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Modelación matemática de frecuencias óptimas de inspecciones de mantenimiento para tornos paralelos en función del contexto operacional

&nbs= p;

Mathematical modeling of optimal maintenance inspection frequencies for parallel lathes = as a function of operational context

= ¹=	Luis Stalin López Telenchana	<= /span>	https://orcid.org/0000-0001-7548-0406
<= /span>	Maestrante en la Universidad Nacional de Chimborazo, Facultad de Ingeniería, Carrera= de Ingeniería Industrial, Universidad Nacional de Chimborazo, Riobamba, Ecua= dor. luis.lopez@unach.edu.ec=
= ²=	Cristi= na Estefanía Ramos Araujo	<= /span>	https://orcid.org//0000-0002-8644-5814
<= /span>	Facultad de Ciencias, Carrera de Estadística, Escuela Superior Politécnica de Chimborazo, Riobamba, Ecuador. cristina.ramos@espoc= h.edu.ec
= ³=	Natalia Alexandra Pérez Londo=	<= /span>	https://orcid.org/0000-0001-9068-879
<= /span>	Facultad de Ciencias, Carrera de Estadística, Escuela Superior Politécnica de Chimborazo, Riobamba, Ecuador. nperez@espoc= h.edu.ec =
= ⁴=	Carmen= del Rocio Moyón Moyón	<= /span>	https://orcid.org/0000-0001-8798-7060
<= /span>	Investigador Independiente, Riobamba, Ecuador. carmy_111@hotmail.com

	Artículo de Investigación Científica y Tecnológi= ca Enviado: 14/06/2023 Revisado: 22/07/2023<= o:p> Aceptado: 09/08/2023<= o:p> Publicado:22/08/2023<= o:p> DOI: https://d= oi.org/10.33262/concienciadigital.v6i3.2.2667
	&= nbsp;
&= nbsp; Cítese= :	&= nbsp;	&= nbsp; López Telenchana, L. S., Ramos Araujo, C. E., Pérez Londo, N. A., & Moyón Moyón, C. del R. (2023). Modelación matemática de frecuencias óptimas de inspecciones de mantenimiento para tornos paralelos en función del contex= to operacional. ConcienciaDigital, 6(3.2), 77-96. https://d= oi.org/10.33262/concienciadigital.v6i3.2.2667 &= nbsp;
	&= nbsp;	CONCIE= NCIA DIGITAL, es una revista multidisciplinar, trimestral, que se publica= rá en soporte electrónico tiene como misión contribuir a la &nbs= p; formación de profesionales competentes con visión humanística y crítica q= ue sean capaces de exponer sus resultados investigativos y científicos en la misma medida que se promueva mediante su intervención cambios positivos e= n la sociedad. https://concienciadigital.org La rev= ista es editada por la Editorial Ciencia Digital (Editorial de prestigio registrada en la Cámara Ecuatoriana de Libro con No de Afiliación 663) www.celibro.org.ec <= span style=3D'text-decoration:none'>
	&= nbsp;	Esta revista está protegida ba= jo una licencia Creative Commons Attribution Non Commercial No Derivatives 4= .0 International. Copia de la licencia: http://creativecommons.org/licenses/by-nc-= nd/4.0/<= /o:p>
Palabras claves: Optimización, frecuencias, mantenimiento, bitácora, modelo, autoregresivo, pronóstico.	&= nbsp;	Resumen La optimización de frecuencias de mantenimiento utilizando el pronóstico de ocurrencia de fallas resultado de modelación matemática y en particular a través del empleo de Modelos Autorregresivos Integrado de Promedio Móvil (ARIMA) es un tema qu= e ha venido siendo investigado y desarrollado en los últimos ańos, debido a que los resultados obtenidos reflejan el aumento de los distintos índices de productividad de las máquinas y equipos intervenidos, es decir se ha comprobado la eficacia, la eficiencia y la efectividad que tiene dichos modelos en la estimación de dichas frecuencias. Se ha aplicado una metodología que parte de la generación de una serie temporal en función de los Tiempos de Buen Funcionamiento (TTF) que se encuentran registrados en= las bitácoras de mantenimiento del torno paralelo TR - 01, esta serie es mode= lada matemáticamente con el objetivo de generar un pronóstico adecuado de la aparición de nuevas fallas, esto permitió reducir indicadores clave de desempeńo a nivel industrial como el Tiempo Medio de Reparación y los Cos= tos de Mantenimiento hasta en un 35%, además la repetibilidad y la reproducibilidad de metodología planteada hace que el estudio se pueda implementar en cualquier activo físico.
Keywords: Optimization, frequencies, maintenance, logbook, model, autoregressive, forecasting.		Abstract The optimization of maintenance frequencies using the prediction of failure occurrence resulting from mathematical modeling and in particular through= the use of Autoregressive Integrated Moving Average Models (ARIMA) is a topic that has been investigated and developed in recent years, because the res= ults obtained reflect the increase of the different productivity indexes of the intervened machines and equipment, that is, the efficiency and effectiven= ess of these models in the estimation of these frequencies has been proven. It has been applied a methodology that starts from the generation of a time series based on the Times of Good Operation (TTF) that are recorded in the maintenance logs of the parallel lathe TR - 01, this series is mathematic= ally modeled with the objective of generating an adequate forecast of the appearance of new failures, this allowed to reduce key performance indica= tors at industrial level as the Average Time of Repair and Maintenance Costs u= p to 35%, also the repeatability and reproducibility of the proposed methodolo= gy makes that the study can be implemented in any physical asset.=

