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PROGRAMACIÓN GENERAL INTRODUCCIÓN A LA DINÁMICA DE SISTEMAS Autoaprendizaje Autoaprendizaje 3 horas horas Unidad 1 MODELOS DE SISTEMAS Autoaprendizaje Autoaprendizaje 3 horas horas Unidad 2 ANÁLISIS DE SISTEMAS DE REALIMENTACIÓN Autoaprendizaje Autoaprendizaje 3 horas horas Unidad 3 DIAGRAMA DE FORRESTER Y MODELOS MATEMÁTICOS Autoaprendizaje Autoaprendizaje 3 horas horas Unidad 4 USO DEL SOFTWARE DE SIMULACIÓN Autoaprendizaje Autoaprendizaje 3 horas horas Unidad 5 RETRASOS EN LOS SISTEMAS Autoaprendizaje Autoaprendizaje 3 horas horas Unidad 6 SIGNIFICANCIA Y ANÁLISIS DE MODELOS DINÁMICOS Autoaprendizaje Autoaprendizaje 3 horas horas Unidad 7 TÓPICOS AVANZADOS Y DESARROLLO DE CASOS Autoaprendizaje Autoaprendizaje 3 horas horas Unidad 8
5
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TABLA DE CONTENIDO Pag. 3 5
Presentación Programa General UNIDAD TEMÁTICA N° 01: INTRODUCCIÓN ALA DINÁMICA DE SISTEMAS ¿Qué es un Sistema? Características Características de los Sistemas Clasificación de los Sistemas ¿Qué es un Sistema Dinámico? Como Nace la Dinámica de Sistemas La Dinámica de Sistemas
9 9 10 11 13 14 16
UNIDAD TEMÁTICA N° 02: MODELADO DE SISTEMAS El Desarrollo de Modelos de Sistemas Desarrollo de Diagramas Causales Los Ciclos Causales Descripción de la Metodología para Desarrollo de Modelos
21 23 25 27
UNIDAD TEMÁTICA N° 03: ANÁLISIS DE SISTEMAS SISTEMAS DE REALIMENTACIÓN REALIMENTACIÓN Bucle de Realimentación Negativa Bucle de Realimentación Positiva Sistemas Complejos y Estructuras Genéricas
33 34 38 41
UNIDAD TEMÁTICA N° 04: Diagrama de Forrester y Modelos Matemáticos Como Iniciar la Construcción de un Diagrama de Forrester Clasificación de Variables Modelo Matemático UNIDAD TEMÁTICA N° 05:
47 47 48 54
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Simulando con el Software Stella UNIDAD TEMÁTICA N° 06:
60 64 68
Retrasos en los Sistemas Retrasos en la Transmisión de Material Retrasos en la Transmisión de Información Reglas para Aplicar Retrasos UNIDAD TEMÁTICA N° 07:
75 77 84 86
Significancia y Análisis de Modelos Dinámicos Significancia Análisis de Sensibilida Sensibilidad d del Modelo Modelo Dinámico Implantación Implantación de Políticas en el Sistema UNIDAD TEMÁTICA N° 08:
89 89 91 92
Tópicos Avanzados y Desarrollo de Casos Realimentación Negativa de Segundo Orden Realimentación Positiva de Segundo Orden Desarrollo de Casos
99 99 101 104
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INTRODUCCIÓN ALA DINÁMICA DE SISTEMAS La sistémica con sus diversas metodologías aporta instrumentos para estudiar aquellos problemas que resultan de las interacciones que se producen en el seno de un sistema complejo. La Dinámica de sistemas, constituyéndose como la metodología mas acertada dentro del mundo sistémico, propone el tratamiento de dichas complejidades considerando la causa y el efecto que las enmarca. En esta primera unidad, vamos a recordar algunos conceptos básicos de la teoría de sistemas, y su relación con la Dinámica de Sistemas, formalizaremos su concepción y veremos su evolución en el tiempo.
Al terminar el estudio de la presente unidad, unidad, el estudiante: estudiante:
Repasa los conceptos generales de la sistémica
Describe las características y elementos de los sistemas
Conoce la historia y precedentes de la Dinámica de Sistemas.
