Revista de Ciencias Tecnológicas (RECIT). Volumen 3 (1): 10-22
Revista de Ciencias Tecnológicas (RECIT). Universidad Autónoma de Baja California ISSN 2594-1925
Volumen 7 (1): e324. Enero-Marzo, 2024. https://doi.org/10.37636/recit.v7n1e324
1 ISSN: 2594-1925
Review Monitoring reliability of man-machine system of
machining area using the Weibull distribution
Monitoreo de la confiabilidad del sistema hombre-máquina del área
de mecanizado mediante la distribución de Weibull
Rosa María Amaya Toral1, Manuel Baro Tijerina2, Martha Patricia García-Martínez1,
Cinthia Judith Valdiviezo Castillo1
1Tecnológico Nacional de México Campus Chihuahua II, Ave. De las Industrias #11101, Chihuahua, Chihuahua,
México, C.P. 31110.
2Tecnológico Nacional de México Campus Nuevo Casas Grandes, Tecnológico Ave #7100, Nuevo Casas Grandes,
Chihuahua, México. C.P. 31700.
Corresponding author: Manuel Baro Tijerina, Tecnológico Nacional de México Campus Nuevo Casas Grandes,
E.mail: mbaro@itsncg.edu.mx. ORCID: 0000-0003-1665-8379.
Submitted: September 1, 2023 Accepted: November 30, 2023 Published: January 1, 2024
Abstract. - This publication presents the development of a method that seeks to monitor the parameters β
(shape) and η (scale) for each component-subsystem combination following the Weibull distribution, necessary
for the calculation of the reliability of the man-machine system in the machining area. This system defines the
workshops of the metal-mechanic, with high-mix and low-volume batch production where conventional and
Computerized Numerical Control (CNC) machines are involved, which share the manufacturing of parts that
sometimes are unique, or their manufacturing period is short. The design of the man-machine system is based
on the analysis of the failures of non-conforming parts in the machining area and on the failure rates, which
the statistical model is developed for its evaluation, considering the 2-parameter Weibull distribution, and a
redundant system with series-parallel configuration. The results obtained were based on the theoretical-
practical, using mathematical and statistical models, as well as the Study Case. With the use of mathematical
and statistical models, it is demonstrated that the probability of failure (risk) of the man-machine system is
time-dependent and is generated by mechanical type stresses, which occur in the manufacture of parts.
Keywords: Reliability; Machine-human; Weibull distribution; Exponential distribution.
Resumen. - Esta publicación presenta el desarrollo de un método que busca monitorear los parámetros β
(forma) y η (escala) para cada combinación componente-subsistema siguiendo la distribución de Weibull,
necesarios para el cálculo de la confiabilidad del sistema hombre-máquina en el área de maquinado. Este
sistema define los talleres de la metalmecánica, con producción por lotes de alta mezcla y bajo volumen donde
intervienen máquinas convencionales y de Control Numérico Computarizado (CNC), que comparten la
fabricación de piezas que en ocasiones son únicas, o su periodo de fabricación es corto. El diseño del sistema
hombre-máquina se basa en el análisis de los fallos de piezas no conformes en el área de mecanizado y en las
tasas de fallo, para cuya evaluación se desarrolla el modelo estadístico, considerando la distribución de
Weibull de 2 parámetros, y un sistema redundante con configuración serie-paralelo. Los resultados obtenidos
se basaron en el teórico-práctico, utilizando modelos matemáticos y estadísticos, así como el Caso de Estudio.
Con el uso de modelos matemáticos y estadísticos, se demuestra que la probabilidad de falla (riesgo) del
sistema hombre-máquina depende del tiempo y es generada por tensiones de tipo mecánico, que ocurren en
la fabricación de las piezas.
Palabras clave: Fiabilidad; Máquina-humano; Distribución de Weibull; Distribución exponencial.
2 ISSN: 2594-1925
Revista de Ciencias Tecnológicas (RECIT). Volumen 6 (4): e324.
1 Introduction
Consider a system that can be characterized as
being in any of a previously specified set of states.
Suppose that the system evolves or changes from
one state to another over time according to a
certain law of motion, and let be the state of the
system at time [1]. If how the system evolved is
not deterministic, but caused by some random
mechanism, then can be a random variable for
each value of the index . This collection of
random variables is the definition of a stochastic
process and serves as a model to represent the
random evolution of a system over time [2]. In
general, the random variables that make up a
process are not independent of each other but are
related to each other in some particular way [3]. A
continuous random variable can take any value in
an interval and it is represented by a set of real
numbers. It is usually associated with
measurements that generate physical quantities.