Introducción

En esta última década se ha venido generando investigaciones que muestran que = la modelación matemática es una herramienta viable para la optimización de las frecuencias de inspección de mantenimiento. Para la modelación matemática se emplean los distintos contextos operacionales y las frecuencias de reparaci= ón, estas dos variables han sido utilizadas exitosamente en el pronóstico de fallas, que es la base de la optimización de las frecuencias de mantenimien= to (= Zdenek & Rudolf, 2003). Además, múltipl= es investigaciones publicadas reflejan el interés por parte de la academia de indagar estos casos y la demanda en la práctica por parte de las empresas de los resultados de la optimización del mantenimiento (= Vanderschueren et al., 2023).

La optimización de las frecuencias de inspección de mantenimiento es una metodología que se ha desarrollado en los últimos ańos, convirtiéndose en u= na herramienta que permite disminuir los costos de intervención de mantenimien= to y anticiparse a la ocurrencia de fallas. Investigaciones previas indican que = el mantenimiento industrial a nivel de producción es complejo, debido a que de= be ser capaz de minimizar fallas imprevistas y evitar revisiones con altos cos= tos para la empresa (Pinciroli et al., 2023).

Por otra parte, la eficacia y la eficiencia de la gestión de mantenimiento está ligada a la adquisición, utilización y rotación adecuada de repuestos y elementos fungibles, descartando de esta manera la política de mantenimiento correctivo. Es por ello, que un modelo matemático que utiliza datos históri= cos de las intervenciones de mantenimiento, el contexto operacional de producci= ón y la rotación de repuesto, es considerado preciso para la optimización de las frecuencias de inspección de mantenimiento, en comparación con otras metodologías reactivas (= Zahedi-Hosseini, 2018).

La importancia de utilizar el contexto operacional de un activo físico en la modelación matemática radica en la influencia que tiene esta condición para= que las máquinas no estén disponibles todo el tiempo, esto se debe a fallos inesperados o programaciones de tareas establecida, cabe recalcar, que las máquinas en producción real tienen distintos lapsos de tiempos en donde est= án paralizadas debido a fallos no programados o intervenciones de inspecciones= de mantenimiento preventivo que buscan anticiparse a la ocurrencia de fallas y evitar que el activo llegue a un estado de avería (Zhang et al., 2021).

En la actualidad el desarrollo de planes de mantenimiento industrial tiene fundamentación en la utilización de herramientas y técnicas propias de ingeniería, que van desde la creación de algorítmicos mediante modelación matemática hasta el pronóstico de posibles escenarios, en estos modelos es indispensable incluir las restricciones de la organización, esto permite qu= e el plan de mantenimiento potencialice tanto la realización y programación de l= as distintas tareas (Parreńo et al., 2021). En este sentido la modelación matemática perm= ite pronosticar la ocurrencia de fallas, optimizando las frecuencias de inspecc= ión de mantenimiento, con esto se puede diseńar e implementar planes adecuados = de mantenimiento, sin que signifique un aumento del uso de recursos de la organización (Rodas & Castrillón, 2019).