Entiende el concepto de Dinámica de Sistemas
¿QUÉ ES UN SISTEMA? Vamos a ver algunas definiciones de lo que es un sistema: L. Von Bertalanffy (1968): "Un sistema es un conjunto de unidades en interrelación." interrelación." Ferdinand de Saussure (1931) : "Sistema es una totalidad organizada, hecha de
elementos solidarios que no pueden ser definidos más que los unos con relación a los otros en función de su lugar en esa totalidad." IEEE Standard Dictionary of Electrical and Electronic Terms : "Sistema es un todo
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sistema ejecuta una función imposible de realizar por cualquiera de las partes individuales. La complejidad de la combinación está implícita." Estándar X3.121970 (ANSI), Estándar 2382/V, VI (ISO) Vocabulary for Information
Processing: "Sistema es una colección organizada de hombres, máquinas y métodos necesarios para cumplir un objetivo específico." Resumiendo y obteniendo aspectos relevantes:
La presencia de elementos heterogéneos pero interconectados. interconectad os.
El concepto de partes formando un todo global.
La presencia presencia del exterior influenciando influenciando en el el sistema. sistema.
El surgimiento surgimient o de nuevas propiedades, resultado de las interrelaciones de las partes.
El objetivo común compartido por todos los elementos participantes participante s en el sistema.
Figura 1.1. Visión sistémica
CARACTERÍSTICAS DE LOS SISTEMAS 1. Globalismo.Globalismo.- Dentro del funcionamiento de los sistemas surge un acontecimiento llamado globalismo, el cual se da cuando un elemento presenta
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2. Entropía.Entropía.- Predisposición que tienen los sistemas al deterioro o desintegración, es decir, a medida que la entropía acrecienta los sistemas se descomponen en estados más simples. El segundo principio de la termodinámica establece el crecimiento de la entropía, es decir, la máxima probabilidad que los sistemas caigan en la desorganización y, finalmente, homogeneización con el ambiente. Los sistemas cerrados están irremediablemente condenados a la desorganización. No obstante, hay sistemas que, al menos temporalmente, revierten esta tendencia al aumentar sus estados de organización (negentropía, información). 3. Homeostasis.Homeostasis.- Equilibrio dinámico entre las partes del sistema, esto es, la tendencia de los sistemas a adecuarse al equilibrio de los cambios internos y externos del ambiente. Este concepto está referido con énfasis a los organismos vivos, por lo tanto, sistemas adaptables. Los procesos homeostáticos operan ante variaciones de las condiciones del ambiente, corresponden a las compensaciones internas al sistema que sustituyen, bloquean o complementan estos cambios con el objeto de mantener invariable la estructura sistémica, es decir, hacia la conservación de su forma. La subsistencia de formas dinámicas o trayectorias se denomina homeorrosis (sistemas cibernéticos) cibernéticos) 4. Equifinalidad.Equifinalidad.- Se refiere al hecho que un sistema vivo a partir de distintas condiciones iniciales y por distintos caminos llega a un mismo estado final. No importa el proceso que reciba, el resultado es el mismo. El fin se refiere a mantener un estado de equilibrio fluyente. “Puede alcanzarse el mismo estado final, la misma meta, partiendo de diferentes condiciones iniciales y siguiendo distintos itinerarios en los procesos orgánicos”. El proceso inverso se denomina multifinalidad, es decir, “condiciones iniciales similares pueden llevar a estados finales diferentes”.
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Sistemas sociales: Integrados sociales: Integrados por personas cuyo objetivo tiene un fin común. Considera el contenido y significado de mensajes, la naturaleza y dimensiones del sistema de valores, la trascripción de imágenes en registros históricos, sutiles simbolizaciones artísticas, música, poesía y la compleja gama de emociones humanas.
Sistemas hombre-máquina: hombre-máquina: Emplean equipos u otra clase de objetos, que a veces se desean logren autosuficiencia. Representa una combinación de uno o más seres humanos y uno o más componentes físicos, que actúan recíprocamente recíprocamente para efectuar, a partir de unas entradas de energía e información determinadas, una producción o salida deseada.
Sistemas abiertos: abiertos: Se caracteriza porque su estado original se transforma constantemente por la acción retroalimentadora del ambiente, desde su creación hasta su declive. Su vida útil depende de su adaptabilidad a las exigencias del ambiente (homeostasis). Intercambian materia y energía con el ambiente continuamente.