The density function, the cumulative function, the
complement function, the expected value of the
average, the expected value of the variance, and
the moments [4]. The random variable in the
Weibull Distribution always takes values greater
than or equal to 0 since it always represents time.
The density function, the cumulative function, the
expected value of the mean, and the variance are
when (1)
(2)
(3)
(4)
To evaluate the reliability of a random variable
with a Weibull distribution, the following
equation is used:
(5)
For the case where no failure times are available,
it is proposed to use the method given in Piña [5].
The method allows estimating the parameters β
and η of the Weibull distribution, from the
observed or predicted principal stresses 1
(maximum) and 2 (minimum), thus allowing
the calculation of the reliability R(t) for any
desired stress value [5].
In the machining work, when performing the
analysis of the causes of non-conforming parts,
some variables are generating failures due to the
applied mechanical stress or strength, which due
to tool wear varies over time. These variables are
represented using the Weibull distribution. On
the other hand, the variables that are generating
the failures and that are not mechanical (human,
procedural, administrative factors, etc.) or time-
dependent, are represented by the Exponential
distribution.
The exponential distribution is applied in
reliability when the failure function is
independent of time. This property of the
exponential distribution is known as
forgetfulness. The density function, the
cumulative function, the expected value of the
mean and variance are:
when t ≥ 0 (6)
(7)
(8)
(9)
(10)
In the machining work, when performing the
analysis of the causes of non-conforming parts,
some variables are generating failures due to the
applied mechanical stress or strain, which due to
tool wear varies over time. These variables are
represented using the Weibull distribution. On
2 ISSN: 2594-1925
Revista de Ciencias Tecnológicas (RECIT). Volumen 6 (4): e324.
the other hand, the variables that are generating
the failures and that are not of a mechanical
nature (human, procedural, administrative
factors, etc.) or time-dependent, are represented
by the Exponential distribution [6].
Because the production process of parts in this
type of machining shop is by high-mix, low-
volume batches, the reliability of the man-
machine system is affected by various products
or part numbers, different types and
characteristics of the material used, the various
machines involved, as well as the different faults
or defects that occur in the parts and produce
scrap [7]. For this reason, reliability monitoring
should be based on the causal factors of low
reliability and the parameters of the distributions
that describe the behavior of failures within the
system components [8].
Some machining shops in the city of Chihuahua
in the metal-mechanic sector have problems that
do not allow them to meet delivery times by not
completing production orders due to the presence
of non-conforming parts in the scrap area. This
generates high costs for them, in addition to the
loss of their customers' confidence because they
are not able to fulfill orders in due time and form.
It was proposed to develop a method to monitor
the reliability of the man-machine system,
monitoring the parameters β (shape parameter)
and η (scale parameter) of the component-
subsystem combinations that follow a Weibull
distribution [9].
This is to incorporate the monitoring information
into the Series-Parallel statistical model, for the
control and improvement of the reliability of the
man-machine system, reducing the number of
failures that generate scrap and can thus fulfill
orders promptly. The monitoring of the changes
in the Weibull parameters was done by
monitoring the magnitudes of the stresses applied
when manufacturing the parts [10].
2 Methodology
2.1 Man-machine system
The worker operates powered equipment, such as
a tool or machine for production. This
configuration is the most widely used in
manufacturing systems. It involves combinations
of one or more workers on one or more pieces of
equipment; workers and machines combine to
take advantage of their respective strengths and
attributes [11].
2.2 Reliability
Suppose that T > 0 is a continuous random
variable that records the failure time of a certain
component that is in operation at time zero. We
will denote by F(t) and f(t) the distribution and
density function of this random variable. In
reliability theory, it is of interest to know the
probability that the component works correctly at
least until any time t or the probability that it
stops working at the next instant of time, given
that the component has worked well until a
certain time [12]. For this reason, the following
two functions are defined: The reliability
function is defined as R(t) = P (T > t), while the
failure rate function is r(t) = f(t)/ (1 - F(t)). The
reliability function is also called the survival
function and the failure rate function is also
known as the hazard function and is so called
because a component that has survived to time t
will fail in the interval (t, t + ∆t] with conditional
probability r(t) ∆t + o(∆t), [13] in effect:
P (t < T ≤ t + ∆t| T > t) =
=
= r(t) ∆t + o(∆t)
r(t) =
= -
ln R(t)
Implying that the reliability function in terms of
the hazard function is given by: R(t) = (- ∫t r(s)
3 ISSN: 2594-1925
Revista de Ciencias Tecnológicas (RECIT). Volumen 6 (4): e324.
ds) exp. In addition, the mean time to failure E(T)
can be written in terms of the function R(t) as:
R(t) = exp (−
).