Varios estudios han determinado que el pronóstico de las frecuencias de mantenimie= nto preventivo es relevante en la planificación a nivel de gestión, las predicciones ayudan a la toma adecuada de decisiones a partir de los datos históricos (Melo & Santana, 2016). La modelación matemática de un procedimiento industrial involucra varias estrategias que permiten optimizar el mantenimi= ento mediante el pronóstico de fallas, con el objetivo de anticiparse a la ocurrencia de estas (= Abdullah = et al., 2017).

En la literatura científica se ha encontrado que los modelos lineales ARIMA (p= , d, q) de Box-Jenkins son aplicados en el pronóstico de series temporales no estacionarias, es decir que la media y la varianza cambia a través del tiem= po. (Ho = et al., 2002). En máquinas de herramientas como los to= rnos paralelos, las frecuencias de inspección de mantenimiento establecen reempl= azos innecesarios, tanto de repuestos como de material fungible, aumentando así = los costos de mantenimiento. Por esta razón, una metodología que combine el pronóstico de la aparición de fallas mediante promedio móvil autorregresivo (ARIMA) y las condiciones habituales de uso de las máquinas generan una optimización correcta, logrando disminuir de forma considerable los costos = de mantenimiento (= Baptista = et al., 2018).

Se ha considerado que mediante modelos autorregresivos se puede describir el comportamiento de series temporales y proyectar valores futuros. Los modelos ARIMA han permitido capturar de forma natural las distintas propiedades empíricas de los datos interrelacionados temporalmente como una herramienta confiable (Mazón & Buńay, 2022). La toma de decisiones en función de las inspecciones de mantenimiento está dividida en dos etapas, la primera el diagnóstico y la segunda el pronóstico (Jimenez-Cortadi et al., 2019).

Metodología

La presente investigación tiene un enfoque cuantitativo, que tiene un conjunto= de procesos secuenciales: recolectar, organizar, analizar y pronosticar la variable en estudio. El diseńo de la investigación es de tipo no experiment= al, se trabaja con datos históricos provenientes de bitácoras de mantenimiento = (Hernán= dez-Sampieri, 2018).

Se ha utilizado 15 bitácoras de mantenimiento de torno paralelo TR-01 que van desde Enero del 2013 hasta Febrero del 2020, en donde se encuentran 86 registros de fallas, de los cuales 47 corresponden a fallas inherentes a las bandas que transmiten la potencia desde el motor principal y que es el elem= ento con mayor criticidad del torno paralelo intervenido, y los 39 registros res= tantes son cambios programados, que fueron recolectados en las bitácora de mantenimiento del Laboratorio de Máquinas Herramientas de la Universidad Nacional de Chimborazo.

La metodología utilizada en la presente investigación consiste en un proceso conformado por una secuencia de procedimientos: recolección de los datos, análisis de datos e implementación del modelo Box-Jenkins más adecuado para= la serie temporal. El modelado de cualquier serie de tiempo tiene como finalid= ad el poder predecir el patrón de la evolución temporal (Taneja et al., 2016). El modelo más utilizado para conseguir un pronóstico adecuado es el modelo autorregresivo integrado de media móvil ARIMA (Box et al., 1994).

La implementación del modelo Box-Jenkins puede ser desarrollado mediante cuatro etapas, como se muestra en la Figura 1:

Figura 1

Fases de implementación del modelo Box-Jenkins

Postular una clase general de Modelos Box - Jenkins

<= span lang=3DES style=3D'mso-ansi-language:ES'>Identificación del Modelo Box - Jenkins

Identifique el Modelo Box - Jenki= ns que se puede emplear.

<= span lang=3DES style=3D'mso-ansi-language:ES'>Determinación de criterios de selección

Determine los criterios AIC Y BIC para seleccionar el Modelo Box - Jenkins

<= span lang=3DES style=3D'mso-ansi-language:ES'>Comprobación de la selección d= el Modelo

żEl Modelo seleccionado es el adecuado para solucionar el problema?