Sistemas cerrados: Se cerrados: Se caracteriza porque no tiene capacidad de cambio por si mismo para adaptarse a los requerimientos del ambiente. Es irreversible y su estado presente y final esta determinado por su estado original. Son efímeros por desgaste (entropía).
Sistemas temporales: Duran cierto periodo de tiempo y posteriormente desaparecen.
Sistemas permanentes: Duran mucho más que las operaciones que en ellos realiza el ser humano, es decir, el factor tiempo es más constante.
Sistemas estables: estables: Un sistema se dice estable cuando puede mantenerse en
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Sistemas probabilísticos: Presentan incertidumbre. incertidumbre.
Subsistemas: Sistemas Subsistemas: Sistemas más pequeños incorporados al sistema original.
Supersistemas: Supersistemas: Sistemas extremadamente grandes y complejos, que pueden referirse a una parte del sistema original.
1. En sus propias palabras defina lo que es un sistema. 2. Mencione ejemplos relevantes de cada uno de los tipos de sistemas. ¿QUÉ ES UN SISTEMA DINÁMICO? Un sistema dinámico es un sistema complejo que presenta un cambio o evolución de su estado en el tiempo, el comportamiento en dicho estado se puede caracterizar determinando los límites del sistema, los elementos y sus relaciones; de esta forma se puede elaborar modelos que buscan representar la estructura del mismo sistema. Al especificar los límites del sistema, en primer lugar, se selecciona a aquellos elementos que contribuyan a establecer las formas de comportamiento, y seguidamente debemos determinar el lugar donde se llevará a cabo el estudio, dejando de lado toda clase de aspectos irrelevantes. En cuanto a la elaboración de los modelos, los elementos y sus relaciones, se debe tener en cuenta que:
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ecuación logística: , donde t denota los pasos discretos del tiempo y x es la variable que cambia con éste. continuo resultante Si el tiempo es medido en forma continua, el sistema dinámico continuo resultante es expresado como una ecuación diferencial ordinaria; por ejemplo: , donde x es la variable que cambia con el tiempo t. La variable cambiante x es normalmente un número real, aunque también puede ser un vector en Rk. Sistemas lineales y no lineales Se distingue entre sistemas dinámicos lineales y sistemas dinámicos no lineales. En los sistemas lineales, el lado derecho de la ecuación es una expresión que depende en forma lineal de x, tal como:
Si se conocen dos soluciones para un sistema lineal, la suma de ellas es también una solución; esto se conoce como principio de superposición. En general, las soluciones provenientes de un espacio vectorial permiten el uso del álgebra lineal y simplifican significativamente significativamente el análisis. Para sistemas lineales continuos, el método de la transformada de Laplace también puede ser usado para transformar la ecuación diferencial en una ecuación algebraica. Los sistemas no lineales son mucho más difíciles de analizar y a menudo exhiben un
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Forrester nació en 1918 en Nebraska, EUA, y después de haber obtenido el título de Ingeniero Eléctrico en la Universidad de Nebraska, continuó sus estudios en el MIT. Forrester fue un pionero del desarrollo de la informática, que participó hacia 1950 en la invención de la memoria de acceso aleatorio RAM y es considerado autor de la primera imagen animada sintética, la representación del bote de una pelota usando un osciloscopio. Jay W. Forrester planteaba que el mundo esta lleno de sistemas, la mayoría de éstos son bastante simples y fácilmente entendibles para el ser humano. No obstante, los problemas sociales son sistemas con una gran cantidad de variables y sumamente complicados.Por esa razón, Forrester plantea el uso de computadores para la simulación de sistemas reales, mediante la enunciación de modelos sencillamente traducibles a programas informáticos. De esta manera, el ser humano aspira a prever el comportamiento de sistemas complejos como los que se presentan en las sociedades. En su libro "Industrial Dynamics" (considerado el punto de partida de la Dinámica de Sistemas), Forrester pone de manifiesto el hecho de que el actual crecimiento de la población es insostenible por más de 100 años. Forrester, junto con otras personalidades fundó el Club de Roma, organización Internacional cuyo objetivo es la concienciación de que el actual sistema es insostenible y está próximo al colapso. Forrester estableció un paralelismo entre los sistemas dinámicos (o en evolución) y los hidrodinámicos, constituido por depósitos, intercomunicados por canales con o sin retardos, variando mediante flujos su nivel, con la concurrencia de fenómenos exógenos. La dinámica de sistemas, permite en estos días ir más allá de los estudios de casos y las teorías descriptivas. La dinámica de sistemas no está restringida a sistemas