2.3 Reliability for serial systems
Consider a system of n components placed in
series as shown in Figure 1. We will assume that
each of these components works independently
of the other. It is intuitively clear that such a
system as a whole work if every component is in
good condition [14]. We are interested in finding
the reliability and failure rate functions of such a
system. Note that the failure time T of such a
system is the smallest failure time of each of the
components, i.e., T =min {T1, Tn}. Thus,
physical schemes of this type help to justify
asking the question of finding the probability
distribution of the minimum of n independent
random variables, which, for time-dependent
homogeneous systems, is the Weibull
distribution [15].
Figure 1. Serial system representation.
The reliability and failure rate functions for series
systems are presented below. We will denote by
R1 (t), Rn (t) the reliability functions of each
component in the system, and the failure rate
functions will be r1(t), rn(t).
The reliability and failure rate functions of the
series system are:
R(t) = R1(t)* R2(t)*· · · Rn(t) (11)
r(t) = r1(t)+ r2(t) + · · · rn(t) (12)
2.4 Reliability for Parallel Systems
In a system of n components placed in parallel,
each of these components operates
independently of the other, as shown in Figure 2.
Such types of systems work as a whole if at least
one of the components is in good condition.
Systems of this type are used when it is desired
to maintain a high level of operational reliability,
such as aircraft flight systems, or nuclear reactor
management systems. The failure time T of such
a system is the largest failure time of the
components, i.e., T = max {T1, . . . . , Tn}, where
the event (T ≤ t) is equivalent to all components
failing before time t. The reliability function of
the parallel system is:
R(t) = 1 – ((1 − R1(t)) * (1- R2(t)) *· · · · (1 − Rn(t)) (13)
Figure 2. Parallel System.
4 ISSN: 2594-1925
Revista de Ciencias Tecnológicas (RECIT). Volumen 6 (4): e324.
2.5 Weibull distribution
The Weibull distribution is used in
Reliability Engineering for failure time data
and is useful to determine the probability
that a part will fail after a certain time t. It is
described by 2 or 3 parameters: shape (β),
scale (η), and location (γ), which are present
in the probability density function (pdf),
where the failure time is given by:
γ
γ
o
(2 parameters)
2.6 Properties of the Weibull
Distribution
The Weibull reliability function starts at β
=1 since it is assumed that at the beginning
of the mission, all the equipment is in good
condition, and as time goes by the reliability
decreases. For values of β less than 1 the
reliability function decreases
asymptotically. Like the density function,
for values of β =1, the reliability function
assumes the exponential form. The
properties of the Weibull distribution are:
For β = 1 the curve decreases monotonically
faster than for 0 < β <1.
2.7 Parameters of the Weibull
distribution
It is useful first to know the effect of the
different parameters of the Weibull
distribution to be able to act:
Effect of the shape parameter. The shape
parameter describes the way the data are
distributed. A shape of β =3 approximates a
normal curve. A low shape value, e.g., β =1,
gives a curve with skewness to the right. A
high shape value, e.g., β =10, gives a curve
with skewness to the left.
Effect of the scale parameter. The scale, or
characteristic life, is the 63.2 failure
percentile of the data. The scale defines the
position of the Weibull curve concerning
the threshold value, which is similar to the
way the mean defines the position of a
normal curve.
Effect of the threshold value parameter. The
threshold value parameter describes a shift
of the distribution away from 0. A negative
threshold value shifts the distribution to the
left, while a positive threshold value shifts
the distribution to the right [16].
3. Study case introduction
Case Study in a Machining Workshop,
which starts with the process of collecting
and analyzing the scrap generated, the
evaluation of its reliability, and the
identification of the worst reliability, as well
as the monitoring of parameters. Research
hypotheses were proposed and, to test them,
different probabilistic and statistical tools
were implemented. The direction of this
research was guided by the testing of four
issues that are shown below:
In the man-machine system there are
activities whose risk or probability of failure
depends on time and/or the failure is
generated by mechanical stresses (friction,
wear, etc.); that is, the damage accumulates
over time.
In the man-machine system, there are
activities whose risk or probability of failure
is constant and does not depend on time, of
which it is possible to determine their mean
and variance.
5 ISSN: 2594-1925
Revista de Ciencias Tecnológicas (RECIT). Volumen 6 (4): e324.
It is possible to combine activities and
determine the reliability of each
combination, in a series-parallel system, to
determine the partial and total reliability of
the system.
It is possible to monitor the parameters of
each component-subsystem combination of
the man-machine system, incorporating
them into the series-parallel statistical
model, due to its configuration, and detect
the worst reliability.