<= span lang=3DES style=3D'mso-ansi-language:ES'>Utilización del Modelos= Box - Jenkins

Utilice el Modelo Box – Jenkins y genere el pronóstico requerido

Nota: Proceso para la implementación= de distintos Modelos Box - Jenkins

Fuente: Adhikari & Agrawal (2013) =

El proceso parte de identificar el modelo apropiado, continúa con la estimació= n de los parámetros, después realiza una comprobación del diagnóstico en la serie temporal y finalmente genera el pronóstico requerido (Adhikari & Agrawa= l, 2013). Para identificar el modelo ARIMA apropiado se verifica la estacionariedad de la serie temporal, esto debido a que no tienen un proceso estocástico específico (Pindyck & Rubinfeld, 1998).

Se aplica el método de diferenciación a la serie temporal con el objetivo de desaparecer la estacionalidad, luego se comprueba la estacionariedad mediante la prueba de raíz unitaria. Se ha utiliza= do el contraste de Dickey-Fuller que detecta estadísticamente la presencia de conducta tendencial estocástica en las series temporales mediante un contra= ste de hipótesis (Dickey & Fuller, 1981). Cuando existe tendencia en una ser= ie temporal en un modelo AR (1), el primer regresor tenderá a ser igual o próx= imo a 1. Esto se debe a la propiedad de reversión a la media de un = proceso estocástico estacionario. Esto implica que cuanto más cerca esté el primer coeficiente de un modelo AR (1), más tardarán las observaciones a volver al valor medio. Esto es sinónimo de no estacionariedad, si el proceso estocást= ico fuera estable, este coeficiente sería menor a 1 o muy próximo a 0.

Se puede diferenciar entre tendencia o no tendencia estocástica en las observaciones en función del número que se le asigne al primer regresor de = la autoregresión, esquemáticamente lo acotado se define como (Rodó, 2019)= :

Mientras que matemáticamente, se parte de un modelo AR (1): <= /p>

Después se resta la variable independiente de ambos lados, tal que:

Se factoriza y se cambia el parámetro para indicar que es una modificación del= original:

Posteriormente se define el incremento:

Determinando el nuevo modelo AR (1) como:

Estableciendo un nuevo contraste de hipótesis:

Cabe resaltar que en los modelos ARIMA (p, d, q), p es el grado del modelo autorregresivo (AR) y q es grado del modelo de promedio móvil (MA), finalmente d es el grado de resta. Después se establecen= los términos AR y MA de los datos con la serie de tiempo estacionaria, dichos términos son establecidos mediante las gráficas de función de autocorrelaci= ón (ACF), se descompone la serie temporal con el método de Descomposición estacional y de Tendencias con Loess (STL) que es más robusto (Clevel= and et al., 1994).

Posteriormente a la identificación de los modelos tentativos, se debe estimar los parámetr= os del modelo, para ello se diagnostica la idoneidad de los modelos preestablecidos, para seleccionar el mejor modelo, se considera el Criterio= de Información de Akaike (AIC), que es una medida de la bondad de ajuste de un modelo estadístico, describe la relación entre el sesgo y varianza en la construcción del modelo, es decir, describe la exactitud y complejidad del modelo (Çankaya & Korbel, 2018 ), y el Criterio de Información Bayesiano (BIC) o criterio Schwarz (SIC), que también es una medida de bond= ad de ajuste de un modelo estadístico, se utiliza como un criterio para para la selección de modelos entre un conjunto finito de modelos y está estrechamen= te relacionado con el (AIC) (Schwarz, 1978).

La aplicación del criterio AIC, se denota como:

(1)=

Donde:

"k" es el número de parámetros de estimació= n, "L" es el valor má= ximo de la función de verosimilitud para el modelo ARIMA correspondiente.

La aplicación del criterio BIC o criterio Schwarz (SIC), se denota como:<= /o:p>

(1)=

En donde:

"n" es el tamańo de la muestra; "k" es el número de parámetros de estimació= n, &= #8203;

&= #8203;

Mediante la prueba Dickey-Fuller se obtuvo un valor p de 0,01 que es menor al 5% de significancia, se rechaza la hipótesis nula y se concluye que la serie es estacionaria.