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Figura 1.2. Jay W. Forrester LA DINÁMICA DE SISTEMAS
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disponga de informaciones cuantitativas de la evolución del sistema real en el pasado. Si el modelo es capaz de generar los comportamientos característicos del sistema real, entonces obtendremos una cierta confianza en la validez del modelo. En Dinámica de Sistemas la simulación permite obtener trayectorias para las variables incluidas en cualquier modelo mediante la aplicación de técnicas de integración numérica. Sin embargo, estas trayectorias nunca se interpretan como predicciones, sino como proyecciones o tendencias. El objeto de los modelos dinámicos es, como ocurre en todas las metodologías de sistemas blandos, llegar a comprender cómo la estructura del sistema es responsable de su comportamiento. Esta comprensión normalmente debe generar un marco favorable para la determinación de las acciones que puedan mejorar el funcionamiento del sistema o resolver los problemas observados. La ventaja de la Dinámica de Sistemas consiste en que estas acciones pueden ser simuladas a bajo coste, con lo que es posible valorar sus resultados sin necesidad de ponerlas en práctica sobre el sistema real. Lo que distingue a la dinámica de sistemas de otras aproximaciones al estudio de los mismos problemas, es el uso de bucles de realimentación. La construcción de modelos por la dinámica de sistemas parte de la distinción esencial entre depósitos y
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También vimos las características, más relevantes, ligados a los sistemas, como lo es el Globalismo, la Entropía, la Homeostasis y finalmente la Equifinalidad, se detallo su implicancia y presencia en la estructura orgánica de los sistemas, se aprecio su necesidad y la diferencia que implicaban cada uno de ellos. Seguidamente especificamos la clasificación de los sistemas, esto se dio en base a su tamaño, comportamiento, tiempo de vida, naturaleza; tomando en cuenta todo tipo de características que permitan ofrecer una amplia gama de tipos de sistemas. Finalmente vimos la historia predecesora a la Dinámica de Sistemas, como es que Jay W. Forrester, su fundador, impulso su crecimiento y su formalización a través del tiempo y como el mundo pudo apreciar su valides a través de las distintas aplicaciones que se desarrollaron. Definimos lo que es la Dinámica de Sistemas, de que trata, cual es la importancia de su estudio e impulso.
Forrester, J. W., Principies of Systems, Wright-AIIen Press, Inc., Cambridge, Massachusetts, 1968.
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La siguiente unidad se encarga de abordar aspectos del modelado de los sistemas, siguiendo las especificaciones que nos ofrece la Dinámica de Sistemas, recuerden que en esta unidad se ha comprendido dicho concepto.
Introducción ala Dinámica de Sistemas – Unidad Nº 1 Responda las siguientes preguntas: 1. Halle aplicaciones en el mundo real de Dinámica de Sistemas (4 ejemplos) 2.
¿Que relación existe entre los sistemas complejos y la Dinámica de Sistemas?
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MODELADO DE SISTEMAS Un aspecto importante del enfoque sistémico es la construcción de modelos. Un modelo es una abstracción de la realidad que captura aspectos relevantes y funcionales de los sistemas, considerando los suficientes detalles como para poder usarse en la investigación y la experimentación en lugar de hacerlo con el sistema
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Los modelos matemáticos son “simbólicos”, ya que para describir un sistema emplean normalmente notaciones matemáticas en forma de ecuaciones. A menudo estas ecuaciones matemáticas tienen representaciones gráficas análogas. Hay tres características de los modelos matemáticos que los hacen particularmente útiles: son precisos, concisos y manejables. Desafortunadamente, por otra parte, estas características tan deseables no son en general apreciadas por los políticos, por los gerentes, por aquellos que tienen la batuta de la organización, ya que no comprenden la simbología o las reglas de utilización. Idealmente, esta barrera en la comunicación entre los autores de los modelos y sus usuarios se salva gracias al uso de modelos verbales o gráficos.