3.1 Study Case
A machining shop in the metal-mechanical
sector, located in the city of Chihuahua,
whose production system is batch, high-
mix-low volume, has semi-automated CNC
machines for the manufacture of parts. The
company has some problems that do not
allow them to meet delivery times, because
they cannot complete production orders due
to the presence of nonconforming parts in the
scrap area, which generates high costs, in
addition to the loss of confidence of its
customers. The workshop wants to improve
its reliability, reducing the number of
failures that generate scrap and thus be able
to fulfill orders promptly.
3.2 Data Analysis
To test the hypotheses formulated and taking
as reference the data from the machining
shop, the following was carried out:
Classification of faults or defects that occur
in the manufacture of parts. Information was
collected on the types of failures or defects
that occurred in the manufacture of parts in
the machining shop, from April 27 to June 2,
2021 [17]. These failures were classified by
their origin (failure mechanism), using the
Cluster Analysis Technique, developed
through brainstorming generated by a work
team made up of 2 operators, 1 maintenance
technician, and a production supervisor.
Table 1 shows the resulting classification of
the type of failure by its origin.
Table 1. Classification of failure type by origin. Own design.
ORIGIN OF THE FAULTS
Mechanic
Mechanic
Error
Human Error
Thermal
Treatments
Material
Improvements
3
22
1
43
26
42
4
23
2
49
30
45
5
24
10
50
33
6
25
16
51
34
7
28
17
52
58
8
29
27
53
9
31
32
55
11
35
36
56
12
37
39
57
13
38
44
14
40
54
15
41
18
46
19
47
20
48
21
6 ISSN: 2594-1925
Revista de Ciencias Tecnológicas (RECIT). Volumen 6 (4): e324.
In Table 1, the failure origins resulting from
the analysis were identified as follows:
mechanical, human error, heat treatments,
material, and unforeseen events. Table 2
shows the frequency of failures by origin.
Table 2. Frequency table of failures by origin.
ORIGIN
NUMBER OF
FAILURES
%
RELATIVE
%
ACCUMULATED
MECHANIC
31
53.45%
53.45%
HUMAN ERROR
11
18.97%
72.42%
THERMAL TREATMENTS
9
15.51%
87.93%
MATERIAL
5
8.62%
96.55%
IMPROVEMENTS
2
3.45%
100%
Total
58
100%
Table 2 shows that one of the origins with
the highest number of failures is the
mechanical type, with 31 out of a total of 58
(53.54%). Figure 3 shows the Pareto
Diagram of the origin of the type of failure,
using the data from Table 2.
Figure 3. Frequency table of faults by origin.
3.3 Man-machine system model, for
the identification of the combinations
present in the failures of non-conforming
parts for scrap.
The man-machine system is made up of the
component-subsystem combinations and
the subsystems that participated in the
failures during the period analyzed. As a
reference to identify the component-
subsystem combinations that participated in
the man-machine system, Figure 4 is
presented. There are 24 possible component-
subsystem combinations possible to occur in
a certain period. Only those combinations
that have contributed to the generation of
failures are considered for the reliability
assessment, using the corresponding
Weibull or exponential distribution,
depending on the origin of the failure.
7 ISSN: 2594-1925
Revista de Ciencias Tecnológicas (RECIT). Volumen 6 (4): e324.
3.4 Component-subsystem
combinations are present in failures
The times to failure generated by a certain
component-subsystem combination
together form a data distribution, either
Weibull or Exponential, with their
corresponding parameters.
Table 3. Combinations are present in the scrap.
WEIBULL/
EXPONENTIAL
PARAMETERS
COMBINATION
DESCRIPTION
β
η
1/λ
1,1
MACHINE MAINTENANCE/SETUP
1
3.1667
1,6
MACHINE-SUPPLIER
1
9.5
2,1
TOOL MAINTENANCE/SETUP
1
9.5
3,1
PROCESS//OPERATION-MAINTENANCE/SET UP
1.0688
3.1268
Considering the parameters in Table 3, these
are used to evaluate the reliability of each
component-subsystem combination of the
subsystems and the system. As can be seen,
all participating combinations were
analyzed with the Weibull distribution,
since only failures of mechanical origin
were involved.
3.5 Evaluation of the Reliability of the
Subsystems and the Man-Machine
System
Considering Equation 11 (series
configuration) the reliability of each
subsystem participating in the failures was
evaluated. Considering Equation 13
(parallel configuration) the reliability of the
man-machine system was evaluated. Table
5 shows the values of the reliability of the
subsystems that were present in the scrap
and the reliability of the man-machine
system during the reviewed period, for 5
periods of time considered within the 9.5 hr.
shift.