Se ha utilizado una variación del algoritmo Hyndman-Khandakar, para determinar los términos (p, d, q) (P, D, = Q) del modelo ARIMA, debido a que esta función combina pruebas de raíces unitarias, minimización el criterio de información de Akaike (AIC) y el criterio de información bayesiano (BIC), para obtener un modelo ARIMA aprop= iado (Hyndman & Khandakar, 2008), do= nde se ha obtenido los siguientes resultados:

ARIMA (1,1,1) (0,1,1) [12]

AIC=3D211.55

AICc=3D212.14

BIC=3D220.71

Las series con tendencia secular y variaciones cíclicas pueden representarse mediante los modelos ARIMA (p, d, q)(P, D, Q). El primer paréntesis (p, d, q) se refiere a la tendencia secul= ar o parte regular de la serie y el segundo paréntesis (P, D, Q) se refiere a las variaciones estacionales, o parte cíclica de la serie temporal. El hecho de= que el modelo identificado sea adecuado no debe descartar la posibilidad de que otro modelo algo más complejo pueda ajustarse mejor a la serie observada (<= span style=3D'background:white'>Burnham & Anderson, 2002).

Para comprobar si algún otro modelo ARIMA se ajusta a conveniencia los términos del modelo ARIMA, como se muest= ra a continuación:

ARIMA 2. (1,1,2) (0,1,1) [12]=

ARIMA 3. (1,1,2) (0,2,1) [12]=

ARIMA 4. (2,1,1) (0,0,1) [12]=

ARIMA 5. (2,1,1) (1,0,1) [12]=

Se ha seleccionado el modelo con m= ejor ajuste, tomando en cuenta el criterio AIC, y el criterio BIC como se muestr= a en la tabla 2.

= Tabla 2

= Comparativo de los criterios AIC Y BIC de los modelos ARIMA propuestos

Modelos propuestos	Criterio AIC	Criterio BIC
ARIMA 1. (1,1,1) (0,1= ,1) [12]	211,5529	220,7147
ARIMA 2. (1,1,2) (0,1= ,1) [12]	212,7154	224,1677
ARIMA 3. (1,1,2) (0,2= ,1) [12]	231,6694	242,2238
ARIMA 4. (2,1,1) (0,0= ,1) [12]	280,4044	292,6176
ARIMA 5. (2,1,1) (1,0= ,1) [12]	253,8389	268,4948

Según los criterios AIC y BIC el m= ejor modelo es ARIMA 1. (1,1,1) (0,1,1) [12], por lo que este modelo es utilizado para generar el pronóstico de los Tiempos de Buen Funcionamiento del torno paralelo TR-01.<= /span>

Por último, se generó el pronóstico del comportamiento de la serie temporal, como se muestra en la figura 7.

Figura 7

<= span style=3D'font-size:12.0pt;line-height:115%;color:black;mso-themecolor:text1; mso-ansi-language:ES-EC;mso-bidi-font-weight:bold;mso-no-proof:no'>Pronósti= co con el ń.modelo ARIMA (1,1,1) (0,1,1) [12]

Los valores obtenidos del pronósti= co se muestran en la tabla 3, tanto para el semestre Abril – Septiembre 2023 y= en la Tabla 4 para el semestre Octubre 2023 – Abril 2024.

= Tabla 3

= Pronóstico de TTF (h) para el semestre Abril – Septiembre 2023

Mes=	Pronóstico (h) <= /o:p>	Alto	Bajo
Abril 2023	120	116	125
Mayo 2023<= /span>	119	114	123
Junio 2023=	118	113	122
Julio 2023=	118	113	123
Agosto 2023	119	114	124
Septiembre 2023<= /o:p>	120	115	125

= Tabla 4

= Pronóstico de TTF (h) para el semestre Octubre 2023 – Marzo 2024.

