = f (t | θ), where t is

the time and θ in both cases is a set of parameters that

depend on the function describing the underlying

process, either degradation or maintenance.

In the case of the maintenance process, a

deterministic concatenation of the maintenance

lifecycles M = f (t | θ) and maintenance duration

D = f (t | θ) composes the maintenance activity

scheduling. Sometimes, the functional variables M

and D use predictive models or are fixed standard

times provided by the overhead crane manufacturer.

In any of them, we deal with functions that depend on

time t and certain parameters θ. Independently, but

following the same idea, the degradation process is a

probabilistic concatenation between the time-to-

failures F = f (t | θ) and time-to-repair R = f (t | θ). This

time, the variables are modeled with frequency

models, i.e., probability distributions. At this point,

this is where the connection and relevance of the

DDSA layer are present. As we can see, the fitting

process ensures a parametric distribution selection

close to the shape of the actual data filtered from the

SAP-SCADA systems. At this point of the description,

the entry points of the integrated digital platform,

which are: degradation data and planned

maintenance data, are also evident.

Crucial for sensible coordination of maintenance

activities is the starting point of the maintenance

scheduling of the component, overhead cranes in our

case, and the degradation modeling process. The

starting point for maintenance scheduling is provided

by the optimization model implemented at the system

level when coordination is requested. The

degradation modeling process is managed by the

DDSA layer using the updated information available

in the monitoring systems. In summary, we can say

that the overhead crane capacity model is a stochastic

Markov chain Monte Carlo (MCMC) process.

Figure 1. Functional view at the component level.

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Figure 2. Functional view at the system level.

Once the number of overhead cranes involved in

the system is known, the modeling steps described

above are applied for each overhead crane

independently z

1 = f (t | θ1), z2 = f (t | θ2), …,

N = f (t | θN).

Fig. 2 shows the system-level modeling

composition.

Once the component-level modeling step is

achieved for each crane, and given the relationship

between them, i.e., the series-parallel block diagram, it

is possible by convolution to build the stochastic

functional system capacity X = f (t | θ

S), where again t

is the time and θ

S is a set of parameters that depends

on other parameters modeled in previous steps. Once

system-level availability A(t) is obtained, which in our

case is a stochastic process X = f (t | θ

S) when Monte

Carlo simulations are used, and knowing the function

that describes the needs of the requested service

Y = f (t | θ

P), again using a convolution process, we

can estimate performance metrics R = f (t | θ

S,P) that

measure the efficiency and adequacy of the system

providing the service, in our case how adequate the

actual system is to fulfill the requested service. As we

can deduce, if the metric performance measures the

system's adequacy, then we can use the metric to find

the best operating point for the system.

Knowing that we are modeling two contributors,

degradation and maintenance and that one is random,

referring to degradation, we can use this approach to

find the best maintenance scheduling for the system

such that the performance metric provides the

lowest/highest possible value. In the end, the

optimization model manages the starting point of the

maintenance schedule for each overhead crane

considered to achieve the goal. Having described the

process from the most granular to the highest point,

we can state that the DDSA layer plays an essential

role in the system performance assessment because it

consequently ensures the search for a sensible

maintenance scheduling for the system.

3 RESULTS AND DISCUSSION

In this section, we apply the proposed fitting

methodology in a case study, i.e., the DDSA layer, as

an example of its implementation in practice. The

starting point is the actual filtered degradation data.

The system under study is made up of 33 different

overhead cranes. The applied fitting process is the

same, following the flow diagram in Fig. 3 of

reference [18] and the algorithm presented in the

previous section. For illustrative purposes, the

selected example and the data described below are

from a specific 50 tons overhead crane. Tab. 1 shows

the raw degradation data for the overhead crane

analyzed, with 59,501 hours of continuous operation

(6.8 years) and 355 hours of the replacement or repair

process.

Table 1. Raw Degradation Data

________________________________________________

Time-to-Failure (hours) Time-to-Repair (hours)

________________________________________________

22; 487; 1,636; 635; 505; 1,044; 49; 4; 5; 7; 4; 6; 7; 2; 29; 6; 3; 4;

98; 913; 79; 18; 170; 333; 484; 7; 2; 30; 7; 4; 4; 2; 1; 4; 1; 1; 10;

249; 430; 50; 65; 2,382; 1,044; 1; 1; 12; 7; 7; 2; 1; 4; 4; 1; 5; 1;

1,150; 2,663; 228; 1,559; 688; 1; 2; 2; 3; 4; 6; 2; 4; 2; 4; 3; 2; 2;