Table 4. Reliability of the combinations presents in the scrap from January 25 to 30, 2021, for 5 time periods.
RELIABILITY OF THE SUBSYSTEMS (Rij)
AND THE SYSTEM RS
PERIODS
(hr.)
R1
R2
R3
RS
1
0.6563
0.9
0.645
0.9878
3
0.2827
0.7292
0.2683
0.8579
5
0.1218
0.5907
0.1116
0.6807
7
0.0524
0.4786
0.0464
0.5289
9
0.0226
0.3877
0.0193
0.4131
Table 4 shows that 7 hours after the start of
the shift, in subsystem 1 (machine), the
probability of not having failures for scrap
was 5.24%, while in subsystem 2 (tool) it
was 47.86% and in subsystem 3
(process/operation) it was 4.64%, being the
worst reliability of subsystem 3.86% and in
subsystem 3 (process/operation) it was
4.64%, the worst reliability being that of
subsystem 3. Concerning the reliability of
the man-machine system, it showed a
probability of 52.89% that no failures for
8 ISSN: 2594-1925
Revista de Ciencias Tecnológicas (RECIT). Volumen 6 (4): e324.
scrap would occur in 7 hours after the start
of the shift.
3.6 Component-subsystem
combinations are present in failures
The times to failure, generated by a certain
component-subsystem combination,
together form a data distribution, either
Weibull or Exponential, with their
corresponding parameters. Table 5 shows
the component-subsystem combinations
that were present in the failure analysis.
Table 5. Combinations are present in the scrap.
As can be seen in Table 5, four of the five
combinations were analyzed with the
Weibull distribution, because they involve
aspects of mechanical origin. The last
combination was analyzed with the
Exponential distribution because it involves
aspects of human error origin. The Weibull
++ software was used to calculate the
parameters of both the Weibull and
Exponential distributions.
3.7 Evaluation of the reliability of the
component-subsystem combinations
Considering the statistical model to evaluate
the reliability with each of the Weibull and
Exponential distributions of Equations 5 and
10 respectively, the reliabilities of each
combination were evaluated. Table 7 shows
the values of the reliabilities of each
component-subsystem combination that
were present in the scrap, during the
analyzed period, for 5 time periods
considered within the 9.5 hr shift.
Table 6. Reliability of the combinations present in the scrap.
RELIABILITY OF COMPONENT-SUBSYSTEM
COMBINATIONS
PERIODS
(hr.)
,
,
,
,
,
1
0.6478
0.7382
0.7509
0.4433
0.9
3
0.2718
0.4023
0.4232
0.0871
0.7292
5
0.1141
0.2193
0.2387
0.0171
0.5907
7
0.0478
0.1195
0.1346
0.0037
0.4786
9
0.02
0.0651
0.0759
0.0006
0.3877
Table 6 shows that the component- subsystem combinations with the lowest
WEIBULL/EXPONENTIAL
PARAMETERS
COMBINATION
DESCRIPTION
β
η
1/λ
1,1
Machine-Manto/set up
3.4615
7.9737
1,6
Machine-Supplier
2.1839
7.1972
2,1
Tool-Manto/set up
1.5263
5.3284
3,1
Process/Operation-Manto/set up
1.9091
2.3472
3,3
Process/Operation-Institutional
Condition/Attitude
9.5
9 ISSN: 2594-1925
Revista de Ciencias Tecnológicas (RECIT). Volumen 6 (4): e324.
reliability value were 3.1 and 1.1, referring
to Process/Operation-Maintenance/Set up
and Machine-Maintenance/Set up,
respectively, with Maintenance/Set up
being the component in common.
3.8 Evaluation of the reliability of the
subsystems and the man-machine system
Considering Equation 11, which considers a
series configuration, the reliability of each
subsystem participating in the failures
was evaluated. Considering Equation 13
(parallel configuration) the reliability of the
man-machine system was evaluated. Table
8 shows the values of the reliability of the
subsystems that were present in the scrap
and the reliability of the man-machine
system, during the period reviewed, for 5
periods of time considered within the 9.5
hrs. shift.
Table 7. Reliability of subsystems and system, present in the scrap.
Table 7 shows that 7 hours after the start of
the shift, in subsystem 1 (machine), the
probability of not having failures for scrap
was 0.57%, while in subsystem 2 (tool) it
was 13.46% and in subsystem 3
(process/operation) it was 0.16%. The worst
reliability was that of subsystem 3.
Concerning the reliability of the man-
machine system, it showed a probability of
14.09% that no scrap failures would occur
in 7 hours after the start of the shift.