Mes=

Pronóstico (h) <= /o:p>

Alto

Bajo

Octubre 2023

122

117

126

Noviembre 2023

123

118

128

Diciembre 2023

124

119

129

Enero 2024

125

120

130

Febrero 2024

124

119

129

Marzo 2024=

123

118

128

Discusión=

Existen múltiples investigaciones como las desarrolladas por Walls & Bendell (1987) y Ho & Xie (1998) , que han empleado modelos ARIMA en series temporales para el análisis y el pronóstic= o de fallo, con base en nuestro estudio se ha evidenciado que la generación de previsiones ha obtenido un rendimiento predictivo satisfactorio en comparac= ión con otros modelos como el Duane, por otro lado, el proceso de construcción = de los modelos ARIMA es iterativos, por lo tanto, la aplicación de estos model= os en series temporales se ha realizado en software estadístico libre. Los mod= elos ARIMA han permitido optimizar frecuencias de mantenimiento mediante un adec= uado pronóstico de fallos, que también se podría aplicar en cualquier activo físico.

En estudios como los publicados por = Ayeleru et al. (2021) y Laurente & Laurente (2019), han aplicado modelos ARIMA para pronostic= ar la producción, contaminación y gestión industrial, además emplean datos anu= ales y para la selección del modelo se basan tanto (AIC) y (BIC), de esta manera= se ha garantizado la selección del modelo con mayor capacidad de capturar el comportamiento y la proyección tanto de la producción como de la contaminac= ión y la gestión industrial , los resultados de estas investigaciones ayudaron a planificar las actividades de producción. En base a previos resultados, en = el presente estudio se ha seleccionado el modelo ARIMA (1,1,1) (0,1,1) [12] que obtuvo el menor valor en la evaluación de los criterios (AIC) y (BIC), dicho modelo ha permitido pronosticar el comportamiento de la serie temporal de l= os tiempos de buen funcionamiento de las bitácoras del torno paralelo 01, este pronóstico permite planificar de manera adecuada las distintas actividades a nivel industrial que están en función al activo intervenido.

En comparación c= on el estudio publicado por Montero et al. (2020), la optimización de las frecuencias de mantenimiento se ha reducido= los costos de mantenimiento entre un 25 % y un 35 %, se ha eliminado los fallos imprevistos entre un 70 % y un 75 %, ha decrecido el Tiempo Medio de Repara= ción (MTTR) entre un 35 % y un 45 % y finalmente la producción ha crecido entre = un 25 % y un 35 %. En el caso de estudio, de la presente investigación en el semestre Abril – Agosto 2023 los costos de mantenimiento se han reducido en= un 35%, se ha eliminado los fallos imprevistos en un 70%, de la misma manera, = el MTTR ha decrecido en un 35%, y ha aumentado en 35% la producción proporcion= al, debe destacarse que el activo físico intervenido ha aumentado su productivi= dad. =

Conclusiones<= /b>

·         Se concluye que mediante modelos ARIMA se ha pronosticado de forma aproximada la aparición de fallas, esto convierte a dichos modelos en una herramienta útil para optimizar las frecuencias de inspección de mantenimie= nto, logrando de esta manera anticipar la acción preventiva de mantenimiento para evitar paros imprevistos en los tornos paralelos.

·         Se determina que utilizando el pronóstico de la modelación matemática= se redujeron indicadores de desempeńo como el MTTR, el número de fallos imprevistos y los costos de mantenimiento, logrando aument= ar los índices de producción hasta un 35%, considerando esta herramienta relev= ante para la productividad en función del contexto operacional de cualquier acti= vo físico.

·&nb= sp;        La presente investigación muestra la utilidad de los datos históricos para los analistas de producción y de mantenimiento en su búsqueda por dise= ńar planes de mantenimiento preventivo, partiendo del estado técnico de un acti= vo y el contexto operacional de las máquinas o equipos.

Conflicto de intereses

Los autores declaramos que no existe conflicto de intereses en relación con el artículo presentado.

Referencias Bibliográficas=

Abdullah, A., Ashutosh, T., & Mark S. (2017), Simulation - based optimisation of maintenance systems: Industrial = case studies, Journal of Manufacturing Systems, 44, Part 1,191-206. h= ttps://doi.org/10.1016/j.jmsy.2017.05.008

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= ISSN: 2600-5859

= Vol. 6 No. 3.2, pp. 77 – 96, ag= osto – septiembre 2023

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Estructura de datos Página 77 | 96

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