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única del mundo, desde el punto de vista de una disciplina particular del desarrollo de modelos, influyendo, de este modo, en el contenido y la forma de sus modelos. Las distintas metodologías de sistemas pueden diferenciarse diferenciarse por la forma en que los desarrolladores consideran el concepto de modelos. Se pone de manifiesto tres tendencias. En la primera, se tiene una fe completa en un tipo de modelo matemático, dentro de cuyas limitaciones tiene que representarse; a programación lineal y los modelos de entradas y salidas (input-output) son ejemplos de esto. En la segunda, se mantiene que los datos lo son todo; en lugar de desarrollar modelos, los miembros de esta tendencia se limitan a obtener curvas que se ajusten a los datos; la econometría es una metodología de este tipo. En la tercera tendencia, se busca la
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relaciones causa-efecto. Una herramienta importante para mostrar interacciones causa-efecto entre las variables claves durante el desarrollo del modelo de un sistema dinámico es el diagrama causal. El primer paso para desarrollar un diagrama causal es identificar las variables clave que describen la situación problema, y registrar la manera de funcionamiento del sistema que está siendo analizado. Estas variables clave, deducidas de un modelo mental o de una descripción verbal de un sistema, se representan en una hoja de papel. Las relaciones causa-efecto entre pares de variables se representan en la segunda fase por medio de flechas. Por ejemplo, la construcción de departamentos causa el aumento de la cantidad de departamentos y la relación causal se
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Excelencia Académica
Polaridad negativa La polaridad negativa indica que el cambio en la variable causa hace que la variable efecto varié en sentido contrario. Es decir que al aumentar la variable causa disminuye la variable efecto y al disminuir la variable causa aumenta la variable efecto. En el mismo problema de los niños abandonados, la respuesta o acciones del gobierno, a través de las entidades encargadas, que se dan para luchar contra este problema, relacionado con el número de niños abandonados y puede ser
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bucle, hasta que la causa inicial produce un efecto indirecto sobre sí misma. Un aspecto clave del análisis de un sistema es centrar la atención sobre los bucles de
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(departamentos / año), y una variable de nivel, tal corno el inventario (departamentos), (departamentos), es un “flujo físico” que estará representado por una línea sólida. No
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Excelencia Académica
Los signos de interrogación que aparecen sobre las variables de nuestro modelo están indicando que sobre ellas hay que definir su relación analítica cuando corresponda, o entregar el valor que tienen. Vamos a fijar nuestra atención en la variable de flujo Nacimientos. Nacimientos. Recordemos que ella está definida extensivamente como “Los Nacimientos están Nacimientos están en relación a la Tasa de Natalidad, Natalidad, que deriva de la Nacimientos, debe Población”. Bien, hagamos doble click sobre la variable de flujo Nacimientos, aparecer la siguiente caja de diálogo como se puede observar en la Figura 5.9,
1 4
2
3
Figura 5.9 Ventana de asignación de relaciones analíticas Sobre esta caja de diálogo (Figura 5.9) haremos algunas precisiones. Hemos marcado cuatro sectores con rotulador de sendos 1,2,3,4, como lo indica la Figura 5.9, que nos servirá para las precisiones. La zona 1 nos indica que la variable de flujo, por defecto y conforme está activado el rótulo UNIFLOW, es un flujo unidireccional, esto es tiene la propiedad de que fluye “material” sobre la variable de estado ( Población Población en nuestro caso) y que no tiene opción de “inhalar” material desde la variable de estado, en términos hidráulicos esta variable de flujo no puede bombear para extraer agua, solo puede depositar agua. El Nacimientos solamente transmite personas hacia la caso es que nuestra variable Nacimientos solamente Población La zona 2 indica lo que ya hemos establecido, que sea la forma analítica que tenga
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Excelencia Académica
La zona 3 está indicando la función analítica con la que quedará definida nuestra variable de flujo, y es en el lugar marcado por el video reverso donde dice “Place right hand side...” donde se debe poner la fórmula. Finalmente la zona 4 entrega una lista de funciones que posee en su estructura el software STELLA, en que eventualmente podamos necesitar. Vamos ahora a definir la variable de flujo. La expresión “Los Nacimientos están Nacimientos están en Natalidad, que deriva de la Población” Población” es equivalente relación a la Tasa de Natalidad, matemáticamente a lo que indica la Figura 5.10
Figura 5.10 Asignando ecuación a la variable Nacimientos Una vez entregada esta instrucción, y convenciéndonos que es la adecuada para definir el comportamiento creciente de la Población, Población, hacemos click en la opción OK. Deberá aparecer el diagrama indicado en la figura 5.8, dónde ha desaparecido el signo de interrogación sobre la variable de flujo Nacimientos. Nacimientos. Ahora vamos a definir la variable de flujo (de salida) Muertes, Muertes, haciendo doble click en esta variable, aparecerá una caja de diálogo similar al de la figura 5.9, y se indica en la figura 5.11.