4. Results
For the verification of the research, these
were based in a theoretical-practical way,
using mathematical and statistical models,
as well as the Case Study. The procedure
performed is described below. In the man-
machine system, there are activities whose
risk or probability of failure depends on
time, and/or the failure is generated by
mechanical efforts (by friction, wear, etc.),
that is, the damage accumulates over time.
time. For the evaluation of the reliability of
the man-machine system, the Weibull
distribution is used, since this distribution is
appropriate when the variables that generate
the failure are based on mechanical efforts,
whose damage accumulates over time. To
prove that in the man-machine system, there
are activities whose probability of failure
depends on time, and are due to mechanical
stress, Table 8 is presented.
SUBSYSTEM RELIABILITY (Rij) AND
SYSTEM
RELIABILITY RS
PERIODS
(hr.)
R1
R2
R3
RS
1
0.4782
0.7509
0.3990
0.9219
3
0.1094
0.4234
0.0635
0.5191
5
0.025
0.2387
0.0101
0.2653
7
0.0057
0.1346
0.0016
0.1409
9
0.0013
0.0759
0.0002
0.0773
10 ISSN: 2594-1925
Revista de Ciencias Tecnológicas (RECIT). Volumen 6 (4): e324.
Table 8. Activities of the man-machine system are dependent on time and mechanical effort.
Activities
Time-dependent
Mechanical stresses
Cut
The cutting tool removes
material in the form of chips
from the workpiece and the
sharpness of the tool suffers
wear over time.
The pressure of the cutting tool against
the metal of the workpiece.
Drilling
The material is cut by rotating a drill bit,
which removes chips from the material,
making a hole.
Turning
The cutting tool approaches the workpiece
radially, which, when rotating, is provided
with the cutting motion.
Milling
The rotary motion of a tool for cutting,
feeding, or depth.
Grinding
The grinding wheel wears the workpiece as
it advances towards the grinding wheel, by
abrasive means.
Table 8 describes 5 activities carried out in
the Machining Workshop that depend on the
edge of the tool. When the cutting tool is
installed new or just sharpened, after some
time of machining parts and it loses its edge,
it makes the efficiency of the cut depend on
time. In addition, the failure is generated by
efforts of the mechanical type, since the
activity is carried out by the pressure of the
cutting tool against the metal of the piece;
therefore, the analysis used is the Weibull.
In the man-machine system, there are
activities whose risk or probability of failure
is constant and does not depend on time,
from which it is possible to determine their
mean and variance. With the use of
mathematical and statistical models, it is
shown that the probability of failure (risk)
of the system may not be a function of time
(Exponential). For the evaluation of the
reliability of the man-machine system, the
Exponential distribution is used, since this
distribution is appropriate when the
variables that generate the failure are not
based on time. To prove that in the man-
machine system, there are activities whose
probability of failure does not depend on
time, Table 9 is presented.
Table 9. Failures in the man-machine system are not dependent on time.
Activities
Not time-dependent
Process errors due to unfamiliarity with new processes and equipment
They occur per event and can be
quantified.
Non-compliance with requirements by subcontracted suppliers
Insufficient power capacity to keep machinery running
Use of incorrect material due to carelessness
Different finishes due to operator carelessness
Table 9 describes 5 failures that occur in the man-
machine system, caused by human error and
unforeseen events, which do not depend on time
and can be quantified to obtain the probability of
failure; then the analysis to use is Exponential.
Table 2 shows that the origin of human error
type, has a frequency of 11, out of a total of 58,
contributing 18.97%. Figure 3 shows the origin
11 ISSN: 2594-1925
Revista de Ciencias Tecnológicas (RECIT). Volumen 6 (4): e324.
of the type of failure, where failures of human-
error origin are identified, together with failures
of mechanical origin, as vital causes,
contributing 72.5%. For reliability monitoring,
the parameters of the Weibull and/or Exponential
distributions were used, based on the occurrence
of failures, according to the data recorded during
2 different periods of the year 2021. Table 3
shows the parameters of the component-
subsystem combinations that were present in the
failure analysis for the period from January 25 to
30, 2021, while Table 6 shows the parameters of
the component-subsystem combinations that
were present in the failure analysis for the period
from May 17 to 28, 2021. Combinations 1,1;1,6;
2,1 and 3,1 contributed to the failures during the
2 periods analyzed, which involved machine-
maintenance/set up; machine-supplier; tool-
maintenance/set up, as well as process/operation-
maintenance/set up. These combinations were
considered to be mechanical in origin and time-
dependent. Combination 3.3 occurred only
during the last period analyzed and involved
process/operation-institutional
conditions/attitude.