134; 1,062; 595; 9; 353; 40; 126; 8; 2; 3; 4; 3; 1; 3; 1; 1; 3; 5; 3; 8;

1,264; 1,244; 67; 2,592; 90; 1,919; 7; 1; 2; 1; 2; 1; 1; 1; 1; 1; 1; 3; 1;

2,943; 383; 324; 89; 1,292; 433; 4; 3; 1; 1; 1

351; 2,109; 1,060; 1,204; 1,054;

1,550; 471; 240; 2,094; 1,405;

3,220; 829; 223; 29; 462; 657;

511; 68; 766; 1,842; 2,476; 280;

486; 50; 206; 279; 917; 104; 24;

237; 660; 125; 127; 763; 205; 332;

204

________________________________________________

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As described in previous sections, F and R

variables are analyzed independently and are the

inputs of the DDSA layer. Consequently, we apply the

same flow diagram for each variable independently.

Tab. 1 highlights that the failure frequency is higher

than ten, so we fit the degradation data to all the

possible single distributions available in the list (the

case study was selected to apply the complete

diagram).

The complexity and size of a 50 tons overhead

crane mean that, for certain failures, more than one

group of workers must fix the unexpected failure

(multiple or parallel actions). This condition of

parallel repair tasks introduces noise into the stored

data. For example, one group of workers completes

the repair task, and the other is still working, or

during the repair time, other potential problems are

found, and the repair time is extended in one of the

groups. All situations described above are treated

with filters in the data just before we calculate the F

and R variables. For example, in cases with multi-

actions (multiple tasks simultaneously), we set the

failure event to the earliest action and the repair event

to the latest action.

The fitting process of the single theoretical

distributions can be defined as a constrained

multivariate non-linear objective function

optimization problem. This investigation solved the

parameter estimation based on the maximum

likelihood estimation (MLE) method for each fitted

distribution with a Sequential Quadratic

Programming method. Knowing the features

underlying the fitting process, Tab. 2 and Tab. 3 list

the final fitted distributions after we apply the infinite

mean and variance filter to the data presented in Tab.

Table 2. Best distribution fitted for historical data F.

________________________________________________

ID Parameters Name AIC Test

________________________________________________

3 µ = 743.7660 Exponential 1219.87

________________________________________________

Table 3. Best distribution fitted for historical data R.

________________________________________________

ID Parameters Name AIC Test

________________________________________________

12 µ = 4.4402 Inverse Gaussian 377.64

λ = 2.7741

________________________________________________

In addition to the AIC criterion for selecting the

final decision, Tab. 4 and 5 also include two well-

established goodness-of-fit tests in the literature, the

Kolmogorov-Smirnov (KS) test and the Anderson-

Darling (AD) test. As expected, not all goodness-of-fit

tests are aligned, and, as we know, all have strengths

and weaknesses depending on the object of study.

While the AD test focuses on how well the tail of the

distribution fits the data, the KS test relies on the full

support of the distribution. However, the AIC

criterion populated stable selections for the historical

degradation data tested.

Table 4. Best distribution for F (contamination vs. data)

________________________________________________

ID Parameters Name AIC KS AD

________________________________________________

3 (Data) µ = 743.77 Exponential 1219.87 0.5782 0.4607

3 (Cont.) µ = 791.07 Exponential 2457.48 0.7694 0.8847

________________________________________________

Table 5. Best distribution for R (contamination vs. data)

________________________________________________

ID Parameters Name AIC KS AD

________________________________________________

12 (Data) µ = 4.44 Inverse 377.64 0.2063 0.3193

λ = 2.77 Gaussian

12 (Cont.) µ = 4.42 Inverse 755.97 0.6189 0.5224

λ = 2.75 Gaussian

________________________________________________

As a result of the implemented fitting process, Tab.

2 and Tab. 3 show the parameters of the estimated

distribution based on the MLE method for the best fit

according to the AIC criterion for the historical F and

R degradation data.

Additionally, Figs. 3 and 4 show the visualization

of the empirical cumulative distribution function

(CDF) versus the theoretically fitted CDF (including

the confidence interval), demonstrating a coherent

approach to selecting the theoretical fit. As a result,

we obtain the closest fit to the degradation data, and

meanwhile, we introduce additional complexity into

the model only when necessary to achieve higher

accuracy (parsimony). The results generated in this

section are evidence of how the implemented DDSA

layer guarantees robust and accurate final selections

in the fitting process, which leaves the database

structure ready to be used by the optimization model

in the final stage of the process (coordination of

maintenance strategies).