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Excelencia Académica
Figura 5.11 Asignando ecuación a la variable Muertes Ahora vamos a definir las constantes Tasa de Natalidad y Tasa de Mortalidad, quedando lo indicado en la Figura 5.12 y 5.13.
Figura 5.12 Asignando valor a la constante Tasa de Natalidad
Figura 5.13 Asignando valor a la constante Tasa de Mortalidad Bien, ¿qué hemos hecho? La verdad es que hemos realizado un programa computacional. computacional . En efecto, observe el icono de la Figura 5.8, y hagamos un click en la flecha inferior, obtendremos el siguiente código computacional (figura 5.14)
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Excelencia Académica
Figura 5.14 Modelo matemático subyacente A este nivel, hemos realizado un programa computacional que nos permitirá hacer una simulación sobre el comportamiento de la Población. Población. Observemos cuidadosamente estas ecuaciones para convencernos que efectivamente corresponden a un sistema de ecuaciones dinámicas que mostrarán la evolución a través del tiempo del sistema dinámico que estamos modelando.
5.1. 1. En la figura 5.9, en la zona 1, observamos que una variable de flujo puede ser Uniflow o Biflow; explique cuando se utilizaría el tipo Biflow, menciones dos ejemplos. 2. En la Figura 5.14, se presenta presenta el modelo matemático matemático del caso visto, realce realce una interpretación interpretación de la siguiente relación analítica: POBLACION(t) = POBLACION(t - dt) + (NACIMIENTOS + MUERTES) * dt
SIMULANDO CON EL SOFTWARE STELLA Vamos a enseñar como realizar simulación para el modelo que hemos realizado. Para esto, fijémonos en el sector izquierdo de la pantalla, aparece el icono . Si ubicamos el mouse sobre ese icono, y hacemos click con el botón izquierdo, sobresale una ventana pequeña, en donde definiremos algunos valores necesarios para la simulación, esto se indicada en la Figura 5.15
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Figura 5.15 Interfaz para la simulación Vamos a enseñar como realizar simulación para el modelo que hemos realizado. Para esto, hacemos clic en Specs elegimos la opción “Time Specs...”, como se ve en la figura 5.15. En esta opción vamos a determinar que el tiempo es discreto, toda vez que la evolución de este sistema es en tiempo discreto, en rigor la unidad de tiempo en “años”, de modo que al hacer un click en la caja de diálogo que aparece lo completamos de la forma que lo indica la figura 5.16.
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Como se puede observar, hemos determinado un periodo entre el año 1995 y 2050, de manera arbitraria, con un DT igual a 1, y hemos elegido la unidad de tiempo “Years” (años). Una vez realizado esto hacemos un click en OK. Con esto volvemos a nuestro diagrama original. Ahora bien, vamos a ver la evolución de la variable Población Población mediante una gráfica, para esto hacemos un clic en el icono arrastrándolo a un sitio conveniente, al fijarla aparecerá la siguiente caja de diálogo que indica la Figura 5.17,
Figura 5.17 Ventana de dialogo para la gráfica del tiempo de simulación Como podemos ver, solo en el eje vertical aparece la dimensión tiempo que hemos indicado anteriormente al definir la “longitud de la simulación”. Para poder definir la variable Población Población que queremos graficar hacemos doble click en cualquier parte gris de este plano cartesiano. Deberá aparecer la siguiente caja de diálogo.