This combination was considered to be of human
error origin and not time-dependent. To evaluate
the reliability of the combinations of both
periods, the statistical models corresponding to
the Weibull (Equation 5) and Exponential
(Equation 10) distribution were used, depending
on the origin of the failures. Table 4 presents the
values of the reliabilities of each component-
subsystem combination that was present in the
scrap in the period from January 25 to 30, 2021,
for 5 time periods considered within the 9.5 hrs.
shift, while Table 6 presents the values of the
reliabilities of each component-subsystem
combination that was present in the scrap in the
period from May 17 to 28, 2021, for 5 time
periods considered within the 9.5 hours shift. For
the 2 periods analyzed, the worst reliabilities
were those of the combinations 3,1 and 1,1,
referring to Process/Operation-Maintenance/Set
up and Machine-Maintenance/Set up.
To evaluate the reliability of the subsystems and
the man-machine system for both periods, the
statistical model for a serial configuration was
used (Equation 11). Table 5 shows the values of
the reliability of the subsystems and the system
for the period from January 25 to 30, 2021, for 5
time periods considered within the 9.5 hours
shift, while Table 7 shows the values of the
reliability of the subsystems and the system for
the period from May 17 to 28, 2021. Table 5
shows that 7 hours after starting the shift, in
subsystem 1 (machine), the probability of not
having failures for scrap was 5.24%, while in
subsystem 2 (tool) it was 47.86%, and in
subsystem 3 (process/operation) it was 4.64%,
with the worst reliability in subsystem 3. Table 8
shows that 7 hours after starting the shift, in
subsystem 1 (machine), the probability of not
having failures for scrap was 0.57%, while in
subsystem 2 (tool) it was 13.46% and in
subsystem 3 (process/operation) it was 0.16%.
The worst reliabilities were those of subsystem 3
for the 2 periods analyzed, referring to the
Process/Operation.
5. Conclusions
With the verification of hypothesis 1, it can be
concluded that, with the use of mathematical and
statistical models, it is demonstrated that the
probability of failure (risk) of the man-machine
system is time-dependent and is generated by
mechanical type stresses, which occur in the
manufacture of parts; the failures of mechanical
origin presented in Table 2 and Figure 3, are an
example of this. The man-machine system, as it
participates in the failures that occur in the
machining area, presents a risk rate or probability
of failure that grows steadily. The Weibull
distribution used to evaluate its reliability,
represents the damage in an additive way, with a
growth rate at a constant rate, through its shape
12 ISSN: 2594-1925
Revista de Ciencias Tecnológicas (RECIT). Volumen 6 (4): e324.
parameter β. This parameter remains constant
and is also representative of the process variation
and reliability that the data represents. The man-
machine system is a dynamic process, so the
location parameter
γ
of the Weibull distribution is
not considered, only the parameters
β
and η.
Failures of human error origin actively
participate in the failures of the machining area
due to the nature of the study, since it is a
workshop that produces high-mix, low-volume
batches, where operators, technicians,
programmers, etc., make voluntary and
involuntary errors when making changes to the
process/operation, machines, tools and materials,
produced by the frequent changes in the number
of parts to be manufactured. For the above, on the
man-machine system, there is a risk or probability
of failure, which is not dependent on time, being
of human error origin, reason enough for which
the Exponential distribution is used for the
evaluation of the reliability of the man-machine
system. Mixing production batches, varying the
type of material and/or the type of tool to be used,
is when wear on the cutting tool can occur and
this is what happens in reality in machining
workshops, since large batches of the same part
are generally not produced. It is also advisable to
use a measuring tool that allows one to appreciate
the wear on the cutting edge of the cutting tool,
such as a microscopic optical comparator since
with the vernier or the micrometer it is not always
possible to appreciate the variation in the cutting
edge of the tool.
6. Author acknowledgments
Rosa Ma. Amaya Toral: Conceptualización; Recursos;
Ideas; Metodología; Análisis formal; Investigación;
Recursos; Análisis de datos; Borrador original. Manuel
Baro Tijerina: Conceptualización; Ideas; Investigación;
Análisis de datos; Escritura. Martha P. García
Martínez: Conceptualización; Ideas; Metodología;
Análisis formal; Investigación; Análisis de datos;
Escritura; Borrador original; Revisión y edición;
Administración de proyecto. Judith Valdiviezo
Castillo: Conceptualización; Ideas; Metodología; Análisis
formal; Investigación; Análisis de datos; Escritura;
Borrador original; Revisión y edición; Administración de
proyecto. References
[1] D. Morin, "The Hamiltonian method," Introduction to
Classical Mechanics, With Problems and Solutions, no.