There are additional validations to evaluate the

performance of the self-analysis layer, which consists

of changing the input data in a controlled way and

then re-evaluating the fitting selection process. Here,

we conduct two tests. The first one consists of

contaminating the input data with new values

generated from the final selections, and the second

one consists of removing one by one the last records

in the dataset. In both cases, the process is re-assessed

after the change. For the first additional validation, we

conducted a simple experiment. We generate a new

sample from the final selections (for both cases, F and

) with the same size as the original data, then

combine both samples (real data and generated data)

into one, and then the DDSA layer is used again

following the same process. Tab. 4 and Tab. 5 show

the results of this experiment. The parameters of the

distribution change, but the final selection is the same.

Moreover, both goodness-of-fit improved their results

as expected.

Figure 3. Empirical versus Theoretical CDF (Best fit) for

historical data F.

In addition, Figs. 5 and 6 show the new fit plots of

the empirical CDF versus theoretical CDF (now

contaminated data), evidencing the reduction of the

confidence intervals compared to Fig. 3 and Fig. 4.

608

Figure 4. Empirical versus Theoretical CDF (Best fit) for

historical data R.

Figure 5. Empirical versus Theoretical CDF (Best fit) for F-

contaminated data.

Figure 6. Empirical versus Theoretical CDF (Best fit) for R-

contaminated data.

In the second additional validation, the experiment

consists of removing the last records in the dataset

one by one and assessing the changes in the final

selected distributions. This experiment is intended to

check what happens when we recalibrate the

parametric distributions, i.e., equivalent to checking

what happens when new failure records appear. In

particular, this experiment applies to failure data. The

test results are shown in Tab. 6. As we can see, even

by removing the last ten records from the dataset, the

final selected distribution remains the same.

Table 6. Best distribution fitted for historical data F

________________________________________________

Scenario Name Parameter AIC Test

________________________________________________

Dataset Exponential µ = 743.77 1,219.87

1 point µ = 750.60 1,206.10

2 points µ = 755.98 1,191.97

3 points µ = 763.13 1,178.16

4 points µ = 763.13 1,162.89

5 points µ = 771.62 1,149.27

6 points µ = 780.36 1,135.64

7 points µ = 780.36 1,135.64

8 points µ = 782.01 1,120.63

9 points µ = 789.58 1,106.70

10 points µ = 800.37 1,093.28

________________________________________________

This experiment is an additional result that

supports the decision diagram implemented for

fitting, which can obtain sensitive results when the

data changes. Moreover, the results show how crucial

online self-calibration is in capturing the changes in

the data. It is clear from Tab. 6 how the

parameterization (µ) changes with the data.

At this point of the investigation, we can conclude

that the data are correctly, accurately, and robustly

filtered, processed, and stored in the database, which

ensures clean inputs for the optimization model used

in the final stage of the investigation (coordination of

maintenance activities). Knowing that the risk model

implemented on the integrated digital platform relies

on historical data to estimate risk indicators, the

approach implemented on the DDSA layer is crucial

to ensure robust data processing to guarantee accurate

predictions.

4 CONCLUSIONS

The research presented here validates the design and

implementation of the DDSA layer that ensures

robustness in the filtering and synthesis of the

degradation data (time-to-failure and time-to-repair)

of the set of overhead cranes considered in this study.

The DDSA layer outcomes are probability

distributions used to simulate probable failures in the

group of overhead cranes, allowing the holistic

coordination of maintenance activities by evaluating

global system risk indicators. The main research result

is validating the robust and accurate filtering and

synthesis of the degradation data used to coordinate

maintenance activities.

Robustness features come from a formal

predefined fitting process (implementation tested in

practice in this contribution using the data presented

in Table 1), which allows us to obtain coherent

distributions to simulate the degradation process due

to system operation, given the online historical

degradation data. The accuracy comes from a formal

technological and modeling connection between

degradation due to the system operation, maintenance

activities schedule, and management process (selected

manufacturing industry needs) through the

intermediary database created for maintenance

activities coordination.

The proposed solution for data filtering, synthesis,

and self-analysis (the DDSA layer) illustrates the

practice of self-decision-making focused on operating

technical objects. It is an example of the process of

adaptation and transition to the digital industry

through machine learning approaches. The results

achieved so far in this paper through the presented

validations (degradation fitting) complement and

guarantee the robustness of other processes of the

engineering solution (coordination of maintenance

activities).

Thanks to the self-contained process designed and

shared in this paper, we will be able to analyze in

future works the impacts introduced by changes in

the data over time, the data window, the criticality of

each overhead crane in the system, and the system

degradation over time.

609

ACKNOWLEDGMENT

The Polish Ministry of Science and Higher Education has

financially supported the work.

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