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Como podemos observar, hay dos cajas principales, una que dice “Allowable” (Accesibles), y otra titulada “Selected” (Seleccionada). Como nuestro interés es graficar la evolución de la variable Población, Población, es que seleccionamos esta variable haciendo doble click a Población en Población en la caja “Allowable”, con esto logramos entonces que Población pase Población pase a la columna “Selected”. De momento solo elijamos esa opción y el resto de los botones, que están por defecto lo dejamos tal cual se ve en la figura 4. Una vez realizado esto hacemos un click en OK, aquí podemos observar que el eje Y está tabulado con la variable Población. Población. Nuevamente nos vamos a Specs y elegimos la opción RUN , debería aparecer una gráfica como la que indica la Figura 5.19.
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Figura 5.20 Tabla de valores para las variables del modelo
5.2. 1.
Investigar Investiga r la existencia de otros programas de simulación para la Dinámica de Sistemas y realizar un cuadro de comparación técnica de dichos programas.
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Direcciones de Internet http://virtualmoodle.uman http://virtualmo odle.umanizales.edu.co izales.edu.co/moodle/mod/ /moodle/mod/resource/vie resource/view.php?id=4 w.php?id=4053 053 http://www.uantof.cl/facultades/csbasicas/M http://www.uantof.cl/facultade s/csbasicas/Matematicas/aca atematicas/academicos/em demicos/emartinez/Dinam artinez/Dinam ica/manualstella/manual.html
Ya conocemos una herramienta que nos permitirá llevar a cabo la simulación, pero aun existen temas mas avanzados en cuanto al modelado de los sistemas dinámicos. En la siguiente unidad vamos a considerar los retrasos en los sistemas, estera será un punto adicional que permitirá enriquecer nuestro modelado.
Uso del software de simulación – Unidad Nº 5 1. Dado el siguiente caso: Usted ha sido contratado para desempeñarse como Ingeniero de Sistemas en "AduranNet", una importante organización que se dedica a la prestación de servicios de acceso a Internet. El Gerente General está preocupado por los
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conoces, averigua un poco acerca de él), de manera que si en el mundo existen unos 500 millones de usuarios de Internet y si consideramos que cada uno de ellos tiene unas 40 direcciones registradas, el proceso de infección es exponencial. La tasa de crecimiento de los usuarios de Internet es de un 12% anual. De los nuevos usuarios, el 10% está consciente de la necesidad de utilizar antivirus. El 90% por ciento restante debe aprender, a veces previa amarga experiencia de pérdida de datos, que los virus y antivirus existen. Ese 90% de usuarios tarda en promedio unos dos meses, después de hacerse usuario, en obtener un antivirus. Los antivirus se pueden producir muy rápidamente, una vez detectado el nuevo virus, unas 48 horas después es liberado el antivirus por las empresas (Symantec, Panda, McAffe, etc.). Los usuarios registrados pueden entonces obtener la versión actualizada de la base de datos del antivirus y proteger su equipo. La capacidad de atención para ofrecer las actualizaciones a través de Internet alcanza a 10.000 usuarios por minuto. Se estima que la tercera parte de los usuarios de antivirus lo actualiza semanalmente, la otra tercera parte cada quince días y la tercera parte restante lo actualiza cada seis semanas. De manera que el proceso de difusión, en muchos casos depende también de que el usuario abra o no el archivo infectado que recibe, lo que ocurre en seis de cada diez correos infectados que llegan a los usuarios sin antivirus, o con él no actualizado. Esto depende de la formación del usuario y del conocimiento que tenga acerca del virus y sus características, las cuales son difundidas por servicios de alerta (ver: www.alerta-antivirus.es) y medios de comunicación unos cinco días después de la aparición del virus. Cada usuario le dice a 10 personas en promedio de la existencia del nuevo virus de manera que la difusión de la existencia del virus es exponencial
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En primer lugar, realiza el diagrama causal.
A partir partir del del diagrama diagrama causal, desarrolla el diagrama diagrama Forrester.
Luego, realiza las ecuaciones correspondientes. correspondie ntes.
Realizar la simulación en el Stella®.
Gracias a su intervención, luego de probar el modelo y realizar simulaciones el gerente sabe que hacer. Ahora él, y usted, tienen trabajo seguro el próximo año. Sugerencias: No olvide identificar e incluir en la concepción del sistema, las costumbres, puntos de vista, modelos mentales o hábitos de las personas que pueden afectar la dinámica del sistema.