October, 2008.
[2] J. C. Pardo, "A brief introduction to self-similar
processes," Department of Mathematical Sciences,
University of …, vol. 1, no. d, pp. 1-20, 2007.
[3] M. Baro-Tijerina, M. Piña-monarrez, and B. Villa-
covarrubias, "Weibull Stress / Strength Analysis with Non-
Constant Shape Parameter," 2017, pp. 1-6.
[4] R. L. Anjum and S. Mumford, Powers, Probability and
Statistics, vol. 417. 2020. https://doi.org/10.1007/978-3-
030-28722-1_8
[5] M. R. Piña-Monarrez, "Weibull stress distribution for
static mechanical stress and its stress/strength analysis,"
Qual Reliab Eng Int, vol. 34, no. 2, pp. 229-244, 2018,
https://doi.org/10.1002/qre.2251
[6] M. W. Fu and J. L. Wang, "Size effects in multi-scale
materials processing and manufacturing," Int J Mach Tools
Manuf, vol. 167, no. May, 2021,
https://doi.org/10.1016/j.ijmachtools.2021.103755
[7] I. Gödri, "Improving Delivery Performance in High-
Mix Low-Volume Manufacturing by Model-Based and
Data-Driven Methods," Applied Sciences (Switzerland),
vol. 12, no. 11, 2022,
https://doi.org/10.3390/app12115618
[8] E. Zio, "Some Challenges and Opportunities in
Reliability Engineering To cite this version: HAL Id: hal-
01550063," IEEE transactions on Reliability, vol. 65, no.
4, pp. 1749-1782, 2016.
https://doi.org/10.1109/TR.2016.2591504
[9] Z. S. Yea and M. Xie, "Stochastic modeling and
analysis of degradation for highly reliable products," Appl
Stoch Models Bus Ind, vol. 31, no. 1, pp. 16-32, 2015,
https://doi.org/10.1002/asmb.2063
[10] R. M. Amaya-Toral et al., "Human-Machine Systems
Reliability: A Series-Parallel Approach for Evaluation and
Improvement in the Field of Machine Tools," Applied
Sciences (Switzerland), vol. 12, no. 3, 2022,
https://doi.org/10.3390/app12031681
13 ISSN: 2594-1925
Revista de Ciencias Tecnológicas (RECIT). Volumen 6 (4): e324.
[11] L. Rainie and J. Anderson, "The Future of Jobs and
Jobs Training," Pew Research Center, no. May, pp. 1-95,
2017.
[12] H. Pham, Handbook of Engineering Statistics
Springer Handbook of Engineering Statistics, vol. 21, no.
4. 1956.
[13] P. Baredar, V. Khare, and S. Nema, "Reliability
assessment of biogas power plant," in Design and
Optimization of Biogas Energy Systems, 2020, pp. 187-
229. https://doi.org/10.1016/B978-0-12-822718-3.00005-
8
[14] Erich Gamma, Design Pattersns. 2006.
https://doi.org/10.1007/978-1-4302-0096-3_10
[15] P. Vassiliou and A. Mettas, "Understanding
accelerated life-testing analysis," Annual Reliability and
Maintainability Symposium, Tutorial Notes, pp. 1-21,
2001.
[16] B. Yaniktepe, O. Kara, I. Aladag, and C. Ozturk,
"Comparison of eight methods of Weibull distribution for
determining the best-fit distribution parameters with wind
data measured from the met-mast," Environmental Science
and Pollution Research, pp. 0-18, 2022,
https://doi.org/10.1007/s11356-022-22777-4
[17] J. Brier and lia dwi jayanti, Research Methodology,
vol. 21, no. 1. 2020.
Derechos de Autor (c) 2024 Rosa María Amaya Toral, Manuel Baro Tijerina, Martha Patricia García-Martínez, Cinthia Judith
Valdiviezo Castillo
Este texto está protegido por una licencia Creative Commons 4.0.
Usted es libre para compartir —copiar y redistribuir el material en cualquier medio o formato — y adaptar el documento —
remezclar, transformar y crear a partir del material— para cualquier propósito, incluso para fines comerciales, siempre que
cumpla la condición de:
Atribución: Usted debe dar crédito a la obra original de manera adecuada, proporcionar un enlace a la licencia, e indicar si se
han realizado cambios. Puede hacerlo en cualquier forma razonable, pero no de forma tal que sugiera que tiene el apoyo del
licenciante o lo recibe por el uso que hace de la obra.
Resumen de licencia - Texto completo de la licencia