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RETRASOS EN LOS SISTEMAS Una característica muy significativa, en la cual se debe reflexionar, en el estudio de sistemas dinámicos son los retrasos que se originan en la transmisión de información o de materiales a lo largo de las relaciones. Al diseñar el diagrama causa - efecto de un sistema se debe considerar que al unir dos variables, dicha relación puede involucrar la transmisión de información o material el cual se da en un lapso de tiempo (retraso). Un retraso es conocido también por retardo o demora. Para formarse una idea sobre la situación de cierto problema es necesario que trascurra cierto tiempo antes de tomar una decisión, y una vez tomada está, debe transcurrir algún tiempo hasta que se observen los efectos en la misma. Los retrasos están presentes en muchos procesos. Por ejemplo, en la construcción de departamentos, dicha construcción se llevara en un periodo de tiempo relativamente largo y no de forma inmediata como se podría traducir al ver el diagrama causal; por ello es necesario incluir retrasos en el modelado de algunos sistemas.
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RETRASOS EN LA TRANSMISIÓN DE MATERIAL Los retrasos de materiales se originan cuando coexisten componentes en el sistema que acumulan el material que fluye por el mismo. En la Figura 6.1, se muestra el diagrama causal que representa un retraso de material de primer orden. El orden viene dado por el número de niveles necesarios para la simulación del mismo. El retraso es producido producido a través de la la acción combinada combinada del nivel y el flujo flujo como se observa en el recuadro de la Figura 6.1.
FLUJO DE ENTRADA
FLUJO DE SALIDA
+
- + MATERIAL EN RETRASO
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Excelencia Académica
Aquí observamos como el flujo de entrada se almacena en la variable de nivel Material en Transito y el flujo de salida es equivalente a lo acumulado dentro del nivel dividido por un tiempo promedio de retraso que precisamente es el encargado de ajustar a la variable de nivel. En el software de simulación Stella®, se cuenta con una función que permite simplificar el diagrama presentado en la Figura 6.1. En esta nueva propuesta (Figura 6.3), la ecuación que rige el comportamiento del flujo de salida es: flujo de salida = SMTH1 (flujo de entrada, tiempo promedio de retraso)
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Excelencia Académica
+
CANTIDAD DE DEPARTAMENTOS DEPARTAMENTOS DESEADOS
- + CONSTRUCCIÓN
+
DISCREPANCIA
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Vamos a mejorar la propuesta anterior, para ello, añadimos el retraso que produce el tiempo de construcción (figura 6.9).
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En las ecuaciones definimos Departamentos con el valor inicial de 8000. La función STEP es utilizada para incrementar los apartamentos deseados de 8000 a 12000, que ocurrirá en un medio año (0.5). El gráfico de la simulación siguiente (Figura 6.11) muestra como el sistema genera un comportamiento oscilatorio, debido al ciclo de realimentación negativo. La Departamentos oscila alrededor de los 12000 y eventualmente variable de nivel Departamentos converge hacia este valor. El incremento en el número de departamentos de 8000 a 12000 retrasa la demanda bastante, de manera que el constructor tiende a tener una razón de construcción máxima cuando la demanda esta descendiendo rápidamente.
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esta dado por la siguiente función PULSE (l5000, l, 0). Deberá presentar el diagrama causal y la simulación en el Stella RETRASOS EN LA TRANSMISIÓN DE INFORMACIÓN Este tipo de retrasos resulta de la necesidad de conservar y almacenar información del sistema antes de tomar una decisión. Los retrasos en la transmisión de información actúan como filtros que son capaces de aislar los picos que presenta la evolución de una variable (Figura 6.12), tomando un valor promedio de la misma.
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Excelencia Académica
La Figura 6.13 muestra el diagrama de Forrester en donde se consideran dos niveles y tres flujos la variable de flujo FSB será la que considera la función especial y esta es igual a SMTH1(llegada, tr1)
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REGLAS PARA APLICAR RETRASOS La introducción de retrasos en la dinámica de sistemas sólo esta justificada por la bondad de los resultados finales que desde un punto de vista práctico se alcanza
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Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.